"Variational techniques with applications to segmentation and

Transcription

Variational techniques with
applications to segmentation and
registration of medical images
by
Monica Hernandez Gimenez
Ph.D. Thesis dissertation submitted in partial fulfillment of the requierements for the
degree of Doctor of Philosophy
Program: Biomedical Engineering
Aragon Institute on Engineering Research (I3A)
University of Zaragoza
Advisor: Salvador Olmos Gasso
c
Monica
Hernandez 2008
A mi abuela
A mis padres
Y a Xavi
Resumen
En las últimas décadas, la utilización de imágenes médicas ha experimentado una
importancia creciente en la práctica clı́nica con el desarrollo de nuevas técnicas de
adquisición de imágenes del cuerpo humano. La gran cantidad de información disponible con las imágenes ha promovido el desarrollo de métodos de procesado de imagen
para su análisis e interpretación de una manera automática a la vez que fiable. Tanto la
segmentación como el registro de imágenes médicas han sido identificados como problemas clave en el análisis de imagen médica llegando a ser áreas activas de investigación.
La presente Tesis tiene como objetivo el desarrollo de métodos para la segmentación
y el registro de imágenes médicas en el marco de importantes aplicaciones. Estos
métodos tienen en común su formulación como problemas variacionales definidos en
sus respectivos espacios de soluciones.
En primer lugar, se ha propuesto un método para la segmentación automática de
la vasculatura cerebral en angiografı́a. El método ha sido formulado como un modelo
deformable implı́cito basado en regiones implementado dentro del paradigma de level
set. La información basada en regiones se ha calculado a partir de las probabilidades
asociadas a los principales tipos de tejido presentes en las imágenes médicas. Nuestro
método asume un modelo de estimación de probabilidad no-paramétrico. El espacio
de caracterı́sticas está compuesto por descriptores diferenciales de imagen de segundo
order en el espacio-escala, los cuales han demostrado ser suficientemente eficientes en la
discriminación de arterias de otras estructuras. El método ha sido aplicado con éxito en
la segmentación de aneurismas cerebrales en 3D-AR y TAC mejorando los resultados
de los modelos deformables implı́citos basados en regiones cuyas prestaciones podrı́an
competir en esta aplicación.
En segundo lugar, nos hemos centrado en el estudio de variedades Riemannianas de
difeomorfismos y en el diseño de métodos eficientes de registro difeomórfico con el objetivo de extender el uso de difeomorfismos en estudios clı́nicos. Nuestro método incluye
la parametrización estacionaria de difeomorfismos dentro del paradigma de Grandes
Deformaciones Difeomórficas (abreviatura en inglés LDDMM). Hemos formulado el
problema variacional asociado al registro y deducido sus ecuaciones de Euler-Lagrange.
Nuestro método ha mostrado prestaciones similares mientras que se ha conseguido una
reduccion considerable en memoria y tiempo de calculo respecto al algoritmo LDDMM
original. Además, hemos propuesto un método de optimización de Gauss-Newton con
la intención de introducir eficiencia y robustez en el algoritmo durante el registro que
ha sido comparado favorablemente con otros algoritmos de registro eficiente propuestos
en la literatura simultáneamente a nuestras publicaciones. Nuestro algoritmo ha sido
utilizado con éxito en la generación de atlas cerebrales 3D a partir de estadı́stica en
poblaciones de difeomorfismos.
Abstract
During the last decades, medical imaging has experienced an increasing importance
in clinical practice with the development of new image acquisition techniques of the
human body. The huge amount of information that becomes available with the images
has promoted the development of image processing methods to automatically analyse
and interpret the images in a reliable way. Both segmentation and registration of medical images have been identified as key problems in medical image analysis becoming
challenging areas of active resarch. This Thesis aims at the development of methods
for the segmentation and registration of medical images in the framework of challenging applications. The common fact existing between these methods is that they are
formulated from variational problems defined in their respective spaces of solutions.
First, we have proposed a method for automatic segmentation of the cerebral vasculature from angiographic data. The method is based on a region-based implicit
deformable model implemented within the level set paradigm. Region-based information is computed from the probabilities associated to the main tissue types present in
the medical images. In our method, a non-parametric model for probability estimation is assumed. The feature space is composed of high-order multi-scale differential
image descriptors that have shown efficient enough to discriminate vessels from other
structures. The method has been successfully applied to the segmentation of cerebral
aneurysms in 3D-RA and CTA showing to outperform some of the region-based implicit
deformable models that could compete in performance for this application.
Second, we have focused on the study of the Riemannian manifold of diffeomorphisms and the devise of efficient methods for diffeomorphic registration intended to
spread the use of diffeomorphisms in clinical research studies. Our method includes
the stationary parameterization of diffeomorphisms in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) paradigm. We have formulated the variational
problem related to the registration scenario and derived the associated Euler-Lagrange
equations. Our method has shown a similar performance while drastically reducing
memory and time requirements with respect to reference LDDMM. In addition, we
have proposed a Gauss-Newton method for optimization in order to introduce efficiency and robustness during registration that has been favorably compared to other
efficient diffeomorphic registration algorithms that were proposed in the literature simultaneously to our conference results. Our algorithm has been successfully used in
the generation of 3D statistical brain atlases from statistics on populations of diffeomorphisms.
Agradecimientos
Durante estos ocho años ha habido muchas personas que me han ayudado tanto en
lo personal como en lo profesional a que esta tesis doctoral siguiera adelante. Como
me resultarı́a imposible nombrar en unas pocas lı́neas a todas ellas, simplemente me
gustaria agradecerles su apoyo y mencionar a aquellas que me han ayudado de una
manera mas especial.
En primer lugar me gustarı́a agradecer a mi director de tesis, Salvador Olmos,
sin cuya ayuda esta tesis doctoral no habrı́a sido posible. Muchı́simas gracias por
depositar tu confianza en mı́ y por proporcionarme todas las oportunidades a tu alcance
para la realización de esta tesis. También me gustarı́a agradecer a las personas que
desinteresadamente me permitieron visitar sus laboratorios y dedicaron su tiempo a que
pudiese aprender parte de sus conocimientos. Me gustarı́a, pues, agradecer a Guillermo
Sapiro por proporcionarme un modelo a seguir tanto en lo profesional como sobre todo
en el trato personal con los estudiantes y a Olivier Faugeras. Muy especialmente me
gustarı́a agradecer a Juan Cebral su comprensión, ayuda y apoyo incondicional sobre
todo en los últimos años del desarrollo de la presente tesis.
En segundo lugar me gustarı́a agradecer a todos a los compañeros de laboratorio
y amigos que he conocido durante mis estancias de investigación por su amabilidad
y dedicación. Muchas gracias a mis compañeros de la Universidad de Minnesota,
sobre todo a Omar (y su encantadora familia), Facundo, Alberto, Liron y Diego por
ayudarme en los duros (y frı́os) comienzos de esta tesis. A los ”roomates” de DinkyTown, muy especialmente a Yuk, Charlotte, Dr. Manuel y nuestros amigos Helena y
Aitor. Muchı́simas gracias a mis compañeros del Inria por ayudarme a sobrevivir en
Francia sin saber hablar francés. Muy especialmente, a Lucero, por ser la persona más
cariñosa que he conocido nunca y a Marie-Cecille por toda tu ayuda dentro y fuera del
laboratorio. También a todas las personas a las que he conocido durante los cursos y
congresos a los que he asistido.
En tercer lugar me gustarı́a agradecer a todas las personas que han venido a
Zaragoza a impartir seminarios (S. Pizer, J. Modersizsky, X. Pennec...) y a los profesores encargados del programa de doctorado de Ingenierı́a Biomédica por hacerlo
posible.
En cuarto lugar me gustarı́a agradecer a los doctores que me han proporcionado las
imágenes y los conocimientos clı́nicos necesarios para poder desarrollar esta tesis y a
todos los pacientes que de manera anónima han formado parte de mi trabajo dı́a a dı́a.
Solo espero que los resultados obtenidos en esta tesis sean mi humilde contribucion en
la mejora del diagnóstico y tratamiento de sus patologı́as.
En quinto lugar, como no, agradecer a todos mis compañeros de la Universidad de
x
Zaragoza. Muchı́simas gracias a Iñaki por ser mas que un compañero, un amigo. A
Mehmet por todo lo que nos hemos reı́do. A Sebastián y Estanislao por todo lo que
hemos compartido en estos años. A Juanma, Santi, Yolanda, Goyo, Javi, Elsa y a los
profes de Fundamentos por ser tan buenos compañeros y hacer el dı́a a dı́a más fácil.
Especialmente a Luis y a Alberto, por la oportunidad de compartir asignatura estos
tres uĺtimos años y vuestra ayuda en estos últimos meses.
Por último agradecer a ”mis chicas” Alicia, Anafru, Bea, Susanita y a Hans por ser
los mejores amigos que nadie pueda tener. A mi familia por su comprensión y apoyo
sobre todo en los malos momentos. Especialmente agradecer a mi abuela los malos
ratos que ha pasado cuando me he marchado ”tan lejos” y a mi abuelo, al que siempre
llevaré en mi corazón. A mis chiquitines Currito y Pi por hacerme sonreir siempre...
...Aunque, sin duda, mi último y más importante gracias es para Xavi. Gracias
porque sin tu paciencia y apoyo esta tesis no hubiese sido posible.
Gracias.
Grants and research projects
The first part of this Thesis was developed under the support of grant AP2001-1678
from the program Formación de Profesorado Universitario (FPU) awarded by the
Spanish Ministry of Sciences, Education and Sports from years 2001 to 2003 and supervised by F.J. Seron and A.F. Frangi. The second part of this Thesis was developed
from years 2004 to 2008 while holding a teaching position as assistant professor at the
Computer Sciences and Systems Department, University of Zaragoza.
In addition, part of the research carried out in this Thesis was performed in the
framework of the research projects mentioned below. I would like to acknowledge to
their principal researchers for giving me the opportunity to participate in these projects
that provided part of the founding needed to carry out my research.
• Diseño y desarrollo de un prototipo para la adquisición, monitorización y análisis
de señales ECG en entorno hospitalario basado en nuevas técnicas de procesado
de señal (CICYT-FEDER project, ref. 2FD97-1197-C02-01). Pablo Laguna,
1997-2000.
• Modelado Tridimensional y simulación de arterias coronarias (MOTRICO) (CICYT project, ref. TIC-2000-1635-C04-01). Francisco J. Seron, 2000-2003.
• Nuevos métodos de análisis y autentificación en biométrica facial mediante modelos activos de forma y apariencia (ieVULTUS) (CICYT project, ref. TIC200204495-C02-02). Alejandro F. Frangi, 2002-2005.
• Consolidación de un Consorcio Nacional Integrado (CNI) y apoyo logı́stico para
la participación nacional en la Red Europea BHEN sobre Análisis y Modelado
Cardiovascular a partir de Imágenes Médicas Multimodalidad (CORCNI) (ref.
TIC2002-10416-E). Alejandro F. Frangi, 2002-2004.
• Imagen médica molecular y multimodalidad (IM3 network) (ISCII project, ref.
G03/185). Alejandro F. Frangi and Manuel Doblare, 2003-2006.
• Optimización de un sistema de radioterapia para el tratamiento del cáncer (DGA
project, ref. PIP061/2005). Juan M. Artacho, 2005-2008.
• Métodos avanzados de procesado de señal para resonancia magnética estructural
y funcional del cerebro humano (CICYT project, ref. TEC2005-07801-C03-02
and TEC2006-13966-C03-02). Salvador Olmos, 2005-2009.
Publications
The main ideas presented in this Thesis have been presented in the following publications:
Journal articles
• M. Hernandez and A.F. Frangi.
Non-parametric geodesic active regions: Method and evaluation for cerebral
aneurysms segmentation in 3DRA and CTA.
Med. Image. Anal., 11(3):224 – 241, 2007.
• M. Hernandez, M.N. Bossa, and S. Olmos.
Registration of anatomical images using paths of diffeomorphisms parameterized
with stationary vector fields.
Int. J. Comput. Vis., in press, 2009.
Peer Reviewed Full Length Conference Papers
• M. Hernandez, G. Sapiro, and A.F. Frangi.
Three-dimensional segmentation of brain aneurysms in CTA using non-parametric
region-based information and implicit deformable models: Method and evaluation.
Proc. of the 6th International Conference on Medical Image Computing and
Computer Assisted Intervention (MICCAI’03), Lecture Notes in Computer Science (LNCS), Springer-Verlag, Berlin, Germany, 2879:594 – 602, 2003.
• M. Hernandez and A. F. Frangi.
Geodesic active regions using non-parametric statistical regional description and
their application to aneurysm segmentation from CTA.
Proc. of the 2nd International Conference on Medical Imaging and Augmented
Reality (MIAR’04), Lecture Notes in Computer Science (LNCS), Springer-Verlag,
Berlin, Germany, 3150:94 – 102, 2004.
• M. Hernandez, M. N. Bossa, and S. Olmos.
Registration of anatomical images using geodesic paths of diffeomorphisms parameterized with stationary vector fields.
xiv
IEEE 11th International Conference on Computer Vision, 2007 (ICCV’07). Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA’07), 2007.
Best MMBIA 2007 Paper Award.
• M. N. Bossa, M. Hernandez, and S. Olmos.
Contributions to 3D diffeomorphic atlas estimation: Application to brain images.
Computer Assisted Intervention (MICCAI’07), Lecture Notes in Computer Science (LNCS), Springer-Verlag, Berlin, Germany, 4791:667 – 674, 2007. MICCAI
2007 Young Scientist Awards runner up.
• M. Hernandez and S. Olmos.
Gauss-Newton optimization in diffeomorphic registration.
5th IEEE International Symposium on Biomedical Imaging (ISBI’08), 2008.
• M. Hernandez, S. Olmos and X. Pennec.
Comparing algorithms for diffeomorphic registration: Stationary LDDMM and
Diffeomorphic Demons.
11th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI’08). 2nd Workshop on Mathematical Foundations
on Computational Anatomy (MFCA’08), 2008.
Book chapters
• M. Hernandez, A.F. Frangi, and G. Sapiro.
Quantification of cerebral aneurysm 3D morphology from CTA based on nonparametric, region-based level-set techniques.
Handbook of Biomedical Image Analysis, Kluwer Academic Press, New York,
2005.
Abstracts in International Conferences
• M. Hernandez, R. Barrena, G. Hernandez, G. Sapiro, and A.F. Frangi.
Pre-clinical evaluation of implicit deformable models for 3D-segmentation of brain
aneurysms from CTA images.
Medical Imaging 2003: Image Processing. Proceedings of SPIE, 5032:1264 – 1274,
2003.
• J. R. Cebral, M. Hernandez, and A.F. Frangi.
Computational analysis of blood flow dynamics in cerebral aneurysms from CTA
and 3D rotational angiography image data.
Proc. of the 1st International Congress on Computational Bioengineering (ICCB’03),
2003.
xv
• J. R. Cebral, M. Hernandez, A. F. Frangi, C. Putman, R. Pergolizzi, and J. Burgess.
Subject-specific modeling of intracranial aneurysms.
Medical Imaging 2004: Physiology, function, and structure from medical images.
Proceedings of SPIE, 5369:319 – 327, 2004.
Brain aneurysm segmentation in CTA and 3DRA using geodesic active regions
based on second order prototype features and non parametric density estimation.
• R. D. Millan, M. Hernandez, D. Gallardo, J. R. Cebral, and A. F. Frangi.
Characterization of cerebral aneurysms using geometric moments.
Internal Reports
Estimation of statistical atlases using groups of diffeomorphisms.
Technical report, Aragon Institute of Engineering Research (I3A), June, 2007.
Insight journal articles
• X. Mellado, M. Hernandez, I. Larrabide, and A.F. Frangi.
Flux driven medial curve extraction.
The Insight Journal, 2007.
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Segmentation and registration of medical images . . . . . . .
1.1.1 Automatic segmentation of vascular structures . . . .
1.1.2 Efficient diffeomorphic registration for CA applications
1.2 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Part I: Segmentation of medical images . . . . . . . .
1.2.2 Part II: Registration of medical images . . . . . . . . .
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Segmentation of medical images
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2 Implicit Deformable Models . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The level set method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Lagrangian vs Eulerian representations for curve and surface evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Variational formulation of the level set method . . . . . . . . .
2.2.3 Numerical aspects of the level set method . . . . . . . . . . . .
2.3 Implicit flows in fluid dynamics . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Flow driven by an external velocity field . . . . . . . . . . . . .
2.3.2 Constant flow in normal direction . . . . . . . . . . . . . . . .
2.3.3 Flows driven by curvature . . . . . . . . . . . . . . . . . . . . .
2.3.4 Combination of simple flows . . . . . . . . . . . . . . . . . . . .
2.4 Implicit flows in image segmentation . . . . . . . . . . . . . . . . . . .
2.4.1 The snake model . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Edge-based models . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Region-based models . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Vessel-specific segmentation models . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Non-parametric vessel enhancement filter . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The multi-scale local structure in 3D images . . . . . . . . .
3.2.1 Scale-space image representation . . . . . . . . . . .
3.2.2 Differential image descriptors. Differential invariants.
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3.2.3 Feature detection and scale selection . . . . . . . . . . . . . . .
Multi-scale representations of the second-order local structure . . . . .
3.3.1 The second-order ellipsoid representation . . . . . . . . . . . .
3.3.2 The second-order prototype representation . . . . . . . . . . . .
3.3.3 The second-order irreducible differential invariant representation
Non-parametric probability estimation in tissue classes . . . . . . . . .
3.4.1 Feature space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Non-parametric probability estimation . . . . . . . . . . . . . .
3.4.3 Training set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Implementation details . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Vessel enhancement in MRA . . . . . . . . . . . . . . . . . . .
3.5.2 Vessel enhancement in CTA . . . . . . . . . . . . . . . . . . . .
3.5.3 Vessel enhancement in 3D-RA . . . . . . . . . . . . . . . . . .
Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . .
4 Non-parametric Geodesic Active Regions . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Region-based implicit deformable models . . . . . . . .
4.2.2 Methods for the segmentation of the cerebral vasculature
4.3 Non-parametric Geodesic Active Regions . . . . . . . . . . . .
4.4 Training set construction . . . . . . . . . . . . . . . . . . . . .
4.4.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Protocol followed for training sets construction . . . . .
4.4.3 Training set selection . . . . . . . . . . . . . . . . . . .
4.5 Parameter and model selection . . . . . . . . . . . . . . . . . .
4.5.1 Selection of the optimal number of neighbors . . . . . .
4.5.2 Model selection . . . . . . . . . . . . . . . . . . . . . . .
4.6 Segmentation results and Evaluation . . . . . . . . . . . . . . .
4.6.1 Datasets and experimental setting . . . . . . . . . . . .
4.6.2 Evaluation framework . . . . . . . . . . . . . . . . . . .
4.6.3 Evaluation results . . . . . . . . . . . . . . . . . . . . .
4.7 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . .
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Registration of medical images
5 Diffeomorphisms in Computational Anatomy . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 Riemannian manifolds of diffeomorphisms . . . . . .
5.2.1 Differentiable structure on infinite dimensional
feomorphisms . . . . . . . . . . . . . . . . . .
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manifolds of dif. . . . . . . . . .
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5.3
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5.2.2 The manifold of H s -diffeomorphisms . . . . . . . . . . . . . .
5.2.3 The ”Lie” group of H s -diffeomorphisms . . . . . . . . . . . .
Diffeomorphic Registration Methods . . . . . . . . . . . . . . . . . .
5.3.1 Large Deformation Kinematics for diffeomorphic registration
5.3.2 Large Deformation Diffeomorphic Metric Mapping (LDDMM)
5.3.3 Numerical aspects of the LDDMM method . . . . . . . . . .
5.3.4 LDDMM from Jacobi Fields . . . . . . . . . . . . . . . . . .
5.3.5 Efficient algorithms for diffeomorphic registration . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 LDDMM from one-parameter subgroups of diffeomorphisms
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Stationary-LDDMM for diffeomorphic registration . . . . . . .
6.2.1 Euler-Lagrange equation for stationary-LDDMM . . . .
6.2.2 Numerical implementation . . . . . . . . . . . . . . . .
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Regularization parameters selection . . . . . . . . . . .
6.3.3 Evaluation framework . . . . . . . . . . . . . . . . . . .
6.3.4 Registration results in real datasets . . . . . . . . . . .
6.3.5 Simulated datasets . . . . . . . . . . . . . . . . . . . . .
6.3.6 Registration results in simulated datasets . . . . . . . .
6.3.7 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Comparing algorithms for efficient diffeomorphic registration
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Gauss-Newton optimization in stationary-LDDMM . . . . . . .
7.2.1 Numerical implementation . . . . . . . . . . . . . . . .
7.3 Stationary-LDDMM vs Diffeomorphic Demons . . . . . . . . .
7.3.1 General variational formulation . . . . . . . . . . . . . .
7.3.2 Characterization of diffeomorphic transformations . . .
7.3.3 Image similarity metric . . . . . . . . . . . . . . . . . .
7.3.4 Regularization energy . . . . . . . . . . . . . . . . . . .
7.3.5 Optimization scheme . . . . . . . . . . . . . . . . . . . .
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Regularization parameters selection . . . . . . . . . . .
7.4.3 Registration results . . . . . . . . . . . . . . . . . . . .
7.4.4 Suitability for the computation of statistics . . . . . . .
7.4.5 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
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xx
8 Generation of 3D anatomical brain atlases . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Statistics on Riemannian manifolds . . . . . . . . . . . . . . . . . . . .
8.2.1 Means in Riemannian manifolds . . . . . . . . . . . . . . . . .
8.2.2 Means in Dif f s (Ω) . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3 Principal Component Analysis on Riemannian manifolds . . . .
8.2.4 Principal Component Analysis in Dif f s (Ω) . . . . . . . . . . .
8.3 Generation of 3D anatomical brain atlases . . . . . . . . . . . . . . .
8.4 Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Statistics on the population of diffeomorphisms and statistical
atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . .
219
220
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232
234
234
9 Contributions, conclusions and perspectives
9.1 Contributions and conclusions . . . . . . . .
9.1.1 Segmentation of medical images . .
9.1.2 Registration of medical images . . .
9.2 Perspectives . . . . . . . . . . . . . . . . .
9.2.1 Segmentation of medical images . .
9.2.2 Registration of medical images . . .
241
241
241
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234
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
List of Tables
3.1
3.2
3.3
4.1
Structure of critical points in Morse theory. . . . . . . . . . . . . . . . . . . .
53
Second-order structure in relation to the eigenvalues of the Hessian |λ1 | ≤ |λ2 | ≤ |λ3 |. 54
Second-order structure in relation to the shape and orientation of the elements in
the prototype space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Summary of the most important characteristics of state of the art methods for segmentation of vascular structures. . . . . . . . . . . . . . . . . . . . . . . . .
88
4.2
Parameters involved in the considered model based techniques (non-parametric Geodesic Active Regions (NP-GAR), parametric Geodesic Active Regions (P-GAR) and
Active Contours Without Edges (ACWE)) and values used in the evaluation experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3
Mean and standard deviation of the DSC values for the methods considered in the
evaluation study: Non-parametric Geodesic Active Regions (NP-GAR), K-Means
(KM), Parametric Geodesic Active Regions (P-GAR), and Active Contours Without
Edges (ACWE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4
Statistics for the study of the similarity between manual and model-based segmentations: Non-parametric Geodesic Active Regions (NP-GAR), Parametric Geodesic
Active Regions (P-GAR), and Active Contours Without Edges (ACWE). The first
row for each model represents the mean and the second row, the standard deviation
of the quantities over the datasets (measured in mm). . . . . . . . . . . . . . . 109
6.1
6.2
Algorithm for non-stationary and stationary diffeomorphic registration. . . . . . . 149
Average and standard deviation of the RSSD (%) (upper row) and Jmin (lower row)
for different values of the regularization parameters α and 1/σ 2 . Metric values associated to the selected parameters are outlined in boldface. Non-diffeomorphic results
are outlined in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3
Average and standard deviation of the RSSD (%) and Jmin values associated to
registration experiments in simulated datasets. . . . . . . . . . . . . . . . . . 165
7.1
Stationary LDDMM registration. Average and standard deviation of the RSSD
(%) (upper row) and Jmin (lower row) for different values of the regularization pa2
. Metric values associated to the selected parameters are
rameters α and 1/σsim
outlined in boldface. Non-diffeomorphic results are outlined in red. Note that the
algorithms do not converge for values α of order 0.0001. . . . . . . . . . . . . . 198
xxii
7.2
7.3
7.4
Diffeomorphic Demons registration. Average and standard deviation of the
RSSD (%) (upper row) and Jmin (lower row) for different values of the regularization
parameters σsim and τ . Metric values associated to the selected parameters are
outlined in boldface. Non-diffeomorphic results are outlined in red. . . . . . . . . 198
Average and standard deviation of the forward and backward RSSD and the distance
− id2 + ϕ−1
◦ ϕi − id2 ) used for the assessment of the
dSSD = 12 (ϕi ◦ ϕ−1
i
i
inverse consistency error. With IC-LDDMM and SG-LDDMM we indicate the inverse
consistent and the symmetric gradient version of LDDMM, respectively. . . . . . 201
Average and standard deviation of the metrics used for the assessment of the smoothness of the diffeomorphisms and the corresponding tangent vectors. With IC-LDDMM
and SG-LDDMM we indicate the inverse consistent and the symmetric gradient version of LDDMM, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 201
List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Different applications from image segmentation. Left, quantification of vessel diameters in a 3D vascular model of the anterior cerebral circulation including a brain
aneurysm. Image obtained from http://www.vmtk.org/Main/Screenshots. Reproduced with permission of Dr. L. Antiga, Mario Negri Institute, Italy. Middle,
radiation dose planning associated to a 3D prostate model. Image courtesy of Dr.
J.M. Artacho, University of Zaragoza, Spain. Right, mean shear stress distribution
of the blood flow simulated in a 3D vascular model of a brain aneurysm located in
the basilar artery. Image courtesy of Dr. J.R. Cebral, George Mason University,
Fairfax, USA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Different applications from image registration. Left, areas of brain activation superimposed on the anatomical image after functional to structural registration. Right,
example of non-rigid intersubject registration. The deforming image is transformed
towards the reference image. The result constitutes the deformed image. The transformation encodes the anatomical differences existing between the reference and the
deforming image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Challenging examples of image segmentation. a) Noise and artifacts in a 3D-RA
image of a cerebral aneurysm acquired after coil implant. b) Intensities of Turkish
saddle in CTA similar to intensities in carotid arteries. c) Lack of strong edges
between blood pool and ventricle walls in cardiac CT. Intensity inhomogeneity and
artifacts inside the auricula due to blood flow concentration. d) Lack of strong edges
between prostate and bladder due to huge partial volume effect. e) Out-of-plane
resolution in thoracic CT. . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Implicit representation of a star-shaped curve. Left, represented curve. Center,
image of the level set isocontours. Right, surface plot of the level set function. . . .
18
Evolution of a sphere driven by a rotation external velocity field defined on the space. 25
Evolution of a dumbell shaped surface in a negative constant velocity field. . . . .
27
Evolution of a star shaped curve under curvature. The tips of the star move inward
while the gaps between the tips move in outward direction. The shape is gradually
transformed to a circle that shrinks towards a point, as stated in Grayson’s theorem.
30
Evolution of a dumbbell-shaped surface under mean curvature. The dumbbell center
narrows until the shape splits in two different evolving interfaces. . . . . . . . . .
30
Evolution of a torus surface under mean curvature. The torus evolves towards its
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
centerline.
31
xxiv
2.8
Evolution of a torus surface under minimum curvature. In this case, the torus shrinks
. . . . . . . . . . . . . . . .
to a point, as expected from Grayson’s theorem.
31
Image from a brain MRI scan and its topographic representation at scale σ = 1.5. .
52
Ellipsoid representation of plate-, line- and blob-like structures, respectively. The
arrows represent the direction of the eigenvectors of the Hessian matrix. The length
of the arrows represents the magnitude of the corresponding eigenvalues. . . . . .
55
Representation of the different second-order shape structures in the prototype space.
Image from Q. Lin, Enhancement, detection and visualization of 3D volume data,
Linkoping University, Sweden, 2003. Ph.D. dissertation. . . . . . . . . . . . . .
58
The arteries of the Circle of Willis. Anterior Cerebral Arteries (ACA) and segments
(A1 and A2) connected by the Anterior Communicant Artery (ACoA), Middle Cerebral Arteries (MCA), Posterior Communicant Arteries (PCoA) connecting the Internal Carotid Arteries (ICA) and the Basilar Artery (BA), and Vertebral Arteries
(VA). Model courtesy of J.R. Cebral, George Mason University, Fairfax, USA. . .
64
3.5
Vessel enhancement in MRA. First row shows axial, coronal and sagittal views
of the image. Second row shows volume rendering images of the results from the
multi-scale vessel enhancement filters and the probability for vessel estimated from
the GMM. Third row shows volume rendering images of the probability for vessel
estimated from our non-parametric method using the ellipsoid, prototype and differential invariant representations, respectively. . . . . . . . . . . . . . . . . . . .
68
3.6
Vessel enhancement in MRA. First row shows axial views of the image and
the corresponding K-Means tissue classification (vessel tissue is labelled in white,
background tissue in grey and partial volume tissue in black). Second row shows the
probability for vessel estimated from the parametric and non-parametric models. .
69
Vessel enhancement in CTA. First row shows axial, coronal and sagittal views of
the image. Second row shows volume rendering images of the results for the multiscale vessel enhancement filters and the probability for vessel estimated from the
GMM. Third row shows volume rendering images of the probability for vessel estimated from our non-parametric method using the ellipsoid, prototype and differential
invariant representations, respectively. . . . . . . . . . . . . . . . . . . . . . .
70
Vessel enhancement in CTA. First row shows sagittal views of the image and the
corresponding K-Means tissue classification (vessel tissue is labelled in grey, background tissue in black and bone tissue in white). It should be noted that, in this
example, K-Means classification is not able to separate the partial volume tissue.
Second row shows the probability for vessel estimated from the parametric and nonparametric models. Third row shows the corresponding probability for bone. . . .
71
Vessel enhancement in 3D-RA (standard example). First row shows axial, coronal and sagittal views of the image. Second row shows volume rendering images of
the results from the multi-scale vessel enhancement filters and the probability for
vessel estimated from the GMM. Third row shows volume rendering images of the
probability for vessel estimated from our non-parametric method using the ellipsoid,
prototype and differential invariant representations, respectively. . . . . . . . . .
72
3.1
3.2
3.3
3.4
3.7
3.8
3.9
xxv
3.10 Vessel enhancement in 3D-RA (standard example). First row shows axial views
of the image and the corresponding K-Means tissue classification (vessel tissue is
labelled in white, background tissue in grey and partial volume tissue in black).
Second row shows the probability for vessel estimated from the parametric and nonparametric models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.11 Vessel enhancement in 3D-RA (low-dose example). First row shows axial, coronal and sagittal views of the image. Second row shows volume rendering images of
the results from the multi-scale vessel enhancement filters and the probability for
vessel estimated from the GMM. Third row shows volume rendering images of the
probability for vessel estimated from our non-parametric method using the ellipsoid,
prototype and differential invariant representations, respectively. . . . . . . . . .
74
3.12 Vessel enhancement in 3D-RA (low-dose example). First row shows coronal
views of the image and the corresponding K-Means tissue classification (vessel tissue
is labelled in white, background tissue in grey and partial volume tissue in black).
Second row shows the probability for vessel estimated from the parametric and nonparametric models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.13 Vessel enhancement in 3D-RA (adquired after coil implant). First row shows
axial, coronal and sagittal views of the image. Second row shows volume rendering
images of the results from the multi-scale vessel enhancement filters and the probability for vessel estimated from the GMM. Third row shows volume rendering images
of the probability for vessel estimated from our non-parametric method using the
ellipsoid, prototype and differential invariant representations, respectively. . . . . .
76
3.14 Vessel enhancement in 3D-RA (acquired after coil implant). First row shows
4.1
4.2
4.3
4.4
coronal views of the image and the corresponding K-Means tissue classification (coil
tissue is labelled in white, vessel tissue in dark grey, background in black and partial
volume tissue in light grey). Second row shows the probability for vessel estimated
from the parametric and non-parametric models. . . . . . . . . . . . . . . . . .
77
Cerebral aneurysm. Image from the book frontcover The brain aneurysm, V. G. Khurana and R. F. Spetzler, Barrow Neurological Institute, Phoenix, USA. Reproduced
with the permission of Dr. V. G. Khurana. . . . . . . . . . . . . . . . . . . .
83
Clip device and scheme of clipping procedure. Once the aneurysm is reached by
craniotomy, the clip is placed across the aneurysm neck in order to stop blood
flow inside the aneurysm. Images obtained from http://www.aesculapusa.com and
http://www.ubneurosurgery.com. . . . . . . . . . . . . . . . . . . . . . . .
84
Guglielmi Detachable Coil device and scheme of coil embolization procedure. The
aneurysm sac is reached from a micro-catheter inserted through the femoral artery.
The platinum coils are deployed into the aneurysm blocking the blood flow into
its dome. Images obtained from http://www.endovascular.jp and http://www.
ubneurosurgery.com. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Examples of 3D-RA images. (a) Slice of an image acquired under standard protocol.
(b) Slice of an image acquired under low contrast dose showing vessel artifacts. (c)
Slice of an image acquired under low contrast dose showing bone tissue. . . . . . .
86
xxvi
4.5
Examples of CTA images. (a) Slice showing a vessel located next to the aneurysm
dome. (b) Slice showing a cerebral aneurysm located next to bone tissue. (c) Slice
showing part of the Turkish saddle next to the Internal Carotid Arteries (ICA). . .
86
Training sets in 3D-RA. Maximum Intensity Projection (MIP) of the 3D-RA
images selected for training. First row corresponds to the datasets selected for the
first training set (Train #1). Second row corresponds to the datasets selected for the
. . . . . . . . . . . . . . . . . . . . . . . .
second training set (Train #2).
96
Training sets in CTA. Volume rendering of the CTA images selected for training.
Upper group corresponds to the datasets selected for the first training set (Train
#1). Lower group corresponds to the datasets selected for the second training set
(Train #2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Probability estimation in 3D-RA. Comparison of the probabilities for vessel
estimated with Train #1 and #2 in the three different feature spaces considered in
the non-parametric estimation. The figure shows the original slice images and the
probabilities associated to the training sets. . . . . . . . . . . . . . . . . . . .
97
Probability estimation in CTA. Comparison of the probabilities for vessel estimated with Train #1 and #2 in the three different feature spaces considered in
the non-parametric estimation. The figure shows the original slice images and the
probabilities associated to the training sets. . . . . . . . . . . . . . . . . . . .
98
4.10 Bias/Variance dilemma in pattern recognition. Illustration of test and training generalization error behavior as the model complexity is increased. . . . . . . . . .
99
4.6
4.7
4.8
4.9
4.11 Upper row, curves of the generalization errors associated to the number of neighbors,
k. Computations from ten different cross-validation experiments. Lower row, curves
of the average computation time spent in the kd-tree search for a volume of size
100 × 100 × 100. Left, curves corresponding to 3D-RA. Right, curves corresponding
to CTA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.12 Probability estimation in 3D-RA. Comparison of the probabilities for vessel
estimated from the three different feature spaces considered in the non-parametric
estimation. The figure shows the absolute differences between probabilities for all
the possible permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.13 Probability estimation in CTA. Comparison of the probabilities for vessel estimated estimated from the three different feature spaces considered in the nonparametric estimation. The figure shows the absolute differences between probabilities for all the possible permutations. . . . . . . . . . . . . . . . . . . . . . . 103
4.14 Comparison of the 3D-RA segmented vascular models with the reference techniques
considered in the evaluation study (ACoA, PCoA, MCA, basilar, and ICA). The
columns show the results from the manual, our non-parametric (NP-GAR), K-Means
(KM), parametric (P-GAR) and Active Contour Without Edges (ACWE) methods,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
xxvii
4.15 Comparison of the CTA segmented vascular models with the reference techniques
considered in the evaluation study (ACoA-1, ACoA-2, PCoA, MCA-1, MCA2). The
columns show the results from the manual, our non-parametric (NP-GAR), K-Means
(KM), parametric (P-GAR), and Active Contour Without Edges (ACWE) methods,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.16 DSC values for the methods considered in the evaluation study: our non-parametric
Geodesic Active Regions (NP-GAR), K-Means (KM), Parametric Geodesic Active
Regions (P-GAR), and Active Contours Without Edges (ACWE). The left figure
presents the results for the 3D-RA and the right figure for the CTA datasets, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1
Diagram of diffeomorphism characterization. Path in Dif f s (Ω) starting at φ(0) = id
parameterized from time-varying vector field flow v(t, φ(t)). . . . . . . . . . . . . 122
6.1
Brain anatomy. Sagittal view of the brain from a cadaver section showing some of
the anatomical regions and subcortical structures that are referenced in this Thesis.
Image courtesy of the Digital Anatomist Project, University of Washington, USA. . 150
6.2
Real datasets (patients 1-7). Top row, sagittal, coronal and axial views of the
image selected as reference, Iref . Left column group, views of the datasets used in this
experimental section, Ii . Right column group, corresponding intensity differences,
Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3
Real datasets (patients 8-14). Top row, sagittal, coronal and axial views of the
image selected as reference, Iref . Left column group, views of the datasets used in this
experimental section, Ii . Right column group, corresponding intensity differences,
Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4
Registration in real datasets (patients 1-7). Illustration of sagittal, coronal and
− Iref .
axial views of the intensity differences after LDDMM registration, Ii ◦ ϕ−1
i
Left column group corresponds to the results obtained with the non-stationary parameterization. Right column group corresponds to the results obtained with the
stationary parameterization. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5
Registration in real datasets (patients 8-14). Illustration of sagittal, coronal
−
and axial views of the intensity differences after LDDMM registration, Ii ◦ ϕ−1
i
Iref . Left column group corresponds to the results obtained with the non-stationary
parameterization. Right column group corresponds to the results obtained with the
6.6
Registration in real datasets. Illustration of sagittal, coronal and axial views
of the intensity variance associated to the populations of transformed images. First
row shows the results for non-stationary LDDMM (max(IV ) = 396.28). Second row
shows the results for stationary-LDDMM (max(IV ) = 350.28). . . . . . . . . . . 159
xxviii
6.7
Registration in real datasets. Illustration of sagittal, coronal and axial views
of the local differences existing between the transformations obtained with nonstationary and stationary LDDMM registration algorithms measured in terms of
the dAI metric. Left column group shows the results from patieents 1 to 7. Right
column group shows the results from patients 8 to 14. . . . . . . . . . . . . . . 160
6.8
Registration in real datasets. First row, illustration of sagittal, coronal and
axial views of the average of the metric dAI between non-stationary and stationaryLDDMM transformations through the database of patients. Second row, average
distance dSSD between corresponding grid points. . . . . . . . . . . . . . . . . 161
6.9
Registration in real datasets (patient #10). First row, superimposed 2D views of
the transformations obtained with non-stationary (white grids) and stationary (blue
grids) LDDMM registration projected onto corresponding sagittal, coronal and axial
planes. Second row, values of distance dAI between both transformations in these
planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.10 Registration in real datasets (patient #10). Upper row, illustration of sagittal views of the transformations obtained with non-stationary (left) and stationary
(right) LDDMM registration algorithms superimposed on the saggital slice image,
I10 . Lower row, generators of the corresponding path parameterizations v(0) and w,
respectivelly. Both grids and glyphs are colored with respect to displacement and
vector magnitude, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.11 Registration in real datasets (patient #10). Illustration of the flow of velocity
fields corresponding to the non-stationary path parameterization, v(t), t = 1, ..., 9.
Glyphs are colored with respect to its magnitude. . . . . . . . . . . . . . . . . 163
6.12 Non-stationary simulated datasets (patients 1-7). Top row, sagittal, coronal and
axial views of the image selected as reference, Iref . Left column group, views of the
datasets used in this experimental section, Ii . Right column group, corresponding
intensity differences Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.13 Non-stationary simulated datasets (patients 8-14). Top row, sagittal, coronal
and axial views of the image selected as reference, Iref . Left column group, views of
the datasets used in this experimental section, Ii . Right column group, corresponding
intensity differences Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.14 Stationary simulated datasets (patients 1-7). Top row, sagittal, coronal and
intensity differences, Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.15 Stationary simulated datasets (patients 8-14). Top row, sagittal, coronal and
intensity differences, Ii − Iref . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
xxix
6.16 Registration in non-stationary simulated datasets (patients 1-7). Illustration
of sagittal, coronal and axial views of the difference images between the reference
and the deformed images obtained with LDDMM registration algorithms. Left column group corresponds to the registration results obtained with the non- stationary
6.17 Registration in non-stationary simulated datasets (patients 8-14). Illustration of sagittal, coronal and axial views of the difference images between the reference
and the deformed images obtained with LDDMM registration algorithms. Left column group corresponds to the registration results obtained with the non- stationary
6.18 Registration in stationary simulated datasets (patients 1-7). Illustration of
sagittal, coronal and axial views of the difference images between the reference and
the deformed images obtained with LDDMM registration algorithms. Left column
group corresponds to the registration results obtained with the non- stationary parameterization. Right column group corresponds to the results obtained with the
6.19 Registration in stationary simulated datasets (patients 8-14). Illustration of
sagittal, coronal and axial views of the difference images between the reference and
the deformed images obtained with LDDMM registration algorithms. Left column
group corresponds to the registration results obtained with the non- stationary parameterization. Right column group corresponds to the results obtained with the
6.20 Registration in non-stationary simulated datasets. Illustration of sagittal,
coronal and axial views of the intensity variance associated to the populations of
transformed images. First row shows the results for non-stationary LDDMM (max(IV )=
84.42). Second row shows the results for stationary-LDDMM (max(IV ) = 72.07). . 174
6.21 Registration in stationary simulated datasets. Illustration of sagittal, coronal
and axial views of the intensity variance associated to the populations of transformed
images. First row shows the results for non-stationary LDDMM (max(IV ) = 50.42).
Second row shows the results for stationary-LDDMM (max(IV ) = 20.27). . . . . . 174
coronal and axial views of the local differences existing between the ground truth
transformations and the results of non-stationary LDDMM algorithm measured in
terms of the dAI metric. Left column group shows the results from patieents 1 to 7.
Right column group shows the results from patients 8 to 14. . . . . . . . . . . . 175
coronal and axial views of the local differences existing between the ground truth
transformations and the results of stationary-LDDMM algorithm measured in terms
of the dAI metric. Left column group shows the results from patieents 1 to 7. Right
xxx
and axial views of the local differences existing between the ground truth transformations and the results of non-stationary LDDMM algorithm measured in terms of
the dAI metric. Left column group shows the results from patieents 1 to 7. Right
and axial views of the local differences existing between the ground truth transformations and the results of stationary-LDDMM algorithm measured in terms of the
dAI metric. Left column group shows the results from patieents 1 to 7. Right column
. . . . . . . . . . . . . . . . . 178
group shows the results from patients 8 to 14.
6.26 Registration in non-stationary simulated datasets. First two rows, illustration
of saggital, coronal and axial views of the average of the metric dAI between ground
truth non-stationary transformations and the results obtained from non-stationary
and stationary-LDDMM, respectivelly. Last two rows, average distance between
corresponding grid points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.27 Registration in stationary simulated datasets. First two rows, illustration of
saggital, coronal and axial views of the average of the metric dAI between ground
truth stationary transformations and the results obtained from non-stationary and
stationary LDDMM, respectivelly. Last two rows, average distance between corresponding grid points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.28 Non-stationary simulated datasets (patient #1). Illustration of sagittal, coronal and axial views of the 2D projections of the transformations obtained with
non-stationary and stationary LDDMM registration (blue grid) superimposed to the
ground truth transformation (white grid). First row shows the results corresponding
to non-stationary LDDMM registration. Second row shows the results corresponding
. . . . . . . . . . . . . . . . . . . . . . 181
to stationary-LDDMM registration.
6.29 Stationary simulated datasets (patient #7). Illustration of sagittal, coronal and
axial views of the 2D projections of the transformations obtained with non-stationary
and stationary LDDMM registration (blue grid) superimposed to the ground truth
transformation (white grid). First row shows the results corresponding to nonstationary LDDMM registration. Second row shows the results corresponding to
stationary-LDDMM registration. . . . . . . . . . . . . . . . . . . . . . . . . 181
7.1
Top row, sagittal, coronal and axial views of the image selected as reference. Left
column, views of the datasets used in this experimental section (patients 1-7). Right
column, intensity differences between the reference and the corresponding dataset on
the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.2
Top row, sagittal, coronal and axial views of the image selected as reference. Left
column, views of the datasets used in this experimental section (patients 8-14). Right
column, intensity differences between the reference and the corresponding dataset on
the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
xxxi
7.3
Curves of the image similarity, Ii ◦ ϕi − Iref 2L2 , during optimization. GD-LDDMM
denotes the curves for stationary-LDDMM with gradient descent optimization. ICLDDMM and SG-LDDMM denote the curves corresponding to stationary-LDDMM
with Gauss-Newton and symmetric gradient optimization. Demons denotes the
. . . . . . . . . . . . . . . . . . . . . . . 204
curves for diffeomorphic Demons.
7.4
Sagittal views of the intensity differences between the reference and the deformed
images obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic
Demons (right). Left column correspond to patients 1-7. Right column corresponds
to patients 8-14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.5
Coronal views of the intensity differences between the reference and the deformed
images obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic
Demons (right). Left column correspond to patients 1-7. Right column corresponds
to patients 8-14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.6
Axial views of the intensity differences between the reference and the deformed images obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic Demons (right). Left column correspond to patients 1-7. Right column corresponds to
patients 8-14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.7
Illustration of sagittal, coronal and axial views of the intensity variance associated to
the populations of transformed images. First row shows the results for IC-LDDMM,
second row shows the results for SG-LDDMM, and third row shows the results for
Diffeomorphic Demons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.8
Illustration of sagittal, coronal and axial views of the average of the metric dAI
between stationary-LDDMM and diffeomorphic Demons transformations through the
database of patients. First row, distance between IC-LDDMM and ESM-LDDMM.
Second row, distance between IC-LDDMM and diffeomorphic Demons. Third row,
distance between ESM-LDDMM and diffeomorphic Demons. . . . . . . . . . . . 209
7.9
Illustration of sagittal, coronal and axial views of the average of the metric dSSD
between stationary-LDDMM and diffeomorphic Demons transformations through the
database of patients. First row, distance between IC-LDDMM and ESM-LDDMM.
Second row, distance between IC-LDDMM and diffeomorphic Demons. Third row,
distance between ESM-LDDMM and diffeomorphic Demons. . . . . . . . . . . . 210
7.10 Illustration of sagittal, coronal and axial views of the 2D projections of the transformations obtained with stationary-LDDMM and diffeomorphic Demons. First group
shows the diffeomorphism and its corresponding inverse obtained with IC-LDDMM.
Second group shows the transformations corresponding to ESM-LDDMM. Third
group shows the transformations corresponding to diffeomorphic Demons. . . . . . 211
7.11 Illustration of sagittal, coronal and axial views of the velocity fields Lw obtained
with stationary-LDDMM and diffeomorphic Demons in a representative example
(patient #10) First row shows the results corresponding to IC-LDDMM. Second row
corresponds to SG-LDDMM. Third row corresponds to diffeomorphic Demons. . . 212
8.1
Diagram of the algorithm for the computation of the average atlas. . . . . . . . . 233
xxxii
8.2
Statistics on a population of diffeomorphisms. Upper row, illustration of sagittal, coronal and axial views of the mean transformation. Next three rows, illustration
of sagittal, coronal and axial views of the first three Principal Components. Both
grids and glyphs are colored with respect to displacement and vector magnitude,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
respectively.
8.3
Statistics on a population of images. Mean atlas and first mode of variation.
8.4
Statistics on a population of images. Mean atlas and second mode of variation. 238
8.5
Statistics on a population of images. Mean atlas and third mode of variation.
237
239
Chapter 1
Introduction
Contents
1.1
1.2
1.1
Segmentation and registration of medical images . . . . . . . . . .
1
1.1.1
Automatic segmentation of vascular structures . . . . . . . . . . . . .
2
1.1.2
Efficient diffeomorphic registration for CA applications
4
Thesis Structure
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Part I: Segmentation of medical images . . . . . . . . . . . . . . . . .
5
1.2.2
Part II: Registration of medical images
6
. . . . . . . . . . . . . . . . .
Segmentation and registration of medical
images
During the last decades, medical imaging has experienced an increasing importance
in clinical practice with the development of new imaging acquisition techniques of the
human body. Medical images have turned into an essential tool for the detection and
diagnosis of multiple diseases facilitating clinicians in tasks such as treatment selection,
surgical intervention guidance, and follow up, among others. The huge amount of
information that becomes available with the images has promoted the development
of image processing methods to automatically analyse and interpret the images in a
reliable way. Both segmentation and registration of medical images have been identified
as key problems in medical image analysis becoming challenging areas of active research.
Segmentation can be defined as the process of delimiting the geometry of the structures of interest in medical images. This task is of great importance in clinical practice,
as the availability of segmented structures allows quantifying clinically relevant parameters or planning for surgical interventions and radiotherapy, as illustrated in Figure 1.1.
In addition, segmentation constitutes an important step in the generation of supporting
models for the simulation of some bio-physical phenomena. Since manual segmentation
is time consuming, subjective, and of low repeatability, automated segmentation arises
as a fast, objective and reproducible alternative for these applications.
Medical image registration aims at finding correspondences between the anatomical
structures found on the images. This task is of great importance in clinical practice,
as it is usual to have images of the same patient acquired at different times or from
different devices that need to be aligned into a common coordinate system prior to be
1.1. Segmentation and registration of medical images
2
Figure 1.1: Different applications from image segmentation. Left, quantification of vessel diameters
in a 3D vascular model of the anterior cerebral circulation including a brain aneurysm. Image obtained
from http://www.vmtk.org/Main/Screenshots. Reproduced with permission of Dr. L. Antiga,
Mario Negri Institute, Italy. Middle, radiation dose planning associated to a 3D prostate model. Image
courtesy of Dr. J.M. Artacho, University of Zaragoza, Spain. Right, mean shear stress distribution
of the blood flow simulated in a 3D vascular model of a brain aneurysm located in the basilar artery.
Image courtesy of Dr. J.R. Cebral, George Mason University, Fairfax, USA.
analyzed. In addition, registration represents a crucial step in morphometric clinical
research studies. In particular, transformations obtained using non-rigid diffeomorphic
registration have shown to be a powerful tool in the study of the evolution of growth
and disease and in the study of the anatomical variability among populations, which
constitute core Computational Anatomy applications. Examples of these fundamental
applications are illustrated in Figure 1.2.
In this Ph.D. Thesis, we aim at the development of methods for the segmentation
and registration of medical images in the framework of challenging applications. The
methods are formulated as variational problems defined in their respective spaces of
solutions. The first part of this Thesis is concentrated on devising an algorithm for as
automatic as possible vascular segmentation with application to the generation of 3D
models of cerebral aneurysms from different angiographic modalities. The second part
is focused on devising an efficient algorithm for non-rigid diffeomorphic registration
intended to compete in performance and computational requirements with reference
registration methods used in Computational Anatomy applications.
1.1.1
Automatic segmentation of vascular structures
Since the development of the different angiographic acquisition techniques, a number
of algorithms have been proposed in the literature to deal with the problems posed
by the automatic segmentation of vascular structures. From them, implicit deformable
models are probably the most effective due to their ability to automatically capture the
topological changes found in complex arterial structures [217]. However, edge-based
models formulated for early segmentation [34], have shown strong limitations when
dealing with images presenting weak and inhomogeneous gradients as those found in
3
Chapter 1. Introduction
reference
deforming
deformed
transformation
Figure 1.2: Different applications from image registration. Left, areas of brain activation superimposed on the anatomical image after functional to structural registration. Right, example of non-rigid
intersubject registration. The deforming image is transformed towards the reference image. The result constitutes the deformed image. The transformation encodes the anatomical differences existing
between the reference and the deforming image.
the angiographic data acquired in clinical practice.
The availability of methods for modeling the probability for different organ tissues
in medical images have made region-based implicit deformable models quite popular
for the segmentation of different structures in medical images [284, 187, 38]. In these
methods, region descriptors are usually defined using probability models estimated
from image intensities. For the segmentation of vascular structures, these models are
often approximated with scale-space or parametric vessel enhancement filters, previously proposed in the literature for tubular structures visualization or segmentation
using statistical thresholding (see the extensive state of the art revision presented in
Chapters 3 and 4). However, these methods use too simple assumptions in the probability model. As shown in this Thesis, relying on these models can lead to very inaccurate
probability estimations introducing severe errors in the segmentation, specially when
dealing with data acquired in clinical practice.
As alternative, in Chapter 3 we propose the use of a non-parametric vessel enhancement filter for the definition of region descriptors. In non-parametric estimation,
no constraint is imposed on the probability model that is learned from a significative
sample of examples. It has been shown that multi-scale second-order differential information constitutes a powerful element in the discrimination of ridges from other
geometric features in images [150]. For this reason, in contrast to parametric estimation that usually considers image intensity as the most discriminant descriptor of
vascular tissues, we propose to rely on up to second-order differential image descriptors
in a multi-scale framework.
1.1. Segmentation and registration of medical images
4
In Chapter 4 we study how to adapt our non-parametric vessel enhancement filter into Geodesic Active Regions (GAR) for the automatic generation of 3D models
of cerebral aneurysms from 3D-Rotational Angiography (3D-RA) and Computed Tomography Angiography (CTA). In addition, the method is evaluated with respect to
manual segmentations and favorably compared with some of the region-based implicit
deformable models that could compete in performance with our algorithm for this
application.
1.1.2
Efficient diffeomorphic registration for Computational
Anatomy applications
Diffeomorphic registration is used in Computational Anatomy to compute the transformations encoding the variability existing between different anatomical images [97].
These transformations have been successfully used in clinical research studies to build
models of anatomical variability intended to identify anatomical differences between
healthy and diseased individuals and improve the diagnosis of pathologies [233, 258,
52, 257, 201]. Moreover, models of growth have been built for the assessment of the
anatomical change over time [232, 19, 170, 87, 55]. The Large Deformation Diffeomorphic Metric Mapping (LDDMM) method and its variants are considered the reference
paradigm for image registration in Computational Anatomy [67, 239, 23]. In LDDMM,
diffeomorphic transformations belong to geodesic paths of diffeomorphisms parameterized by time-varying flows of vector fields and computed from the solution of the nonstationary transport equation associated to these flows. Although the computation
of diffeomorphisms in this framework is well suited for the analysis of the statistical
variability among populations, the slow rate of convergence of gradient descent optimization together with the large computational complexity inherent to the use of the
non-stationary parameterization constitute the main limitations for the extensive use
of diffeomorphisms in clinical studies.
Recently, an alternative parameterization of paths of diffeomorphisms that uses
constant-time flows of vector fields has been proposed in the literature [8]. With this
parameterization, diffeomorphisms constitute solutions of stationary ODEs. This representation is closely related to the group structure of diffeomorphisms as the paths
that can be parameterized using stationary vector field flows are exactly identified with
the one-parameter subgroups. In Chapter 6 we include the stationary parameterization
of diffeomorphisms for registration in the LDDMM framework. In our method, computations are restricted to the space of one-parameter subgroup generators allowing great
memory and time savings with respect to non-stationary LDDMM while achieving
similar registration results.
In addition, stationary-LDDMM is improved in Chapter 7 with the incorporation
of efficient second-order optimization methods. During this Thesis, similar methods
for efficient diffeomorphic registration were proposed simultaneously to our work [11,
251], motivating its comparison with respect to these alternative methods performed
in Chapter 7.
5
Finally, in Chapter 8 we focus on the generation of statistical models of anatomical variability with application to the construction of statistical atlases associated to
a population of anatomical images. Instead of working with the usual Riemannian
structure defined on the manifold of diffeomorphisms [14, 21, 257], we rely on algebraic
oriented calculus [8]. It has been shown in the literature that working with the group
structure is appropriate for situations where either the Riemannian framework results
quite computationally expensive or the resulting statistics lack of some desirable properties [7]. As shown in this Thesis, this is precisely the case of Riemannian calculus
on diffeomorphisms. The methods developed through this Thesis together with the
obtained results, leave open the possibility of dealing with the algebraic structure in
Computational Anatomy applications.
1.2
Thesis Structure
This manuscript is structured in two differentiated parts. The first part is devoted
to segmentation while the second part deals with diffeomorphic registration. Each
part is divided into an introductory chapter that presents the mathematical tools used
through the rest of chapters and details the state of the art techniques most related
to the methods developed in this Thesis. They are introduced in the central chapters.
The last chapter of each part focuses on the application of the methods to challenging
problems arising from clinical practice or research studies.
1.2.1
Part I: Segmentation of medical images
• Chapter 2 is focused on the study of the theory of implicit deformable models.
The level set method and the most remarkable details regarding level set calculus
in its physical context are thoroughly described. In addition, the most popular
variational problems for edge- and region-based segmentation in closest relationship with the methods developed in this Thesis are overviewed with special attention to those models specifically devised for dealing with the segmentation of
thin structures.
• In Chapter 3 we present our novel method for non-parametric vessel enhancement. We begin introducing the need for improving the visualization of arterial structures in clinical practice and reviewing the state of the art techniques
for vessel enhancement. We point out the limitations of these techniques when
dealing with clinical data or images presenting vascular pathologies as cerebral
aneurysms. Then, we review the most important ideas of scale-space theory and
present the three different characterizations of the multi-scale second-order differential image structure considered for the discrimination of arterial vs non-arterial
features in our method. In addition, our non-parametric method for probability
estimation is detailed. Finally, the method is compared with reference vessel en-
1.2. Thesis Structure
6
hancement techniques in three different angiographic modalities that are usually
acquired in clinical practice for the diagnosis of cerebral vascular pathologies.
• In Chapter 4 we present our method for non-parametric Geodesic Active Regions
with application to automatic segmentation of cerebral aneurysms in 3D-RA
and CTA. We begin introducing the benefits of the availability of 3D models of
cerebral aneurysms in clinical practice procedures and its potential for providing
geometric shape descriptors or hemodynamic parameters that would help in the
prediction of growth or risk of rupture. Then, we present an extensive state
of the art review of the methods for vascular segmentation that points out the
lack of effective algorithms for the segmentation of cerebral aneurysms in clinical
data. Then, we proceed to present the method for Geodesic Active Regions
segmentation adapted to our application. Finally, the method is evaluated using
ground truth manual segmentations and compared with alternative region-based
implicit deformable models.
1.2.2
Part II: Registration of medical images
• Chapter 5 introduces how Computational Anatomy studies the anatomical differences between different populations from the spatial transformations existing
between the anatomy of individuals and a template selected as anatomical reference and motivates the use of certain groups of diffeomorphisms for modeling
anatomical variability in this framework. The fundamental aspects of the differential and algebraic structures of infinite dimensional Riemannian manifolds of
diffeomorphisms are thoroughly studied. In addition, the reference variational
problems for registration arisen in the Large Deformation Diffeomorphic Metric
Mapping (LDDMM) paradigm are thoroughly described due to its close relationship with the methods developed in the rest of the Thesis. Finally, special
attention is given to the methods for efficient second-order diffeomorphic registration that have been proposed in the literature simultaneously to ours.
• In Chapter 6 we present our method for diffeomorphic registration. We begin
introducing the need for simple and efficient algorithms for the computation
of diffeomorphisms in extensive clinical research studies. Then, we detail how
to introduce the stationary parameterization of diffeomorphisms in the classical LDDMM variational problem and the computations of the associated EulerLagrange equations for gradient descent optimization. Finally, the performance
and efficiency of non-stationary and stationary LDDMM methods are compared
and evaluated in real and simulated databases of cerebral Magnetic Resonance
Images (MRI) commonly used in clinical research for the study of the cerebral
anatomical variability.
• In Chapter 7 we improve our stationary-LDDMM in efficiency and robustness
proposing a method for Gauss-Newton optimization. We begin pointing out the
7
lack of attention paid to the optimization strategy in the LDDMM framework.
Then, we detail our method for second-order optimization. In addition, we provide a detailed theoretical and practical comparison between our second-order
stationary-LDDMM method and the alternative techniques for efficient registration that have been proposed in the literature simultaneously to our work.
• In Chapter 8 our efficient stationary-LDDMM method is included into a framework for the generation of statistical models of anatomical variability with application to the construction of statistical atlases. We begin revisiting the theory
related to statistical calculus on finite dimensional manifolds and studying the
feasibility of its extension to diffeomorphisms. Then, we detail our algorithm
for the computation of the statistical atlas associated to a population of MRI
anatomical images.
In Chapter 9, we summarize and discuss the most remarkable contributions of this
Thesis and the conclusions that can be inferred from the presented results. Finally, we
point out the future work that will be tackled from our developed methods.
Part I
Chapter 2
Implicit Deformable Models
Abstract
Implicit deformable models is probably one of the most important paradigms
in image segmentation. These models are based on the theory of curve and
surface evolution. The segmentation is defined as the interface realizing the
minimum of a variational problem where the energy functional depends on
some image knowledge. The Euler-Lagrange equation associated to this energy constitutes a partial differential equation that drives the evolution of the
interface towards the minimum. The numerical implementation of the evolution is performed in an Eulerian framework provided by the level set method.
Thus, the model is parameterization independent and able to capture topological changes during the interface evolution in a natural way. This last property
makes implicit deformable models a powerful technique in the segmentation
of complex anatomical structures from medical images. In particular, it has
been widely applied in the segmentation of arterial structures. In this Chapter, we focus on studying the theory of implicit deformable models. The level
set method and the most remarkable details regarding level set calculus in its
physical context are thoroughly described. In addition, the variational problems for edge- and region-based segmentation in closest relationship with the
methods developed in this Thesis are reviewed. Finally, the handicaps of geometric implicit deformable models with narrow structures segmentation are
stated and some of the most remarkable specific methods for dealing with such
structures are studied.
2.1. Introduction
12
Contents
2.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
The level set method . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3
2.4
2.5
2.1
2.2.1
Lagrangian vs Eulerian representations for curve and surface evolution
16
2.2.2
Variational formulation of the level set method . . . . . . . . . . . . .
19
2.2.3
Numerical aspects of the level set method . . . . . . . . . . . . . . . .
20
Implicit flows in fluid dynamics
. . . . . . . . . . . . . . . . . . . .
2.3.1
Flow driven by an external velocity field
2.3.2
Constant flow in normal direction
2.3.3
Flows driven by curvature
2.3.4
Combination of simple flows
24
. . . . . . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . .
26
. . . . . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . . . . . . . . . .
29
Implicit flows in image segmentation
. . . . . . . . . . . . . . . . .
32
2.4.1
The snake model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.4.2
Edge-based models
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.4.3
Region-based models
. . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4.4
Vessel-specific segmentation models
. . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
44
Introduction
Since the development of medical imaging acquisition techniques, the analysis of medical imagery in clinical practice has become an important task in the detection, diagnosis and treatment of multiple diseases. In most clinical applications, segmentation
is probably one of the most important steps in this analysis, as it allows delimitating
the geometry and quantifying the dimensions of the structures of interest. The large
number of images that need to be analyzed in clinical routine and the large subjectivity
introduced by manual segmentation have motivated the development of as automatic
as possible computerized segmentation algorithms.
Medical image segmentation has become one of the most important and challenging issues in Computer Vision. As can be appreciated from the examples shown in
Figure 2.1, medical images acquired in clinical practice are often noisy and suffer from
partial volume effects. Artifacts may be present in the image due to suboptimal but
clinically feasible acquisition parameters or the presence of implants within the location
of interest. In the majority of cases, the voxel size of the images in out-of-plane resolution is much bigger than at in-plane resolution difficulting an accurate delineation
in this direction. The structures of interest usually lack of strong edges and present
13
Chapter 2. Implicit Deformable Models
a
b
d
c
e
Figure 2.1: Challenging examples of image segmentation. a) Noise and artifacts in a 3D-RA image
of a cerebral aneurysm acquired after coil implant. b) Intensities of Turkish saddle in CTA similar
to intensities in carotid arteries. c) Lack of strong edges between blood pool and ventricle walls in
cardiac CT. Intensity inhomogeneity and artifacts inside the auricula due to blood flow concentration.
d) Lack of strong edges between prostate and bladder due to huge partial volume effect. e) Out-ofplane resolution in thoracic CT.
intensity inhomogeneity inside. In certain locations, there exist adjacent objects with
intensity patterns similar to the structure of interest. All these issues make even manual segmentation a difficult task for experienced radiologists providing some of the
algorithmic challenges that automatic segmentation methods should overcome.
A wide range of algorithms have been proposed in order to tackle with the problems
posed by medical image segmentation [281]. The most remarkable methods can be
classified into:
Thresholding, classifiers and clustering
These methods use pixel-wise information in order to decide if a pixel belongs or not
to the structure of interest.
• Thresholding allows creating a partition of the image domain from pixels belonging to user-supplied intensity intervals.
• Clustering methods are unsupervised pattern recognition techniques. Segmentation is obtained from a partition on a feature space based on the proximity
2.1. Introduction
14
of similar image features according to some defined distance. The most commonly used clustering algorithms are K-means, fuzzy C-means and ExpectationMaximization [65, 101].
• Classifiers are supervised pattern recognition techniques. Whereas in the case of
unsupervised methods the algorithms train themselves using the available data
from the whole image, in the case of supervised classifiers, the training data
is user-supplied from manual segmentation, thresholding or unsupervised methods. The most popular classifiers are Nearest Neighbors, Markov Random Fields,
Support Vector Machines and Neural Networks [65, 101].
Region growing and watershed
These methods consider the information from the neighborhood of a pixel in order to
decide if that pixel belongs to the structure of interest.
• Region growing allows extracting a region of the image that is connected to a set
of user-supplied seed points based on some predefined criteria.
• Watershed segmentation considers the image as a topographic surface and simulates the flooding from sources of water located within the valleys while preventing
the merge of water comming from different sources [254]. The watershed lines
provide a segmentation of the image domain enclosing the homogeneous grey
level regions of the image.
Model fitting
These methods make use of parametric models to characterize the shape of the structure
of interest. Model parameters are estimated through a minimization process where
shape instances are fitted to an example of the modelled structure in a new image.
• Typical examples include the representation of vessels as cylindrical shapes with
fixed or varying radius or the representation of the bladder and prostate organs
using super-ellipsoids (see [131, 272] and [210] and references therein).
• The most popular model fitting algorithms are Active Shape and Active Appearance Models (ASMs and AAMs), where statistical models of shape and appearance of the objects are built from linear combinations of the average and the
most energetic modes of variation associated to a representative population of
samples [47, 48].
• Segmentation based on deformable templates (also known as atlas based segmentation) can be also considered a model fitting method. In this case, the model is
provided by a ground truth segmentation of the structure of interest performed
on a representative image of the anatomy. Given a new image, this reference
15
is deformed into the target image using registration techniques. The resulting
transformation is used to warp the labels from the ground truth segmentation
to the coordinate system of the target image. These warped labels constitute
the segmentation of the structure of interest in the target image (see [205] and
references therein).
Deformable models
These methods are closely related to the theory of curve and surface evolution extensively developed in Fluid Dynamics during the 80’s. The model of the structure of
interest can be represented as an evolving front where the external forces depend on
the information extracted from the image.
• Curve evolution was introduced for medical image segmentation in the seminal
work of Kass et al. [126]. The model is defined as a curve called snake that evolves
towards the minimization of an energy functional depending on the overlap of
the snake with the image edges and a regularization term that constraints contour smoothness. The most important limitations of this approach are that the
solution is parameterization dependent and the model is not able to deal with
changes in the topology of the snake during evolution.
• The level set method for implicit curve and surface evolution was introduced from
Fluid Dynamics to image segmentation in order to overcome with the limitations
of the snake model [185]. This method deals with the implicit representation of
curve and surface evolution. Thus, the solution is parameterization independent
and topological changes on the evolving interface can be automatically handled.
This property has made the level set method one of the most popular techniques
for the segmentation of complex anatomical structures.
Graph cuts
Graph cuts method allows efficiently searching for a partition of a weighted graph with
two terminals using combinatorial optimization [26]. Recently, it has been pointed out
the close relationship between graph cuts and deformable models [135]. The weights
of the graph can be related to the energy functional associated to the deformable
model evolution and the cut of the graph can be interpreted as a curve or surface.
Thus, the minimum graph cut can be identified with the interface that minimizes the
energy functional. Whereas the level set method for curve and surface evolution usually
allows finding a local minimum, graph cuts does obtain a global one. Therefore, graph
cuts is being increasingly considered as alternative or complementary optimization
technique for solving the variational problems that have been traditionally associated
to deformable models.
From the segmentation algorithms mentioned above, implicit deformable models are
probably the most popular methods for arterial structures segmentation. The ability
2.2. The level set method
16
to capture topological changes in a natural way makes this a really powerful technique
for the segmentation of complex arterial structures from angiographic data. However,
there exist still some limitations that need to be overcome. In this Thesis we will
tackle with the development of region-based deformable models specifically designed
for working with arterial structures in angiographic images.
This Chapter is focused on the study of segmentation methods based on implicit
deformable models. In Section 2.2 the theory of implicit curve and surface evolution
is described pointing out the most remarkable theoretical and practical points with
regard to the calculus of the level set equation. In Section 2.3, we study the basic
physical flows involved in the implicit deformable models for image segmentation. In
Section 2.4, the most popular edge- and region-based implicit deformable models in
closest relationship with the methods developed in this Thesis are overviewed with
special attention to those models that have been specifically devised for dealing with
thin structure segmentation. Finally, Section 2.5 summarizes the most remarkable
aspects studied in the Chapter.
2.2
The level set method
A significant number of problems in Physics and Computer Vision can be studied from
the evolution of curves and surfaces. The motion of these interfaces is usually driven
by external forces associated to the physical model of evolution and internal forces
depending on the geometry of the interface. These interfaces can break up or merge,
arise or dissappear during the course of time evolution. The level set method has
emerged as a powerful paradigm for the computation of front evolution while handling
topological changes in a natural way. Since its first introduction in a physical context,
an extensive level set calculus theory has been developed in order to solve a large
variety of problems in Fluid Dynamics, Material Sciences and Computer Vision.
In this section we describe the key components of the level set theory for implicit
curve and surface evolution. We focus on the method, the variational formulations
from which the level set equation arises and the most remarkable numerical details
involved in level set calculus theory. We have based on the books by J.A. Sethian, G.
Sapiro, S. Osher, R. Kimmel and collaborators [217, 209, 184, 130] and the compact
reviews provided in [73, 240].
2.2.1
Lagrangian vs Eulerian representations for curve and
surface evolution
The Lagrangian representation of curve evolution is given by a family of parameterized
curves C(p; t) ⊆ Ω × I ⊆ R2 × R+ . The evolution is driven by the partial differential
equation (PDE)
17
−
→
−
→
∂C
(p) = F (p) · N (p)
∂t
(2.1)
−
→
−
→
with initial condition C0 = C(p; 0), where F is a velocity field defined on Ω and N (p)
denotes the normal vector to the curve at a given point (x(p), y(p)). Analogously, the
Lagrangian representation of surface evolution is given by a family of parameterized
surfaces S(u, v; t) ⊆ Ω × I ⊆ R3 × R+ . The evolution is driven by the PDE
→
−
−
→
∂S
(u, v) = F (u, v) · N (u, v)
∂t
(2.2)
−
→
with initial condition S0 = S(u, v; 0). In this case, F is a velocity field defined on Ω
−
→
and N (u, v) denotes the normal to the surface at a given point (x(u, v), y(u, v), z(u, v)).
By abuse of notation the same variable names are used for curve and surface evolution.
The numerical implementation of curve and surface evolution may be performed
discretizing the evolving interface into a finite number of points and tracking these
points with a motion driven by the corresponding PDE. The resulting algorithm is not
so challenging if the velocity field provides a smooth motion of the points on the interface. Unfortunately, even the most simple velocity fields can cause abrupt motions. In
those cases, the evolving interface can deteriorate quickly if the discretization is not periodically smoothed in problematic locations. Moreover, the algorithm is often required
to handle with topological changes on the evolving interface. Thus, the algorithm has
to be able to detect possible merging or splittings on the interface and continue with
the evolution in a consistent way after topological changes occur. This has been accomplished by external procedures using information from the physical phenomenon
related to the PDE for the selection of the correct solution [215]. However, the algorithm is prone to develop solutions with high curvature and sharp corners resulting not
much physically realistic.
The level set method uses an alternative representation of curve and surface evolution. This method was introduced in Fluid Dynamics at the end of the 70’s [163, 58, 59].
Later on, Osher and Sethian provided a solid mathematical foundation with application
to gas dynamics and flame propagation [185] that was introduced for image segmentation in [33, 159]. This framework is highly appropriate for curve and surface evolution.
Hence, it has become the standard way of representing interfaces evolution in Fluid
Dynamics and Computer Vision, among other disciplines [184].
Given a curve defined on the plane C ⊆ Ω ⊆ R2 , the level set method deals with its
implicit representation that is defined as the zero level set of a function φ : Ω ⊆ R2 → R
C = {(x, y) ∈ Ω : φ(x, y) = 0}
(2.3)
As example, the implicit representation of a star-shaped curve is shown in Figure 2.2.
18
Figure 2.2: Implicit representation of a star-shaped curve. Left, represented curve. Center, image
of the level set isocontours. Right, surface plot of the level set function.
Analogously, the implicit representation of a surface defined in the space S ⊆ Ω ⊆ R3
is defined as the zero level set of a function φ : Ω ⊆ R3 → R
S = {(x, y, z) ∈ Ω : φ(x, y, z) = 0}
(2.4)
The level set method allows expressing the evolution of the interface in terms of the
evolution of the corresponding implicit function. For example, if the motion of the
interface is given by Equation 2.1 in the case of curves, the motion of the corresponding
implicit representation φ is driven by the PDE
→
∂φ −
+ F · ∇φ = 0
∂t
(2.5)
with initial condition φ0 . Identical expression except for the dimensionality of the
−
→
definition domain for φ and F arises in the case of surfaces. Viceversa, the evolution
of the original interface can be captured from the evolution of the implicit function by
the computation of the associated zero level set. Equation 2.5 is traditionally known
as the level set equation and constitutes the Eulerian formulation for curve and surface
evolution.
The numerical implementation for curve and surface evolution is performed solving the level set equation on a regular sample of the computation domain Ω. As the
implicit function evolves, its level sets can merge and split. Therefore, this formulation allows automatically handling with topological changes in the evolving interface.
Moreover, the intrinsic geometric properties of the evolving interface can be easily determined from the implicit representation. The temporal and spatial derivatives can
be approximated by finite difference schemes introducing stability in the numerical
computations. Well posedness and numerical stability is provided by the theory of
viscosity solutions [51, 218, 209, 13]. Furthermore, this formulation can be straightforwardly extended to any dimension. Without loss of generality, in the following we will
focus on surfaces evolving in the space.
19
2.2.2
Variational formulation of the level set method
Curve and surface evolution can be embedded into a variational formulation. The
evolving interface at a given time provides a partition of the computation domain into
two regions Ω = Ωin ∪ Ωout . The motion of the interface results from the contribution
−
→
−
→
∇φ
∇φ
of the regional forces 1 F in = Fin · |∇φ|
and F out = Fout · |∇φ|
associated to Ωin and
−
→
−
→
Ωout , respectively, and the contour forces F S = FS · N associated to the interface
S. The interface motion arises from the Euler-Lagrange equation associated to the
minimization of the energy functional
Fin dw + α
Fout dw + γ FS dσ
(2.6)
E(S) = α
Ωin
Ωout
S
that corresponds to
−
→ −
→
∂S
= (α(Fin − Fout ) + γ(FS Kmean + ∇FS · N )) · N
∂t
with associated level set equation
∂φ
∇φ
+ α(Fin − Fout ) + γ FS Kmean + ∇FS ·
· |∇φ| = 0
∂t
|∇φ|
(2.7)
(2.8)
In the same way, the evolution of the implicit representation φ can be embedded into
a variational formulation [72]. The regions Ωin and Ωout can be characterized from the
Heaviside distribution associated to the implicit representation
0 if φ(x, y, z) ≤ 0
H(φ(x, y, z)) =
(2.9)
1 if φ(x, y, z) > 0
The volume of these regions can be computed from the integrals
vol(Ωin ) =
H(φ)dw
and
(1 − H(φ))dw
vol(Ωout ) =
Ω
(2.10)
Ω
while the area of the evolving surface is provided by
δ(φ)|∇φ|dw
area(S) =
(2.11)
Ω
where δ(φ) is Dirac’s delta function.
1
By abuse of notation, the absolute value symbol is used in this Chapter to denote the Euclidean
norm of vectors. We thus follow classical notation used in works related to level set methods.
20
The level set equation arises from the Euler-Lagrange equation associated to the minimization of the energy functional
E(φ) = α
Fout (1 − H (φ))dw + γ
Fin H (φ)dw + α
Ω
Ω
FS δ (φ)|∇φ|dw (2.12)
Ω
that corresponds to
∇φ
∂φ
+ δ (φ) α(Fin − Fout ) + γ FS Kmean + ∇FS ·
· |∇φ| = 0
∂t
|∇φ|
(2.13)
where H denotes a differentiable -approximation of the Heaviside distribution and δ
corresponds to its first derivative. It should be noted that both variational formulations
provide level set equations that differ in the δ (φ) factor. Therefore, strictly speaking,
both formulations will not produce equivalent results in locations far from the evolving
interface. However, in the level set method it is enough to capture the motion of the
closest isophotes to the interface and, therefore, both variational formulations can be
considered to be equivalent.
2.2.3
Numerical aspects of the level set method
The general level set algorithm is an iterative method where each iteration can be
summarized into three fundamental steps
1. Initialization-reinitialization of the implicit function φ
−
→
2. Numerical approximation of F · ∇φ
3. Update of the implicit function φ
In this section we discuss the most remarkable details of the numerical aspects involved
in each of these steps.
Initialization
The implicit function φ is usually initialized to the signed distance transform of the
initial evolving interface S0
⎧
if (x, y, z) ∈ S0
⎨ 0
d((x, y, z), S0 ) if (x, y, z) ∈ Ωin
φ0 (x, y, z) =
(2.14)
⎩
−d((x, y, z), S0 ) if (x, y, z) ∈ Ωout
where Ωin is the region inside S0 and Ωout is the region outside S0 .
Although other functions (such as Heaviside type functions) could be used for initialization, it has been shown that the selection of a distance transform avoids the
development of too step or flat gradients as well as discontinuities in the smoothness
21
of the interface [178]. Moreover, the signed distance transform satisfies the Eikonal
equation
|∇φ(x, y, z)| = F (x, y, z)
(2.15)
with F (x, y, z) = 1. Often, the level set equation can be simplified using |∇φ(x, y, z)| =
1 introducing efficiency and stability into the numerical computations.
The most popular method for the computation of the distance transform is the Fast
Marching method. It was originally proposed by Tsitsiklis et al. [241] and rediscovered
for the level set community by Sethian et al. [216]. This method constitutes a computationally efficient algorithm for the solution of the Eikonal equation and allows for
the computation of weighted distances on manifolds with metric distance defined from
F . In the case F = 1 the fast marching method provides a highly efficient method for
computing the Euclidean distance.
Discretization
The numerical solution of the level set equation is computed on a regular sample of the
computation domain Ω. Points on the numerical domain are denoted by (xh , yh , zh ) and
Δx, Δy, Δz represent the space sampling. The size of time sampling is represented
by Δt. For the numerical approximation of temporal and spatial derivatives finite
difference schemes are used in this computation domain. Discretization in time is
usually approached using a forward Euler scheme
n
φn+1
−
→
ijk − φijk
= − F nijk · ∇φnijk
Δt
(2.16)
leading to the update of the implicit function
−
→n
n
n
φn+1
ijk = φijk − Δt F ijk · ∇φijk
(2.17)
−
→
where F nijk corresponds to the velocity discretization in the computation domain at
time tn and ∇φnijk approximates the spatial gradient at the same time. Alternative
higher order Runge-Kutta or semi-implicit schemes have been proposed in (specially
physical) applications where a more accurate temporal scheme is critical in the numerical computations [92, 70, 273, 229].
The spatial derivatives are computed using finite difference approximations. The
resulting numerical schemes are usually first-order combinations of the two point stencils
• Forward differences: Dx+ φ =
• Backward differences: Dx− φ =
φ(xh+1 ,yh ,zh )−φ(xh ,yh ,zh )
Δx
φ(xh ,yh ,zh )−φ(xh−1 ,yh ,zh )
Δx
• Central differences:
22
Dx0 φ =
φ(xh+1 ,yh ,zh )−φ(xh−1 ,yh ,zh )
2Δx
that can be straightforwardly extended to y and z directions. For the selection of
the most convenient numerical scheme, the type of evolution and the corresponding
PDE class (usually hyperbolic or parabolic) need to be identified. In the case of
hyperbolic equations, different problem-specific finite difference approximations such as
Lax-Friedrichs, Roe-Fix or Godunov stencils have been proposed in the literature [145,
184]. All these numerical schemes are based on the method of characteristics. The
case of parabolic equations is more simple, as finite difference approximations based
on centered stencils can be commonly used.
These schemes could be improved using higher-order approximations for Dx+ φ, Dx− φ
and Dx0 φ. These approximations use adaptive stencils based on the local smoothness
of the numerical solution and the non-oscillatory property near discontinuities. Essentially non-oscillatory (ENO) schemes use six point stencils being able to achieve
third-order accuracy [99, 185, 219, 220]. Weighted essentially non-oscillatory (WENO)
schemes use convex combinations of ENO approximations achieving up to fifth-order
accuracy with the same stencil length than ENO [186, 152, 122, 121].
It should be noted that, in contrast to physical applications where a high-resolution
computation domain is usually available, the image size and spacing is used to build
the computation domain in clinical applications. Hence, high-order schemes may degrade the accuracy of the numerical scheme in these low-resolution domains. This
phenomenon could be specially harmful in applications as thin or folded structures
segmentation, like cerebral vessels or brain cortex, where there are not enough voxels
inside or among the objects to properly compute a high-order approximation of the
implicit function derivatives.
Stability. Selection of the time step.
Stability in the numerical scheme for the level set equation is imposed from the CFL
condition stated by Courant, Friedreichs and Lewy in 1928 [50]. This condition is based
on the observation that a necessary stability condition for a numerical scheme is that
the domain of dependence of each point in the numerical domain should include the
domain of dependence of the PDE. As a consequence, the time step Δt should be less
−
→
than a bound that depends on the grid size and the maximum value reached by | F | on
the computational domain. Otherwise, the numerical solution would become unstable
producing incorrect results.
Reinitialization
−
→
In the general case of a non-uniform velocity field F , the level set equation leads to
the motion of different patches of the same level set with different velocity values.
In consequence, the implicit function does not remain a distance function during the
course of evolution. Although the zero level set remains being correct, it is necessary
a reinitialization procedure in order to preserve the distance function, at least in a
23
neighborhood of the zero level set. Otherwise, undesirable numerical errors may arise
during the evolution affecting to the final accuracy of the algorithm.
The most simple and fast method for reinitialization is to compute the solution
of the Eikonal equation using the fast marching method. The zero level set of the
implicit function is obtained by thresholding and used as initial condition of the Eikonal
equation. However, the implicit function is not exactly equal to zero at the grid points
neighboring the interface. Hence, this method does not preserve the subvoxel accuracy
that characterizes deformable models. As alternative, it has been proposed to initialize
the fast marching algorithm with the values of the implicit function of the grid points
neighboring the interface [140]. This method has shown to maintain subvoxel accuracy
with similar efficiency to the fast marching algorithm. This algorithm is currently
used by the Insight Toolkit community (http://www.itk.org) for reinitialization in
segmentation with the level set method.
In fluid dynamics, a commonly used method for the reinitialization of the implicit
function is the so called PDE approach proposed by Sussman et al. in [228]. The
implicit function is replaced by the solution of the PDE
∂ψ
+ sign(φ) · (|∇ψ| − 1) = 0
∂t
(2.18)
with initial condition ψ0 = φ, where the sign function is approximated by
φ
sign(φ) = 2
φ + Δx2
(2.19)
At the steady state, the expression sign(φ) · |∇ψ| is equal to 1, so the solution ψ is
a signed distance transform with the same zero level set as φ. Further improvements
including the definition of different sign functions, the inclusion of volume-preserving
constraints and the use of ENO or WENO schemes for the numerical approximation
of the derivatives in ∇ψ have been proposed in [228, 227, 226, 192].
Moreover, some efforts in the development of level set algorithms that avoid the
reinitialization of the implicit function have been done. For example, Gomes et al.
proposed a modification of the level set equation extending the speed values on the
zero level set through isophotes [93]. This way, the velocity remains uniform in the
computation domain and the distance transform is properly preserved. More recently,
Li et al. have proposed to include a term in the level set PDE that penalizes the
deviation of the level set function from a signed distance [146].
Narrow band
In the level set method, it is enough to capture the motion of the zero level set and
the closest isophotes to the interface. The narrow band level set implementation introduced by Adalsteinsson et al., restricts the numerical computations to a thin band
of voxels neighboring the interface [1]. Thus, only the values of the level set function
2.3. Implicit flows in fluid dynamics
24
at the grid points inside the narrow band are updated in order to solve the level set
equation whereas the values of the boundary of the band are frozen. The reinitialization procedure of the level set function is also performed inside this band. When
the zero level set moves near this boundary the band is recomputed according to the
new position of the interface. The time complexity of the algorithm is reduced up to a
factor that depends on the band width and the number of band reinitializations. The
efficiency of the narrow band algorithm was further improved with the sparse field level
set implementation where the values of the level set function are updated just on the
interface and its two neighboring level sets [261].
2.3
Implicit flows in fluid dynamics
In this section, we describe the families of simple flows in Fluid Dynamics arising as
part of the more complicated flows involved in image segmentation. Most of the details
can be found in S. Osher et al.’s book [184]. The examples have been simulated with
the help of I. Mitchell’s level set toolbox [174].
2.3.1
Flow driven by an external velocity field
This flow results from the evolution of an interface leaded by an external vector field.
It was introduced for representing the evolution of a flame front in [163]. The interface
moving with this flow results transported with the external velocity field, hence this
flow is also known as advection.
The evolution equation is expressed as
→
−
−
→
∂S
(u, v) = F (u, v) · N (u, v)
∂t
(2.20)
→
−
where F corresponds to the external velocity field. The associated level set equation
is given by
−
→
∂φ
(x, y, z) + F (x, y, z) · ∇φ(x, y, z) = 0
∂t
(2.21)
and the numerical discretization corresponds to
→
−n
n
n
φn+1
ijk = φijk − Δt F ijk · ∇φijk .
(2.22)
−
→
The expression F nijk · ∇φnijk represents the spatial discretization of the scalar product
−
→
F ·∇φ, that is computed extending the discretization of the corresponding 1D-equation
to 3D. Thus, let be
25
Figure 2.3: Evolution of a sphere driven by a rotation external velocity field defined on the space.
∂φ
∂φ
(x) + f (x) ·
(x) = 0
∂t
∂x
(2.23)
the 1D advection equation. This equation is a particular case of a one-dimensional
conservation law. It has been shown that Godunov schemes provide the exact numerical
solutions for this kind of PDEs [184]. As conservation laws are also special cases of
hyperbolic equations, the same numerical approximation can be obtained from the
application of the method of characteristics to hyperbolic PDEs. Thus, if f (x) > 0
the Godunov scheme indicates that backward finite differences should be used for the
approximation of ∂φ/∂x at point x. Otherwise, if f (x) < 0 the Godunov scheme
indicates that forward finite differences should be used for the approximation of the
spatial derivative. Hence, the numerical solution of the 1D advection corresponds to
φi − φi−1
, if fi > 0
Δx
φi+1 − φi
, if fi < 0
= φni − Δt fi ·
Δx
= φni − Δt fi ·
φn+1
i
(2.24)
φn+1
i
(2.25)
The resulting approximation is popularly known as upwind scheme as the information
used to approximate the spatial derivatives is propagated upwind the direction of the
corresponding PDE characteristics. The extension to 3D arises from the discretization
26
of fx · ∂φ
, fy · ∂φ
, and fz · ∂φ
from the corresponding 1D equations. The stability of the
∂x
∂y
∂z
numerical scheme is enforced with the CFL condition
→
−
→
−
→
−
|F | |F | |F |
+
+
Δt · max
<1
(2.26)
Ω
Δx
Δy
Δz
An example of this flow is given in Figure 2.3 where a sphere is transported under a
rotation vector field defined on the space.
2.3.2
Constant flow in normal direction
The equation for constant motion in normal direction can be considered as a particular
case of the motion driven by an external velocity field where the velocity is constant.
This flow derives from a velocity field associated to a balloon force. The evolution
equation is expressed as
→
−
∂S
(u, v) = F (u, v) · N (u, v)
∂t
(2.27)
where F (u, v) is a constant scalar function. If F > 0 the interface moves in the
normal direction whereas if F < 0 the motion is opposite to the normal direction. The
associated level set equation is given by
∂φ
(x, y, z) + F (x, y, z) · |∇φ(x, y, z)| = 0
∂t
(2.28)
and the numerical discretization corresponds to
n
n
φn+1
ijk = φijk − Δt Fijk · |∇φijk |
(2.29)
In this case, Godunov scheme provides the following approximation for the associated
1D partial differential equation
∂φ
∂x
∂φ
∂x
2
≈ max(max(Dx− φ, 0)2 , min(Dx+ φ, 0)2 ), if f > 0
(2.30)
≈ max(min(Dx− φ, 0)2 , max(Dx+ φ, 0)2 ), if f < 0.
(2.31)
2
It should be noted that this approximation differs from the one provided by upwind
schemes if the equation is considered as an advection flow. However, Godunov scheme
is preferred as it has been shown that the upwind scheme presents numerical problems
27
Figure 2.4: Evolution of a dumbell shaped surface in a negative constant velocity field.
in cases where the signs of Dx− φ and Dx+ φ are different. The stability of the numerical
scheme is enforced using the CFL condition
Δt · |F |
1
1
1
+
+
Δx Δy Δz
<1
(2.32)
Figure 2.4 shows an example of a dumbbell moving inwards with constant velocity
F = −1.
2.3.3
Flows driven by curvature
This family of flows is characterized by a motion in normal direction that depends on
the curvatures of the interface. These flows are usually involved in diffusive processes as
they are in close relationship to the heat equation. In contrast to advective and balloon
flows, the velocity depends on geometric properties of the evolving interface. Hence,
these flows are also known as geometric flows. The evolution equation is expressed as
→
−
∂S
(u, v) = K(u, v) · N (u, v)
∂t
(2.33)
If > 0 the interface moves in the direction of the surface concavity. If < 0 numerical
instabilities in the computation of the flow lead to an ill-posed problem. The associated
level set equation is given by
∂φ
(x, y, z) + K(x, y, z) · |∇φ(x, y, z)| = 0
∂t
28
(2.34)
The numerical discretization of the level set equation corresponds to
n
n
n
φn+1
ijk = φijk − Δt Kijk · |∇φijk |
(2.35)
This equation is a parabolic PDE. Therefore, characteristics include information from
both forward and backward directions. Derivatives cannot be discretized with upwind
or Godunov schemes but with central finite difference schemes [40]. The stability of
the numerical scheme is enforced using the CFL condition
2
2
2
+
+
<1
(2.36)
Δt ·
(Δx)2 (Δy)2 (Δz)2
As pointed out in Section 2.2, the intrinsic geometric properties of the evolving interface
can be determined from its implicit representation φ. In particular, the curvatures of
the evolving interface can be expressed in terms of the spatial derivatives of φ. Thus,
mean and Gaussian curvatures can be computed from the formulas [223]
Kmean =
φxx (φ2y + φ2z ) + φyy (φ2x + φ2z ) + φzz (φ2x + φ2y )
2(φ2x + φ2y + φ2z )3/2
φxy φx φy + φxz φx φz + φyz φy φz
−
(φ2x + φ2y + φ2z )3/2
(2.37)
φ2x (φyy φzz − φ2yz ) + φ2y (φxx φzz − φ2xz ) + φ2z (φxx φyy − φ2xy )
KGauss =
(2.38)
(φ2x + φ2y + φ2z )2
φx φy (φxz φyz − φxy φzz ) + φy φz (φxy φxz − φyz φxx ) + φx φz (φxy φyz − φxz φyy )
+2
(φ2x + φ2y + φ2z )2
and the minimum curvature is provided by the positive solution of the second degree
equation
2
− 2Kmean · Kmin + KGauss = 0
Kmin
(2.39)
In the case of curves, curvature flow is popularly known as the curve shortening flow
since it is solution of the minimization of
1
∂C (2.40)
length(C) =
∂t (p) dp
0
29
This problem was studied by Grayson et al. who showed that smooth closed curves in
the plane shrink to a point under curvature motion [95]. Figure 2.5 shows an example
of this evolution for a star-shaped curve.
The extension of this flow to 3D has been studied under the theory of minimal surfaces.
In this case, mean curvature flow is associated to the minimization of the surface area.
Although the theorems of existence, stability, and uniqueness from 2D still hold in 3D,
Grayson’s theorem of planar curves can be extended only for some specific surfaces. For
example, Figure 2.6 shows a dumbbell-shaped surface where mean curvature motion
leads the interface to shrink towards two different points instead of one, thus leading
to a counter-example for Grayson’s theorem. The most pathological case can be found
in the geometric evolution of thin structures under mean curvature, where the motion
leads the interface to shrink towards its centerline, as shown in Figure 2.7 for a thin
torus.
Mean curvature flow in 3D corresponds to the geometric motion of hyperfurfaces in the
space (co-dimension 1 flow). This motion is not suitable for describing the evolution
of curves in the space under curvature. The problem of evolving curves in higher
co-dimension was approached by Ambrosio et al. within the level set framework [3].
Defining φ as an implicit representation of the curve in the space, the evolution under
curvature corresponds to the level set equation
∂φ
= λ(∇φ, ∇2 φ)
∂t
(2.41)
where λ(∇φ, ∇2 φ) is the smallest non-zero eigenvalue of the matrix P∇φ · ∇2 φ · P∇φ
and Pv represents the projection matrix associated to v. This flow is equivalent to
considering a revolution surface of thin radius and the given curve as axis and evolving
this surface under minimum curvature. In this case, minimum curvature flow extends
Grayson’s theorem for the evolution of elongated structures, as shown in Figure 2.8 for
the thin torus experiment. This is the reason why mean curvature is usually replaced
by minimum curvature in geometric flows for thin structure segmentation.
2.3.4
Combination of simple flows
Balloon-diffusion equation
The balloon-diffusion equation is associated to the combination of a constant motion
in normal direction and a mean curvature flow. This equation represents physical
phenomena of heat transfer such as the motion of a flame in the normal direction
plus extra heating and cooling effects due to the curvature of the flame front. The
corresponding implicit equation is expressed as
∂φ
(x, y, z) + F (x, y, z) · |∇φ(x, y, z)| + Kmean (x, y, z) · |∇φ(x, y, z)| = 0
∂t
(2.42)
30
Figure 2.5: Evolution of a star shaped curve under curvature. The tips of the star move inward
while the gaps between the tips move in outward direction. The shape is gradually transformed to a
circle that shrinks towards a point, as stated in Grayson’s theorem.
Figure 2.6: Evolution of a dumbbell-shaped surface under mean curvature. The dumbbell center
narrows until the shape splits in two different evolving interfaces.
31
Figure 2.7: Evolution of a torus surface under mean curvature. The torus evolves towards its
centerline.
Figure 2.8: Evolution of a torus surface under minimum curvature. In this case, the torus shrinks
to a point, as expected from Grayson’s theorem.
2.4. Implicit flows in image segmentation
32
The numerical approximation of the spatial derivatives is performed using a Godunov
scheme for the balloon term and central differences for the diffusive term. The stability
of the numerical scheme is guaranteed by the CFL condition
|F | |F | |F |
2
2
2
+
+
+
Δt ·
+
+
<1
(2.43)
Δx Δy Δz (Δx)2 (Δy)2 (Δz)2
Advection-diffusion equation
A more general flow that accounts for the resistance of the unburnt material is provided by the advection-diffusion, also known as convection equation. This equation
is associated to the combination of a motion from a externally generated vector field
and a mean curvature flow. This flow represents physical phenomena of heat and mass
transfer as, for example, the motion of a flame in the normal direction burning through
a material. The level set equation is expressed as
−
→
∂φ
(x, y, z) + F (x, y, z) · ∇φ(x, y, z) + Kmean (x, y, z) · |∇φ(x, y, z)| = 0
∂t
(2.44)
The numerical approximation of the spatial derivatives is performed using an upwind
scheme for the advective term and central differences for the diffusive term. The
stability of the numerical scheme is guaranteed by the CFL condition
−
−
→
→
−
→
2
2
2
|F | |F | |F |
+
+
+
+
+
<1
(2.45)
Δt · max
Ω
Δx
Δy
Δz
(Δx)2 (Δy)2 (Δz)2
2.4
Implicit flows in image segmentation
In this section, we focus on the families of implicit flows that have been proposed in
the literature for image segmentation. We first study the snake model that introduced
the variational formulation for curve evolution on the theory of deformable models.
Then, the implicit methods that include edge- and region-based information in close
relationship with the methods developed through Chapters 3 and 4 are studied. We
end with the flows that have been specifically designed for thin structure segmentation.
2.4.1
The snake model
The snake model constitutes the classical formulation for segmentation methods based
on explicit deformable models. Although this model is actually not embedded into the
33
implicit formulation of curve evolution, it is studied here as the snake model can be
considered the precursor of implicit deformable models for image segmentation. This
method was first proposed by Kass et al. within a variational formulation for the
segmentation of 2D-images [126]. The evolution of the snake model is defined from the
1
Eint (C(p))dp + β
E(C(p)) = α
0
1
1
Eimg (C(p))dp + γ
0
Eext (C(p))dp
(2.46)
0
where
• C(p) is the evolving curve parameterized in the plane, C : [0, 1] → R2 .
• The internal energy term Eint regularizes the curve smoothness during evolution.
• The image energy term Eimg guides the curve towards the desired image properties.
• The external energy term Eext is used to introduce user- defined constraints or
prior knowledge on the structure to be recovered.
• The parameters α, β and γ lead to a compromise between smoothness and external constraints.
The internal energy is defined as the contribution of two terms
2
∂C 2
∂ C 2
Eint (C(p)) = wtension (C(p)) (p) + wstiffness (C(p)) 2 (p)
∂p ∂p
(2.47)
where the first order derivative term makes the snake to behave as a membrane and
the second-order term makes the snake act like a thin plate. The image energy term
depends on the properties used for the definition of the boundaries of the structure to
be recovered. Although the squared image gradient norm Eimg (C(p)) = |∇I(C(p))|2
is the most spread representation, alternatives as the use of isophotes, region-based
representations or combinations of any other image features may be used.
The solution of the snake variational problem proceeds by computing the EulerLagrange equation associated to the energy functional (Equation 2.46) which provides
the expression of the PDE for the evolution of the snake model
∂C
∂ 2C
∂4C
(p) = α wtension 2 (p) − wstiffness 4 (p) − β ∇Eimg (C(p))
∂t
∂p
∂p
(2.48)
34
The numerical implementation of the snake motion is performed within the Lagrangian
framework for front evolution. Thus, the algorithm proceeds defining an initial curve
parameterized using a certain number of control points accompanied by a set of basis
functions. The position of these control points is updated using the snake model deformation that drives the initial curve towards the desired image properties. After every
update, the evolving contour should be re-parameterized. The process is continued
until convergence.
The resulting algorithm enjoys of low complexity and allows user interactivity. In
addition, it is quite easy to introduce a priori knowledge. Moreover, it is able to deal
with both open and closed structures and the extension to surfaces results straightforward. However, the main drawbacks of the approach are
• The energy functional is non-convex. Therefore, a good global minimum is difficult to find. Moreover, spurious edges may stop the evolution far from the
boundaries of the desired object.
• The algorithm is quite sensitive to initial conditions. The user must specify an
initial shape that is close to the boundaries of the final object. Depending on the
application, this may be a tedious task, specially in 3D.
• The algorithm is strongly dependent on curve parameterization. In addition, the
final segmentation results depend on the selection of control points and basis
functions.
• The snake model is unable to support changes on the topology of the object. For
example, this method is unable to segment several objects simultaneously or to
merge different shapes.
2.4.2
Edge-based models
Implicit deformable models borrow the idea from geodesic snakes of evolving an initial
contour towards a local minimum of an energy functional that depends both on internal
and external energies. Contour evolution is approached with the level set representation, thus overcoming with some of the limitations of the Lagrangian representation
used by the snake model. Early attempts use the image gradient as external force to
attract the active contour towards the object boundaries.
From Fluid Dynamics to image segmentation
The implicit approach for deformable models was introduced by Caselles et al. [33] and
Malladi et al. [159] extending the level set method for curve and surface evolution from
Fluid Dynamics to image segmentation. The evolution of the model is defined from
the level set equation
∂φ
= g(|∇I|)(Fint (x, y, z) + Fext (x, y, z)) · |∇φ(x, y, z)|
∂t
(2.49)
35
where
• The image term g enhances strong image gradients.
• The internal term Fint is a force depending on geometric properties of the zero
level set.
• The external term Fext is used to either expand or shrink constantly the interface
acting like a balloon force.
Geodesic Active Contours
Caselles et al. [34] and Kichenassamy et al. [129] simultaneously provided the connection between implicit deformable models and the variational problem associated
to the snake model. Geodesic Active Contours (GAC) model was defined from the
1
1
∂C 2
g(|∇I(C(p))|)dp
(2.50)
E(C(p)) = α
∂p (p) dp + β
0
0
This energy can be seen as a simplified snake model where the stiffness and the external
terms are missing and the image term is a monotonically decreasing function of the
image gradient. The term geodesic in GAC comes from the fact that the active contour
can be interpreted as a problem of finding local geodesics in a Riemannian space where
the metric distance is defined from the image.
Inserting the regularization inside the image term and using the arc-length parameterization of the curve, the GAC energy functional results into the expression
E(C(p)) =
0
1
1
∂C (p) dp ≡ E(C(s)) =
g(|∇I(C(p))|) g(|∇I(C(s))|)ds
∂p 0
By analogy, the GAC energy functional for surfaces is defined as
E(S(u, v)) =
g(|∇I(S(u, v))|)dudv
(2.51)
(2.52)
Ω
The Euler-Lagrange equation associated to this energy functional provides the expression of the PDE for GAC evolution. In the case of surfaces,
−
→
−
→ −
→
∂S
= g(|∇I|)Kmean · N + (∇g(|∇I|) · N ) N
∂t
This flow is implemented using the level set equation
(2.53)
∂φ
+ g(|∇I|)Kmean · |∇φ| + ∇g · ∇φ = 0
∂t
36
(2.54)
that corresponds to an advection-diffusion equation where the curvature term is weighted by the image gradient. In locations of the image where the gradient is plain, the
front moves regularized by the curvature term. As the gradient becomes more steep
the curvature term has lower effect on the flow and the advection term contributes to
attract the front towards the boundaries of the objects. A balloon force is usually incorporated to this PDE in order to avoid the front getting trapped into a local minima
at locations far from the edges. This method circumvents some of the limitations of
the snake model
• The energy functional is convex. In consequence, the Euler-Lagrange equation is
guaranteed to converge to the minimum with standard optimization techniques.
• The initial shape can be a little circle/sphere either inside or outside the object.
• The Eulerian formulation makes the algorithm not dependent on curve parameterization.
• The algorithm automatically supports changes in the topology being able to
segment several objects from a single initial contour or to merge different shapes.
Compared to Caselles-Malladi’s evolution (Equation 2.49), the incorporation of the
advection term in the GAC model generates a potential field in regions close to the edges
of the objects. Whereas Caselles-Malladi’s method has to be manually stopped using
heuristic criteria, the GAC equation reaches the steady-state once the front evolution
gets close to the boundaries of the objects leading the interface to stop at the edges.
However, since advection makes effect only near the image edges, the balloon force
becomes necessary in most applications in order to push the interface towards the
edges. If the balloon force is selected to be too large, the active contour passes through
weak edges where the advection force is not strong enough. On the contrary, if the
selected balloon force is not large enough, the curvature term leads the front evolution.
In addition, in places with plain gradient, weak edges, or high curvature values
(such as elongated structures), mean curvature motion leads the evolution making
the front shrink until disappear. This is the reason why edge-based models do not
provide satisfactory results when dealing with thin vessels segmentation. Hence, the
combination of these local models with more global ones becomes necessary.
2.4.3
Region-based models
Region-based models are suitable for the segmentation of objects defined by weak
boundaries. These models use global information that depends on the probability
distribution estimated in the different regions located in the image. Combined with
37
local edge information, these methods are able to deal with some of the limitations of
edge-based models.
Mumford-Shah model
Mumford-Shah model assumes that the image can be approximated by a piecewise
smooth function simulating a weak membrane model. Under this assumption, MumfordShah method searches for a partitioning of an image I into multiple regions that minimizes the length of the region boundaries while approximates the image inside each
region by smooth functions [179]. Let Ω denote the image domain. Then, the MumfordShah method searches for a pair (I, δR), where δR is the set of region boundaries
minimizing the energy
2
2
(I − I) dw + β
|∇I| dw + γ
dσ
(2.55)
E(I, δR) = α
Ω\δR
Ω\δR
δR
where α, β and γ balance the contribution of each term to the evolution and δR dσ
represents the length or area of the interface for curves and surfaces, respectively.
In practice, the lack of differentiability of the Mumford-Shah energy functional does
not allow to compute the associated Euler-Lagrange equation. Moreover, the discretization of the pair (I, δR) is a very complex task. Therefore, multiple simplifications of
the original Mumford-Shah model have been proposed in the literature instead.
Active contours without edges
A quite simplified version of the Mumford-Shah method was proposed by Chan et
al. [38]. Two regions Ωin and Ωout are considered and the image is approximated by a
piecewise constant function
if (x, y, z) ∈ Ωin
μin ,
(2.56)
I(x, y, z) =
μout , if (x, y, z) ∈ Ωout
Under these assumptions, |∇I|2 = 0 on Ω\δR and the Mumford-Shah energy functional
is simplified to
(μin − I) dw + α
E(I, δR) = α
Ωin
(μout − I) dw + γ
2
2
Ωout
dσ
(2.57)
δR
The Euler-Lagrange equation associated to the energy functional corresponds to
−
→
→
−
∂(δR)
= α((μin − I)2 − (μout − I)2 ) · N + γKmean · N
∂t
This flow is implemented using the corresponding level set equation
(2.58)
38
∂φ
+ α((μin − I)2 − (μout − I)2 ) · |∇φ| + γKmean · |∇φ| = 0
∂t
(2.59)
that can be seen as an adaptive piecewise balloon motion combined with a diffusive
term. The algorithm provides a partition of the image domain presenting the maximum
variance among regions. The corresponding piecewise approximation of the image
is equal to the average of the image intensities inside each region. Therefore, this
algorithm can be interpreted as a deformable model version of K-Means clustering
algorithm [206].
The extension of the piecewise constant model to a more general piecewise smooth
model was proposed within the level set framework in [253]. For this method, the
variational approximation of the level set problem presented in Section 2.2.2 was introduced for the definition of the level set equation. Thus, two regions are considered
and the image is approximated by the piecewise smooth function
I(x, y, z) =
I + (x, y, z) · H (φ(x, y, z)),
I − (x, y, z) · (1 − H (φ(x, y, z))),
if (x, y, z) ∈ Ωin
if (x, y, z) ∈ Ωout
(2.60)
where H is a smooth -approximation of the Heaviside function.
The Mumford-Shah energy functional corresponding to this image model can be written
as
E(I, δR) = α |I − I| H (φ(w))dw + α |I − − I|2 (1 − H (φ(w)))dw
Ω
Ω
+β |∇I + | H (φ(w))dw + β |∇I − | (1 − H (φ(w)))dw
Ω
Ω
(2.61)
+γ δ (φ) |∇φ|dw
+
2
Ω
The smoothness of H function allows to compute the Euler-Lagrange equation associated to this energy functional. Therefore, the corresponding level set equation is given
by
∂φ
+ δ (φ)(α|I + − I|2 − α|I − − I|2 + β|∇I + |2 − β|∇I − |2 + γKmean ) · |∇φ| = 0 (2.62)
∂t
where δ is a smooth approximation to the Dirac’s delta function resulting from the
spatial derivatives of Heaviside function.
Both the piecewise constant and smooth methods have been applied not only to
image segmentation but also to image denoising and restoration. However, both model
39
assumptions are so simple that, in the majority of the cases, they do not provide a
good approximation of the image. Moreover, the statistical model is so simplistic that
it may not be a good indicator for region statistics. Therefore, these methods may not
achieve an accurate delineation of the boundary of the objects, specially in medical
image applications.
Bayesian Active Regions
The Bayesian formulation for image segmentation proposed by Zhu and Yuille, assumes
that the intensities inside and outside the objects presented in the image are result of
a random process with a given density function [284]. The maximum a posteriori
partitioning of the image can be obtained by the minimization of the energy functional
N Ri (αi , w)dw + γ
α
E(δR) =
i=1
Ωi
dσ
(2.63)
δΩi
where Ω = ∪N
i=1 Ωi constitutes the partitioning of the image into the different regions
of interest and Ri corresponds to the region descriptor associated to region Ωi . Region
descriptors usually represent a priori error functions depending on the probability
distribution of each region. In general, these descriptors may depend on some unknown
parameters αi associated to the parameterization of the probability distribution.
In the case of considering two regions Ωin and Ωout and a constant dependence of the
parameters αi on the image partition δR, the Euler-Lagrange equation associated to
the energy functional is simplified to
−
→
−
→
∂(δR)
= γKmean · N + α(Rin − Rout ) · N
∂t
(2.64)
This flow is implemented using the level set equation
∂φ
+ γKmean · |∇φ| + α(Rin − Rout ) · |∇φ| = 0
∂t
(2.65)
that corresponds to a balloon motion combined with a diffusive term. The front evolves
in order to minimize the a priori error contribution inside while the diffusive term
contributes to regularize the front smoothness.
Geodesic Active Regions
The Geodesic Active Regions (GAR) model combines Bayesian formulation for image
segmentation with Geodesic Active Contours (GAC) model [188]. In the case of two
regions partition, the energy functional is defined as
E(δR) = α
Ωin
40
-log(Pin (I(w)))dw + α
-log(Pout (I(w)))dw
Ωout
+γ
g(|∇I|)dσ
(2.66)
δR
where the region descriptors Ri are computed from the minus logarithm of the probabilities associated to each region. This flow is implemented using the corresponding
level set equation
∂φ
+ α(log(Pout ) − log(Pin )) · |∇φ| + γ(gKmean · |∇φ| + ∇φ · ∇g) = 0
∂t
(2.67)
Inside the objects of interest, the front evolves in order to minimize the corresponding
a priori probability error while remaining smooth. When the front reaches the edges,
the advection term leads the evolution towards the object boundaries.
The availability of methods for modeling the probability for different organ tissues
in medical images have made both Bayesian and Geodesic Active Regions quite popular for the segmentation of different structures in medical images. However, in the
majority of cases, parametric models have been used for computing the probabilities
associated to region descriptors. These methods make strong assumptions on the model
distribution so that only a small set of parameters need to be estimated. However, the
choice of a specific model (often Gaussian) may restrict the applicability to a limited set
of images that should satisfy the underlying statistical assumptions. In fact, the work
developed in Chapters 3 and 4 shows that more general non-parametric estimation
models outperform parametric ones when applied to arterial structures segmentation.
Simultaneously or subsequently to our conference results, non-parametric models have
been also incorporated into GAR for the segmentation of medical images [196, 197, 177]
and textures [207].
2.4.4
Vessel-specific segmentation models
Due to the effect of mean curvature, most geometric flows are not able to capture
elongated low contrast structures. At those places, edge gradients are usually weak
due to partial volume effects and the contribution of the mean curvature leads the evolution making the front shrink towards its centerline until dissappear, similarly to the
evolution shown in Figure 2.7, Section 2.3. Apart from the vessel-specific region-based
implicit deformable models that will be discussed in the next Chapters, some other
vessel-specific models have been proposed in the literature. These methods include
either geometric or intensity-based forces that favor the flow of the interface inside
narrow structures.
41
Co-dimension two Geodesic Active Contours
Fronts involved in vessel segmentation are thin cylindrical shaped surfaces evolving
in the space. In this case, the maximum curvature is equal to the inverse of the
cylinder radius and the minimum curvature is equal to the curvature of the centerline
associated to the cylindrical shape. In segmentation of thin vessels with Geodesic
Active Contours (GAC), the mean curvature in the geometric flow is approximately
equal to the maximum curvature that usually reaches large values. In consequence,
the motion under mean curvature makes the surface dissappear or provides unstable
numerical results. Therefore, geometric flows involving mean curvature do not show
good results when dealing with the evolution of these elongated fronts. In contrast, the
motion under minimum curvature in the geometric flow can be considered equivalent to
the motion of the centerline of the front under curvature in co-dimension two providing
a stable geometric regularization of the evolving front [3]. This idea was introduced by
Lorigo et al. for thin vessels segmentation improving segmentation results compared
to classical GAC [155, 156].
Capillary Active Contours
Yan et al. provided a very elegant framework for implementing flows inside narrow
structures with application to vessel segmentation [275, 276]. In Capillary Active
Contours (CAC), the interface evolution is modelled as the flow of a liquid inside a
capilar tube delimited by vessel walls. The capillary action allows the fluid to evolve
without any other external force favoring the evolution inside thin structures. If thin
blood vessels are considered as capillary tubes this mechanism favors segmentation of
thin structures in low contrast situations.
In the CAC model the evolving front is divided into three regions
• The free surface, Sf , is the part of the front that is not in contact with the solid
boundaries
• The wetted surface, Sw , is the part of the front that is in contact with the solid
boundaries
• The contact line, C, is the boundary of the free surface Sf
The energy functional associated to the variational problem is defined from the combination of
E(S(u, v)) = αEvol + βEGAC + γECAC
(2.68)
where
• Evol =
S
dudv imposes a regularization on the volume of the evolving surface
42
• EGAC =
g(|∇I|)dudv is the energy functional associated to the GAC segΩ
mentation problem
• ECAC = C(p)dp + |∂C/∂p|dp is the capillary energy, that constraints the
length of the contact line and the area of the free surface, respectively
The Euler-Lagrange equation associated to this energy functional is given by
−
→
→
− −
→
−
→
∂S
∇g
= αN + β(g Kmean + ∇g · N )N + γ(1 + Kmin ) N −
cos θ
∂t
|∇g|
(2.69)
where θ is the angle between the free flow surface and the vessel boundary.
This flow is implemented using the corresponding level set equation
∂φ
∇g
+α|∇φ|+β(g Kmean ·|∇φ|+∇g·∇φ)+γ(1+Kmin ) |∇φ| −
cos θ = 0 (2.70)
∂t
|∇g|
where cos θ can be computed from the implicit representation φ as
cos θ =
∇φ · ∇g
|∇φ||∇g|
(2.71)
The term associated to the volume constraint corresponds to a balloon force. The terms
associated to the GAC evolution correspond to an advection-diffusion equation. From
the terms associated to the capillary action, the minimum curvature term corresponds
to a diffusive equation that regularizes the curvature of the contact line. The term
associated to the capilar force corresponds to a scalar flow in normal direction. This
term is responsible of the surface evolution in the tangential direction of the boundaries
of the vessel favoring the flow inside thin structures. The inclusion of the capilar force
has shown to outperform co-dimension two method for brain vessels segmentation in
Magnetic Resonance Angiography (MRA).
Flux Maximizing Flow
This model, proposed by Vasilevskiy et al. [247, 248], consists in a flow that evolves
favoring the alignment of the normals of the interface with a given vector field. Although the model does not take into account the geometry of thin structures, it has
shown to be suitable for the segmentation of organs with low contrast as, for example,
cerebral thin vessels.
−
→
The inward flux associated to a vector field V and a surface S is defined from the
integral
−
→ →
−
V , N dudv
(2.72)
flux(u, v) =
S
43
The flux maximizing flow seeks for increasing the inward flux of the vector field through
the boundary S as fast as possible. Thus, the PDE associated to the maximization of
the flux is given by
→
− −
→
∂S
= div( V ) · N
∂t
(2.73)
→
−
∂φ
+ div( V ) · |∇φ| = 0
∂t
(2.74)
with associated level set equation
This PDE is associated to a balloon motion where the velocity is given by the divergence
−
→
of the external vector field V . Considering this vector field equal to the normalized
gradient of the image, the flux maximizing flow aligns the normals of the interface
with the normals of the image isophotes. Then, the level set evolution is given by the
equation
∂φ
∇I
= −div
(2.75)
· |∇φ| = −KI · |∇φ|
∂t
|∇I|
where KI represents the mean curvature of the corresponding image isophote at each
point.
Haralick-Canny Active Regions
This model, proposed by Holtzman-Gazit et al. in [117, 118], consists in a flow that
evolves the contour towards the Haralick-Canny edges. The Haralick-Canny edge detector is based on the fact that the second derivative of the image in the gradient
direction
∇I
Iξξ = ΔI − div
|∇I|
(2.76)
|∇I|
is equal to zero. The variational problem associated to Haralick-Canny active contours
comes from the minimization of the energy functional
−
→
∇I
∇I · N dudv − div
E(S) =
|∇I|dw
(2.77)
|∇I|
S
Ω
The Euler-Lagrange equation associated to this energy provides the PDE associated
to the flow of a front moving towards the Haralick-Canny edges
−
→
∂S
= Iξξ · N
∂t
(2.78)
2.5. Summary
44
that corresponds to an evolution in normal direction constrained by the values of Iξξ .
The effect of the Haralick-Canny flow is the alignment of the front along the edges
of the desired object. As happens with flux maximizing, this flow is well suited for
tracking edges of objects with low contrast with respect to the background or adjacent
structures. It has shown to provide good results in combination with Active Contours
Without Edges (ACWE) model for thin structure segmentation.
2.5
Summary
In this Chapter we have studied the most remarkable aspects of the theory of implicit deformable models. These models are based on the theory of curve and surface
evolution where the level set method has emerged as a powerful paradigm for the computation of the interface evolution while handling topological changes in a natural way.
This ability has made implicit deformable models rather suitable for the segmentation
of complex structures and it is the reason why this methodology has been extensively
applied for the segmentation of arterial structures from medical images.
Although the level set method arose in a physical context, it has been extended to a
great variety of problems not only in Fluid Dynamics and Material Sciences but also in
Computer Graphics and Computer Vision, among others. The implicit representation
of curve and surface evolution allows the computation of the solution of the PDE
that drives the evolution in an Eulerian domain. Partial derivatives can be efficiently
computed using finite difference schemes. Stable numerical algorithms can be designed
from the combination of the numerical schemes studied for basic flows associated to
well known physical phenomena.
Implicit deformable models for segmentation are associated to a variational formulation where the Euler-Lagrange equation from the minimization of the energy functional
provides the PDE that drives the evolution of the interface towards the final segmentation of the structure. In this Chapter we have studied the variational problems
associated to edge and region-based information most related to the methods devised
in this Thesis for cerebral vascular structures segmentation. Moreover, we have studied
the specific models for thin structure segmentation that were proposed parallely to the
development of this part of the Thesis.
Edge-based models were formulated for early segmentation algorithms. The use of
edge information alone has shown strong limitations when dealing with images presenting weak and inhomogeneous gradients as those acquired in clinical practice. Regionbased methods include a global statistical model associated to the intensity distribution
of the regions present in the image. From them, Geodesic Active Regions (GAR) combines the benefits from both edge and region-based information. In places with weak
gradient, region-based information drives the evolution of the interface overcomming
with the limitations of edge-based models in these problematic locations. Finally, some
of the specific models for thin structure segmentation have pointed out the adverse ef-
45
fect of the mean curvature on geometric flows for narrow structures segmentation.
These methods overcome with this limitation by replacing the mean curvature motion
with flows that favor the evolution inside narrow structures. The most popular method
includes minimum curvature evolution. Recently, a method that simulates the physical phenomenon of the capillary action has been proposed showing promissing results.
Some other methods try to align the surface normals with the gradient of edge-related
functions. Although these methods do not specifically take into account the geometry
of thin structures, they have shown to provide good results in low contrast situations.
In conclusion, region-based implicit deformable models show a high potential for the
segmentation of arterial structures. In addition, their combination with vessel specific
deformable models may even improve the performance of each individual method.
However, in the majority of cases, the proposed methods for probability estimation use
too simplistic assumptions that are not well fitted to model probability distribution of
arterial structures in angiographic images. This has motivated in this Thesis the design
of alternative methods for probability estimation with more general assumptions that
better adapt to the features present in arterial structures.
Chapter 3
Non-parametric vessel enhancement
filter
Abstract
Visualization of vascular structures from 3D medical images is crucial in
clinical practice for diagnosis, treatment, surgical planning and follow-up.
Vessel enhancement techniques are aimed to improve vessel visualization in
medium- or low-quality clinical data with the potential of facilitating the tasks
of segmentation and centerline extraction. Among them, vessel enhancement
techniques based on scale-space theory are specifically designed for the detection of tubular structures but usually fail at bifurcations, adjacent objects
or vascular pathologies. Probabilistic techniques constitute a more general
framework for vessel enhancement. These methods usually consider image
intensity as the most discriminant tissue descriptor and model the intensity
distribution using parametric estimators. However, in some image modalities,
these model assumptions can lead to very inaccurate probability estimations
and introduce severe errors in vessel detection. In this Chapter we present a
novel method for probabilistic vessel enhancement. Instead of considering the
intensity as the only tissue descriptor, up to second-order differential image
descriptors are used in a multi-scale framework. In addition, a non-parametric
model for probability estimation is assumed. The method has been tested in
different angiographic modalities from the Circle of Willis territory (MRA,
CTA and 3D-RA), showing to outperform reference multi-scale and parametric vessel enhancement filters, specially in the most challenging examples.
3.1. Introduction
48
Contents
3.1
Introduction
3.2
The multi-scale local structure in 3D images
3.3
3.4
3.5
3.6
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
48
50
3.2.1
Scale-space image representation . . . . . . . . . . . . . . . . . . . . .
51
3.2.2
Differential image descriptors. Differential invariants.
. . . . . . . . .
52
3.2.3
Feature detection and scale selection
. . . . . . . . . . . . . . . . . .
53
Multi-scale representations of the second-order local structure
.
54
3.3.1
The second-order ellipsoid representation . . . . . . . . . . . . . . . .
54
3.3.2
The second-order prototype representation . . . . . . . . . . . . . . .
55
3.3.3
The second-order irreducible differential invariant representation . . .
59
Non-parametric probability estimation in tissue classes . . . . . .
3.4.1
Feature space
3.4.2
Non-parametric probability estimation
3.4.3
Training set
3.4.4
Implementation details
Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
60
. . . . . . . . . . . . . . . . .
61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
63
3.5.1
Vessel enhancement in MRA . . . . . . . . . . . . . . . . . . . . . . .
63
3.5.2
Vessel enhancement in CTA
. . . . . . . . . . . . . . . . . . . . . . .
65
3.5.3
Vessel enhancement in 3D-RA
. . . . . . . . . . . . . . . . . . . . . .
65
Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . .
78
Introduction
In clinical practice, visualization and quantification of cerebral vascular structures from
3D medical images is crucial for diagnosis, treatment, surgical planning and follow-up.
Images of the brain vascular system are acquired using different imaging modalities as
ultrasound, X-ray angiography, Magnetic Resonance Angiography (MRA), Computed
Tomography Angiography (CTA) and 3D-Rotational Angiography (3D-RA). In the
last years, these imaging modalities have been increasingly incorporated into clinical
protocols as they are non invasive and provide useful information in the diagnosis and
treatment of cerebral vascular diseases like stenosis, aneurysm development or vascular
malformations.
However, the cerebral images acquired in clinical practice are often noisy and present
reconstruction artifacts. Vessel thickness in the brain widely varies providing a large
range of intensities in the imaging data. Due to partial volume effects, the boundaries
49
Chapter 3. Non-parametric vessel enhancement filter
between vessels and surrounding tissues are often difficult to find even for an experienced observer. This makes image interpretation and quantification quite difficult and
subjective tasks. Vessel enhancement techniques are aimed to improve vessel visualization and, therefore, they make image interpretation and quantification more objective.
Besides, they have the potential to facilitate the tasks of segmentation and centerline
extraction.
The majority of the techniques for vessel enhancement have been proposed in the
framework of scale-space theory. These techniques exploit the second-order differential
structure of the image at multiple scales in order to construct a vesselness measurement.
Two different approaches have been proposed in the literature. The first one is based
on the theory of cores that defines a medialness measure from the similarity of the
local structure of the image at a given point with an ideal ridge [198]. This technique
has been applied to Contrast Enhanced MRA (CE-MRA) of the cerebral vascular
system [17, 84]. The second approach use the eigenvalues of the Hessian in the design of
filters that provide maximum response at tubular shapes [134, 153, 211, 81, 53, 212, 24].
The great majority of these techniques have been applied to different MRA modalities.
Probabilistic techniques are used to estimate the probability distribution of the
intensity values corresponding to the different tissues presented in the image. These
methods are inspired on brain MRI tissue classification techniques [103, 142, 270].
Vesselness measurements can be obtained from the probability corresponding to vessel
tissue class. Usually, a parametric distribution of Finite Mixtures (FM) is assumed for
probability estimation. These methods have been applied to the enhancement of the
cerebral vascular tree and cerebral aneurysms in Time of Flight and Phase Contrast
MRA (TOF-MRA and PC-MRA) [264, 265, 100, 45], and 3D-RA [60, 85]. More ad
hoc techniques use analytic models for approximating the probability distribution of
vessel intensities or vessel profiles in MRA and cardiac CT [271, 272, 69, 200].
Vessel enhancement filters have been extensively used for segmentation and centerline extraction. Scale-space based vessel enhancement techniques have been applied
to the segmentation of the cerebral vasculature in [80, 139, 17, 39, 61]. The great
majority of the probabilistic vessel enhancement techniques have been presented with
application to segmentation [264, 265, 60, 100, 45, 85]. Vessel enhancement techniques
have been used for centerline extraction in [266, 60, 17] among others. Furthermore,
the design of anisotropic diffusion filters focused in the denoising of tubular structures
is closely related to vessel enhancement techniques. These methods usually include
vesselness metrics in the anisotropic diffusion equation in order to smooth the image
while preserving vessel structures [222, 138, 30, 161].
Multi-scale vessel enhancement filters are specifically designed for the detection of
tubular structures and usually fail at bifurcations or vascular pathologies like stenosis
or cerebral aneurysms. Besides, due to the effect of Gaussian blurring, they show poor
performance in the detection of adjacent objects even though scale selection procedures
are commonly applied. The majority of these methods show good results in high quality
MRA but fail when dealing with medium- or low-quality data.
Probabilistic enhancement usually assume a Gaussian Mixture Model (GMM) for
3.2. The multi-scale local structure in 3D images
50
probability estimation. However, as pointed out in [85], vascular tissues occupy a very
small proportion (usually < 1%) compared to the background, bone and brain tissues
found in the different imaging modalities. In images containing brain aneurysms, this
proportion may hugely vary from one dataset to another depending on the aneurysm
size and location. For these reasons, parametric models can lead to very inaccurate
probability estimations and introduce severe errors in vessel detection.
In this Chapter, we present a novel method for probabilistic vessel enhancement.
Instead of using a parametric assumption, we propose a non-parametric model for
probability estimation. This way, no constraint is imposed on the statistical estimation, that is performed by learning the distribution of probabilities associated to the
main tissue types presented in the medical images from a significative sample of examples. Parzen windows is the machine learning method selected for non-parametric
estimation. The feature space is composed of up to second-order differential image
descriptors in a multi-scale framework. Related ideas were previously introduced by
van Ginneken et al. in the context of landmark-based statistical shape models for the
segmentation of ribs in 2D computed tomography scans and aortic aneurysms in 3DCTA [245, 56]. In these works, a gray level appearance model at object boundaries is
constructed based on the first-order multi-scale local jet of the image and an Active
Shape Model (ASM) is fitted using a non-parametric k-Nearest Neighbors classifier. In
our work, we have studied three spaces of features based on different representations of
the second-order local structure of the image. Our method has been tested in different
angiographic modalities (MRA, CTA and 3D-RA) located at the Circle of Willis territory. It has shown to outperform reference multi-scale and parametric probabilistic
vessel enhancement filters, specially in the most challenging examples.
The method for non-parametric vessel enhancement was presented at the international conference MICCAI’03 [113] with application to the segmentation of cerebral
aneurysms using region-based implicit deformable models. Further improvements on
the method including the design of different feature spaces were presented in MIAR’04
and SPIE’05 [108, 109]. The most relevant results were finally published in the journal
Medical Image Analysis (MEDIA) [110].
The rest of the chapter is organized as follows. In Section 3.2 we revisit the theory
of multi-scale local structure of 3D images. In Section 3.3 we describe the multiscale second-order representations considered as potential feature space candidates. In
Section 3.4 we present our method for non-parametric probability estimation. Experiments are gathered and discussed in Section 3.5. Finally, the main conclusions and
perspectives from this work are given in Section 3.6.
3.2
The multi-scale local structure in 3D images
The motivation for the study of the multi-scale local structure of images comes from the
need in Computer Vision community of analysing and interpreting real-world images
by automatic methods imitating biological vision. In real world, objects are composed
51
of different structures at different scales. The local structure of objects depends on the
scale of observation. For a machine vision system analysing an unknown scene, there
is no way to a priori know which scales are appropriate for describing the interesting
structures. Hence, the only reasonable approach is to consider the local description at
all scales.
Scale-space theory provides a framework for handling with image structure at multiple scales in a consistent manner. The basic idea is to embed the image into a
one-parameter family of gradually smoothed images in which the finer details are successively suppressed. It has been shown that scale-space operators show a high degree
of similarity with receptive profiles found in the mammalian retina and the first stages
of the visual cortex. Therefore, the scale-space framework can be seen as a theoretically
well-founded paradigm for early vision.
In this section we revisit the fundamental aspects of scale-space theory that have
been taken into consideration during the design of our algorithm for non-parametric
probability estimation. This review is based on the scale-space works by Lindeberg,
Koenderink, Florack, and collaborators [149, 150, 132, 77, 78, 79] where the fundamentals of the scale-space theory have been studied or revisited.
3.2.1
Scale-space image representation
In scale-space theory, an image is represented as a family of smoothed images parameterized by the size σ of the Gaussian smoothing kernel
G(x, σ) =
x2
1
− 2
2σ
e
(2πσ 2 )3/2
(3.1)
where x = (x, y, z) ∈ R3 . The parameter σ is called scale. The scale-space representation of an image I is denoted as
L = {L(x, σ) | σ ≥ 0}
(3.2)
where L(x, 0) = I(x) and L(x, σ) = I(x) ∗ G(x, σ) [268]. Image differentiation is
defined as the convolution of the image with normalized Gaussian derivatives
Li (x, σ) =
∂
∂
L(x, σ) = L(x) ∗ σ γ
G(x, σ)
∂xi
∂xi
(3.3)
where γ ≥ 0 is a parameter related to the dimensionality of the image. The notion
of γ-normalized derivative was introduced by Lindeberg et al. in order to allow the
comparison of the response of the differential operators at multiple scales [149].
The analysis of the local structure of an image is performed considering the image
as a topographic manifold in the scale-space (Figure 3.1). This representation allows to
translate the study of the local structure of the image to the study of the local geometric
3.2. The multi-scale local structure in 3D images
52
Figure 3.1: Image from a brain MRI scan and its topographic representation at scale σ = 1.5.
properties of this manifold. In practice, the image manifold at a point (x, σ) ∈ R3 × R+
is approximated by its second-order Taylor expansion
L(x + δx, σ) ≈ L(x, σ) + δxT ∇L(x, σ) + δxT HL(x, σ)δx
(3.4)
where ∇L(x, σ), and HL(x, σ) denote the image gradient and Hessian computed using
the normalized derivatives at scale σ. Using this approximation, image properties
can be described in terms of differential geometric descriptors from linear and nonlinear combinations of derivatives. Although the framework could be extended to
higher orders, it is restricted here to second-order due to our interest in the study of
tubular objects which can be appropriately described with multi-scale second-order
information.
3.2.2
Differential image descriptors. Differential invariants.
The multi-scale N -jet representation of an image provides the most simple local representation of the image manifold [133]. The N -jet of a general function is defined
as the set of spatial derivatives up to order N . In particular, the second-order N -jet
representation of an image at scale t is denoted as
(L, Li , Lj , Lk , Lij , Lik , Ljk )
2
(3.5)
and Li,j = ∂∂xL(x,σ)
.
where L = L(x, σ), Li = ∂L(x,σ)
∂xi
i ∂xj
However, this representation depends on the coordinate system. For the study of
the local structure it is necessary to base the analysis on differential descriptors invariant with respect to primitive transformations of the original coordinate system such
as translations, rotations, and scale and affine changes. These differential invariant
descriptors can be derived in an axiomatic manner in terms of directional derivatives
53
along certain preferred directions [78, 79]. Moreover, there exist a family of differential
invariants that constitute a complete and irreducible set of differential operators appropriate for the description of the local image structure up to any desired order [77].
As any other algebraic invariant can be reduced to a combination of elements in this
minimal set, local image properties are usually represented in terms of such invariants.
3.2.3
Feature detection and scale selection
The singularities of the differential image descriptors play a crucial role for feature
detection. In particular, the gradient of the image allows to distinguish between regular
points (points x s.t. ∇I(x) = 0) and critical points (x s.t. ∇I(x) = 0) where geometric
features like blobs, ridges, corners, junctions or edges are located. The most primitive
features (blobs and ridges) can be identified from the sign of the determinant and the
trace of the Hessian matrix analogously to the classification of manifold points in Morse
theory [173]. Table 3.1 shows this classification.
Table 3.1: Structure of critical points in Morse theory.
det(HI) trace(HI)
structure
<0
<0
blob (maximum)
>0
>0
blob (minimum)
≈0
<0
ridge saddle
≈0
>0
valley saddle
any
≈0
symmetric saddle
An important problem in feature detection from singularities is that local extrema
are very sensitive to noise. The computation of local extrema in the scale-space provides
a robust approach that circumvents this problem. If there is no prior reason for looking
at specific image structures, then the feature detector must be able to handle with
all the scales at the deep structure representation of the image. Fortunatelly, any
real world image is limited to pixel and image sizes and, in practice, the scale-space
representation can be therefore limited by these two values [132, 77]. The outer scale
of an image is given by the size of the image whereas the inner scale is given by the
pixel size or image resolution.
This principle also applies to image objects. The outer scale of an object corresponds to the size of the object while the inner scale corresponds to the size at which
substructures of the object begin to appear. The outer and inner scales of image objects are bounded by the outer and inner scales of the image. Automatic scale selection
is based on finding the local maxima or minima not only on the spatial but also on the
scale components [151].
3.3. Multi-scale representations of the second-order local structure
54
Table 3.2: Second-order structure in relation to the eigenvalues of the Hessian |λ1 | ≤ |λ2 | ≤ |λ3 |.
Structure Polarity
bright
λ1 0, λ2 0, λ3 0
blob
dark
λ1 0, λ2 0, λ3 0
ridge
bright
λ2 ≈ λ3 0, λ1 ≈ 0
valley
dark
λ1 ≈ 0, λ2 ≈ λ3 0
plane
bright
λ3 0, λ1 ≈ λ2 ≈ 0
plane
dark
λ1 ≈ λ2 ≈ 0, λ3 0
blob
3.3
Eigenvalues
Multi-scale representations of the
second-order local structure
Once a critical point has been detected, the multi-scale second-order information provided by the Hessian constitutes a powerful element in the discrimination of ridges
from other geometric features. In the majority of the imaging modalities used in clinical practice for the study of the vasculature, vessels constitute ridges in the image
at scales of size equal to vessel diameters. In this section, we describe three different
representations of the second-order structure of the image. These representations have
been considered in this work as potential descriptors that, incorporated into a learning
system, would allow to discriminate between vessel and non-vessel tissue classes.
3.3.1
The second-order ellipsoid representation
The ellipsoid representation locally approximates the second-order structure of the
image by an ellipsoid with main axes aligned with the eigenvectors of the Hessian
matrix and the corresponding eigenvalues as axes semi-lengths [153, 211, 81]. This is
the simplest approximation of the second-order structure. The ellipsoid shape depends
on the sign and magnitude of the eigenvalues. The ratios between the eigenvalues
magnitude allow to distinguish between a plate-like, line-like or blob-like structures,
and their sign indicates if the structure is either brighter or darker in relation to its
surroundings. Table 3.2 summarizes the relations between the eigenvalues for these
different structures that are also illustrated in Figure 3.2. Considering |λ1 | ≤ |λ2 | ≤
|λ3 |, an ideal tubular shape verifies |λ1 | ≈ 0, |λ1 | |λ2 | and λ2 ≈ λ3 . When all the
eigenvalues have a relatively low absolute value, the image has no evident second-order
structure.
The eigenvalues of the Hessian matrix have been extensively used in the design of
different vessel enhancement filters [153, 211, 81, 24]. Probably, the most widespread
vessel enhancement method is the one introduced by Frangi et al. [81]. This method
has been extensively evaluated in different image modalities and applications and it
is nowadays considered as the reference technique for vessel enhancement in medical
image processing community [182]. The filter is defined as the maximum on the scale-
55
Figure 3.2: Ellipsoid representation of plate-, line- and blob-like structures, respectively. The arrows
represent the direction of the eigenvectors of the Hessian matrix. The length of the arrows represents
the magnitude of the corresponding eigenvalues.
space
V(x) =
max
σmin ≤σ≤σmax
V(x, σ)
(3.6)
of the expression
V(x, σ) =
0 if λ2 > 0 or λ3
>0 2 2
RL
R2
S
exp − 2βB2
1 − exp − 2c
else
1 − exp − 2γ 2
2
(3.7)
where RL = |λ2 |/|λ3 | is equal to 1 for ideal edges and 0 for planar shapes. Therefore,
the first factor in the non-null definition of V measures the deviation from line shape
allowing to distinguish between plate or line-like structures. The deviation from blob
shapes is measuredvia RB = |λ1 |/ |λ2 · λ3 | from the second factor in the definition
of V. Finally, S = λ21 + λ22 + λ23 provides the energy of the intensity pattern at point
x. The parameters γ, β and c are user-supplied and control the sensitivity of the filter
to the measures of RL , RB and S, respectively.
3.3.2
The second-order prototype representation
The prototype representation arises from the fact that the second-order derivative operator set is not orthogonal and, therefore, the fundamental local second-order features
(magnitude, shape, and orientation) of the local pattern of the image are entangled.
To obtain discriminators of orientation and shape independent from magnitude, the
image responses have to be computed by an orthogonal operator set. The prototype
representation allows to compute these descriptors from the image responses in an orthogonal operator set constituted by spherical harmonics of order zero and two. With
this representation, the magnitude, orientation and shape can be disentangled.
The technique that allows computing these local second-order features is called
derotation and was proposed by Danielsson et al. in [53]. In the process of derotation,
the orientation of an arbitrary pattern f is identified as the rotation required to align
56
the pattern with its prototype p. This is done by mapping the derivatives of the pattern
f onto the harmonic operator set and identifying the eigenvalues of the Hessian with
the derivatives of the prototype p. More precisely, let denote with
g2 = (g20 , g21 , g22 , g23 , g24 , g25 )T
(3.8)
to the second-order derivative operator set. The action of g2 on a pattern f is defined
as
fg2 = f ∗ g2 = (fxx , fyy , fzz , fxy , fxz , fyz )T
(3.9)
The spherical harmonic operator set
c2 = (c20 , c21 , c22 , c23 , c24 , c25 )T
(3.10)
can be expressed as a linear combination of the second-order operator set
c 2 = M · g2
(3.11)
where M is the matrix that transforms the second-order derivative operator set into
the harmonic operator set
⎛ ⎞
1
1
1
0
0
0
⎜ 6
⎟
6
6
⎜
⎟
5
5
5
0
0
0 ⎟
⎜ − 24 − 24
6
⎜ ⎟
⎜
⎟
5
5
− 8
0
0
0
0 ⎟
⎜
8
⎟
M=⎜
(3.12)
⎜
⎟
5
⎜
⎟
0
0
0
0
0
2
⎜
⎟
⎜
⎟
5
⎜
0 ⎟
0
0
0
0
2
⎝
⎠
5
0
0
0
0
0
2
Hence, the mapping of a pattern f onto the harmonic operator set can be expressed as
fc2 = f ∗ c2 = f ∗ (M · g2 ) = M · fg2 = M · (fxx , fyy , fzz , fxy , fxz , fyz )T .
(3.13)
The projection pc2 of fc2 onto the prototype space is computed as
pc2 = (p20 , p21 , p22 )T = M · pg2 = M · (pxx , pyy , pzz )T
(3.14)
57
where M is the upper 3x3 matrix in M and pxx = λ1 , pyy = λ3 and, pzz = λ2 with
λ1 ≤ λ2 ≤ λ3 the eigenvalues of the Hessian matrix. Thus, the prototype p associated
to an arbitrary pattern f is identified with pc2 .
From the prototype response pc2 , the three basic quantities for describing the local
second-order structure can be derived. Namely,
magnitude : m = pc2 2
shape : τ1 = arg( p221 + p222 , p20 )
orientation : τ2 = arg(p21 , p22 )
The values of the prototype shape and orientation allow to distinguish among platelike, line-like or blob-like structures. The sign of τ1 indicates if the structure is either
brighter or darker in relation to its surroundings. In addition, this representation
allows to discriminate from structures similar to a double cone that appear in vessels
presenting stenosis or touching arteries. Table 3.3 summarizes the values of shape
and orientation for these different structures. Figure 3.3 shows the space of prototype
shapes.
Table 3.3: Second-order structure in relation to the shape and orientation of the elements in the
prototype space.
Structure
Polarity
τ1
τ2
bright
0
2π
3
π
3
π
3
2π
3
2π
3
π
3
blob
dark
− π2
π
2
ridge
bright
−atan( √2 )
valley
dark
atan( √2 )
plane
bright
−atan( √1 )
plane
dark
atan( √1 )
double cone
bright
0
double cone
dark
0
blob
0
5
5
5
5
The prototype representation has been used in the design of a vessel enhancement
filter that has shown to provide better results than filters based in the ellipsoid representation for the enhancement of vascular structures presenting stenosis [148]. The
filter is defined as the maximum on the scale-space of
V(x) =
where
max
σmin ≤σ≤σmax
σ 2 K(x, σ) · fc2 (x, σ)2
(3.15)
58
Figure 3.3: Representation of the different second-order shape structures in the prototype space. Image from Q. Lin, Enhancement, detection and visualization of 3D volume data, Linkoping University,
Sweden, 2003. Ph.D. dissertation.
59
⎧
⎨ 0
K(x, σ) =
⎩ − √4
p20
√
if
p221 +p222
5 p220 + 45 (p221 +p222
f20
fc2 2
√ p21 else
3 p22
·
−
)
∈ − 23 − δ, − 23 + δ
(3.16)
The first factor in the non-null definition of K is equal to 1 for ideal bright ridges,
0 for blobs and dual cones and −1 for dark valleys. The second factor accounts for
the differences between planes and bright ridges. The term fc2 (x, σ)2 measures the
energy of the intensity pattern at point x.
3.3.3
The second-order irreducible differential invariant
representation
As discussed in Section 3.2.2, the second-order irreducible differential invariants are the
minimal family of second-order differential descriptors that are invariant to primitive
transformations and provide a complete description of the second-order differential
structure of images [78].
This family consists of the descriptors
(ΔL(x, σ), trace(HL(x, σ)2 ), ∇L(x, σ)T HL(x, σ)∇L(x, σ))
(3.17)
where ΔL(x, σ), HL(x, σ), and ∇L(x, σ) respectively denote the image Laplacian, the
Hessian and the gradient computed using the normalized derivatives at scale σ.
This constitutes a general representation for the description of second-order information in contrast to the ellipsoid and prototype representations that are specific for
line-like structures. To our knowledge, the work presented in this Thesis has introduced
the use of irreducible differential invariants for the local representation of angiographic
images in the literature [110].
3.4
Non-parametric probability estimation in
tissue classes
In this section we describe our method for non-parametric probability estimation. The
method, as any other supervised learning method, involves the definition of the feature
space that, in our case, best represents the structure of the different tissues in the image,
the technique for non-parametric estimation, the selection of the set of examples that
constitute the training set and the selection of the model parameters that provides the
best generalization capability.
3.4. Non-parametric probability estimation in tissue classes
3.4.1
60
Feature space
The problem of the selection of the best representation for the observations is crucial
in the design of any learning algorithm. It is well known that if the starting point
of a pattern recognition problem is not well defined, this cannot be improved later in
the process of learning [66]. For this reason, we have focused on the study of the best
feature spaces for our application.
In our method, the image feature vector for a point x is defined from
f (x) = (fσ0 , ..., fσd−1 )(x)
(3.18)
where fσi represents the local structure of the image at point x and scale σi . As we are
interested in the discrimination of bright tubular- and blob-like objects, three different
feature spaces based on different second-order local structure representations have been
investigated in this work
• The ellipsoid feature space
fσi (x) = (Iσi , ∇σi I2 , λ1σi , λ2σi , λ3σi )
(3.19)
• The prototype feature space
fσi (x) = (Iσi , ∇σi I2 , p20 σi , p21 σi , p22 σi )
(3.20)
• The irreducible differential invariants feature space
fσi (x) = (Iσi , ∇σi I2 , ΔIσi , trace(Hσ2i ), (∇σi I)T Hσi (∇σi I))
(3.21)
All these representations include the intensity and the gradient magnitude as loworder descriptors for the discrimination of tissues in angiographic data. The secondorder descriptors depend on the adopted second-order image structure representation.
In the ellipsoid space second-order information is approximated by the eigenvalues of
the Hessian. In the prototype space second-order information is extracted from the
projection of the shape vector in the space of prototypes. The feature space of minimal
differential invariants use the irreducible set of second-order differential descriptors to
approximate second-order information of the image.
61
3.4.2
Non-parametric probability estimation
In our method, the probability associated to a tissue class C at a point x is considered within a Bayesian framework as a conditional probability P (x ∈ C|f (x)), where
f (x) is the vector of features used to characterize the local structure of the image. At
this point, different statistical models could be used for the estimation of this conditional probability. Among them, the non-parametric models do not make any a priori
assumption about the features distribution that is learned directly from the image.
The most popular non-parametric estimation technique is Parzen windows [190] as
it is a consistent estimator of any continuous probability [246]. In applications like ours
where the number of available samples compared with respect to the dimensionality of
the feature space provides a sparse representation of the domain of all possible patterns,
k-Nearest Neighbor (kNN) rule [54] can be interpreted as a suitable approximation of
Parzen windows. Assuming that points with similar local image structure belong to
the same tissue class, our method integrates a kNN estimator for the approximation
of the probability associated to a tissue.
Thus, for a voxel x, the feature vector f (x) is computed as in Equation (3.18). Then,
the k nearest feature vectors are found in a training set according to the Euclidean
distance. The probability for a voxel x to belong to a tissue class Ci is computed from
the formula
x̂∈T ∩N (x) Kγ (f (x), f (x̂))
P (x ∈ Ci |f (x)) = i k
x̂∈Nk (x) Kγ (f (x), f (x̂))
(3.22)
where Ti represents the set of points in the training set that belong to the class Ci , Nk (x)
is the set of the k nearest neighbors and Kγ is a Gaussian kernel with standard deviation
equal to the Euclidean distance to the k-th nearest neighbor (i.e. γ = d(x0 , x[k] ) where
x[k] is the k-th nearest neighbor). In this work, the metric used for measuring distances
on the feature space is associated to the Euclidean norm between vectors in R5d .
3.4.3
Training set
A learning method is not only defined by the representation of the observations and
the algorithm used for the estimation but also by the set of examples selected for
training. In general, the selection of the candidates for training is of great importance
in the learning stage of any supervised pattern recognition method. Often, the overall
performance of the method strongly depends on the selection of these candidates, that
has to be carefully carried out after a deep empirical study of the available data. It
remains an open issue how to develop a general methodology for the construction of
training sets that, associated to the pattern recognition method, would provide the
learning system with minimal generalization error [66].
In this work, we started with a preliminary exploration of the set of samples that
may lead to the removal of candidates that would provide undesirable effects during
3.4. Non-parametric probability estimation in tissue classes
62
learning. After gaining some insights into the learning method, a selection procedure
based on the empirical observations found on this preliminary exploration was devised.
The construction of the training sets involved the selection of image samples, tissue
labeling, and point sampling. Image samples were obtained from cropping the data in
the locations of interest. These crops should include a representative sample of vessel
patterns and typical non-vessel features. Crop size was selected in order to make tissue
labeling and point sampling as automatic as possible. If necessary, several crops from
the same image in different locations of interest were used. Tissue labeling involved
the identification of vessel, background, and partial volume tissues. It is important to
remark that partial volume tissues need to be identified as it was detected that their
inclusion reduced considerably the performance of the learning method and, therefore,
they should be neglected in the selection of training points. Several algorithms were
considered for automatic labeling as manual thresholding, region growing or K-Means
combined with mathematical morphology operators. In this work we selected K-Means
as this algorithm showed to provide satisfactory labeling results. The number of clusters
and label to tissue assignment were manually performed by the user. If necessary, the
cropped images were pre-processed using anisotropic diffusion filters in order to reduce
noise inside vessel and background structures and improve labeling results. Finally,
the training points were randomly selected from vessel and background tissues, and
the corresponding feature vectors were computed and stored.
3.4.4
Implementation details
Multi-scale image features
The definition of the feature space depends on the scales in which the computations
of the normalized derivatives are performed. The set of scales is chosen according to
an exponential sampling, σi = σ0 · eiρ with i = 0, . . . , d − 1, as suggested by scalespace theory [151]. Since vessels present different sizes in the images, it is important
to introduce a range of scales according to these vessel sizes in the feature space. In
our experiments, the inner scale has been set equal to the in-plane voxel size of the
image. The outer scale has been selected equal to the maximum radius of the vessels
of interest. The number of scales d has been typically selected equal to 10.
Feature rescaling using z-scores
As the nearest neighbors are computed from the Euclidean distance among feature
vectors and the range of magnitudes of each vector component is pretty heterogeneous,
the features in the training and test sets are previously converted into z-scores [65]. This
is equivalent to consider a weighted Euclidean metric for the computation of distances
in the feature space having into account the heterogeneity of vector components. Thus,
being the feature vector f and fi its i-th component, the z-score fî is computed from
the formula
63
fi − mi
fî =
si
(3.23)
where mi and si are the mean and standard deviation of the sample of the i-th feature
in the training set.
3.5
Results
We carried out a comparison of our non-parametric probabilistic vessel enhancement
filter with the most related techniques proposed in the literature. Among multi-scale
filters, we considered the methods proposed by Frangi et al. and Lin et al.’s due to their
close relationship with the ellipsoid and prototype representations, respectively [81,
148]. We also considered the parametric probabilistic filter based on Gaussian Mixture
Models (GMM) used in [60] for probabilistic enhancement of 3D-RA data. We present
results on the enhancement of the vessels in the Circle of Willis (shown in Figure 3.4
for anatomical reference) from three different angiographic modalities (MRA, CTA
and 3D-RA). The images were collected and kindly handed over by Dr. Juan R.
Cebral from George Mason University, Virginia, USA. The comparison was performed
based on visual assessment. Complementing these results, a thorough evaluation of
the technique in the framework of segmentation with Geodesic Active Regions can be
found in the next Chapter.
The set of scales for both multi-scale and non-parametric filters was selected as
explained in Section 3.4.4. Additional parameters for Frangi et al.’s method were
selected equal to typical values proposed in the original work [81] (i.e. γ = 0.5, β = 0.5
and c estimated from the 25% of the typical intensities inside the tissue of interest). For
the non-parametric method, patient-specific training sets were generated. The number
of samples in the train sets for each tissue was selected equal to the 10% of the correctly
labelled vessel points provided by K-Means clustering. As discussed in Chapter 4, the
number of neighbors k should be selected as the parameter providing the model with
minimal generalization error computed from cross-validation [101]. However, due to
the specificity and small size of the training sets constructed for this experimental
section, the estimated generalization errors resulted negligible in all cases and useless
for parameter selection. Hence, the number of neighbors k were (ad hoc) selected equal
to 30 as this selection provided satisfactory results in all cases.
3.5.1
Vessel enhancement in MRA
Figures 3.5 and 3.6 show the results in an example of Phase-Contrast MRA of the Circle of Willis with a PCoA aneurysm. This image modality constitutes a flow-sensitive
acquisition technique. In consequence, the images usually show low intensity values
inside vessel segments where the blood flow slows down as can be appreciated at the
vessel segments shown in the axial and coronal views of Figure 3.5. Moreover, due
3.5. Results
64
Figure 3.4: The arteries of the Circle of Willis. Anterior Cerebral Arteries (ACA) and segments
(A1 and A2) connected by the Anterior Communicant Artery (ACoA), Middle Cerebral Arteries
(MCA), Posterior Communicant Arteries (PCoA) connecting the Internal Carotid Arteries (ICA) and
the Basilar Artery (BA), and Vertebral Arteries (VA). Model courtesy of J.R. Cebral, George Mason
University, Fairfax, USA.
to flow velocity inhomogeneity, the intensity values inside brain aneurysms are significantly reduced with respect to the parent vessels showing abrupt intensity changes.
This causes problems in vessel visualization and segmentation at those locations.
Multi-scale filters showed to enhance the thickest vessel segments in the image in a
similar way. However, both methods were unable to enhance the thinnest vessels found
in Middle Cerebral and Basilar top territories. Moreover, both filters showed a high
response at background tissues due to noise or the existence of features with line-like
structure. It should be noted that Lin et al.’s filter showed a larger wrong response at
these locations. In addition, these filters showed a low response inside the aneurysm
dome as its multi-scale structure corresponded to ablob-like shape with low values for
the energy of the intensity pattern (i.e. low S = λ21 + λ22 + λ23 in Frangi et al. and
low m = fc2 (x, σ)2 in Lin et al. vessel enhancement filters, respectively).
The parametric probabilistic filter showed to enhance vessels in a wide range of
diameters. However, as in the case of multi-scale filters, some background structures
were incorrectly enhanced. This can be appreciated in detail in Figure 3.6. Moreover,
this filter was not able to enhance the lowest intensity values inside the aneurysm dome.
65
In contrast, our non-parametric filter showed to enhance correctly all vessel regions.
In addition, it showed to minimize the incorrect enhancement in background tissue
outperforming the rest of considered techniques. It should be noted that for this image
modality and geometry, the three feature spaces provided similar results.
3.5.2
Vessel enhancement in CTA
Figures 3.7 and 3.8 show the results in an example of CTA acquired from the same
patient than the MRA example. This image modality is a Computed Tomography (CT)
where the vessels of interest have been injected with a contrast agent. As a result,
vessels present high intensity values similar to those found in tissues with density
close to bone structures. Although this image modality allows to visualize a wide
range of vessels in the brain, the enhancement of vessels next to the skull basis (as
those belonging to the Circle of Willis) results problematic and constitutes a really
challenging task.
Multi-scale filters showed to enhance not only vessel segments but also the brightest
ridge structures located at bone tissues difficulting vessel identification. From them,
Frangi et al.’s filter showed a higher wrong response at these locations. Hence, multiscale filters could be considered of limited applicability for visualization in this image
modality.
The parametric probabilistic filter showed to enhance thick vessels correctly. However, this filter was unable to the enhance medium and small size vessel segments as,
for example, those arising from the Basilar or Middle Cerebral arteries. In addition, it
enhanced the whole region of partial volume voxels surrounding bone tissues as can be
appreciated from Figure 3.8.
In contrast, our non-parametric filter showed to enhance vessel tissues in a wider
range of diameters. The three feature spaces showed to provide similar results for
vessel regions unleashed from the bone tissue. Although the problem of enhancement
at partial volume voxels attached to the bone persisted, our method was able to reduce
it to the voxels surrounding the carotid grooves and the Turkish saddle. From the
three feature spaces, probabilities associated to prototype space seemed to minimize
this effect.
3.5.3
Vessel enhancement in 3D-RA
Finally, we show results in 3D-RA. This imaging technique involves the acquisition of
a sequence of 2D digital X-ray projection images after the injection of a contrast agent.
The sequence is acquired during a 180 degrees rotation of the C-arm on which the
X-ray detector is mounted. The final 3D-RA volume is obtained using a reconstruction
algorithm from the sequence of projected images. The reconstructed image quality is
sensitive to the number of projections and contrast injection leading to noise increment
and reconstruction artifacts that difficult vessel visualization and segmentation. We
present results in three different examples acquired at the Circle of Willis.
3.5. Results
66
Standard 3D-RA acquisition
Figures 3.9 and 3.10 show the results in an example of 3D-RA medium size ACoA
aneurysm. This image could be considered a representative example of 3D-RA acquisition without the reconstruction problems mentioned above. The image presents high
intensity values inside vessel tissues and the partial volume region between air and
bone tissue. It should be noted that bone tissue is not visible.
Multi-scale filters showed to enhance vessel segments in a wide range of sizes. As
happened with MRA, these filters showed a high response at background tissues due
to noise or features with a tube-like structure. It should be noted that Lin et al.’s filter
also showed a higher wrong response at these locations. In contrast, the filters showed
high response inside the aneurysm dome despite the blob-like multi-scale structure,
that may be due to the contribution of high intensity values to the energy components
S and m in the filters.
The parametric probabilistic filter showed to enhance correctly all vessel regions.
However, many of the air to bone partial volume voxels showed a high probability for
vessel.
In contrast, our non-parametric filter showed to enhance correctly all vessel regions
outperforming the rest of considered techniques. For this image example, the three
feature spaces seemed to provide similar results.
Low dose 3D-RA acquisition
Figures 3.11 and 3.12 show the results in an example of 3D-RA small size PCoA
aneurysm. This image was acquired at low-dose contrast injection due to patient
problems for delivering contrast. As a result, bone tissue is visible in the image with
intensity values in the same range than those found inside vessels.
As in the standard case, multi-scale filters showed to enhance vessel segments in a
wide range of sizes. These filters showed a high response at background tissues with
similar results to the standard 3D-RA case. Bone tissue was incorrectly enhanced.
The parametric probabilistic filter showed to enhance correctly all vessel regions.
However, many of the bone tissue voxels showed a high probability for vessel.
In contrast, our method was able to reduce incorrect enhancement of bone tissues
while correctly enhancing vessel tissues. It should be noted that prototype and differential invariant based feature spaces seemed to provide better results than ellipsoid
based space for the smallest vessels.
3D-RA acquisition after coil implant
Finally, Figures 3.13 and 3.14 show the results in an example of 3D-RA MCA aneurysm
after coil insertion. The image shows huge intensity values compared to the range of
intensities found in the rest of image regions due to the high density of coil material.
Therefore, visualization and segmentation result problematic and constitute really challenging tasks.
67
Multi-scale filters showed similar results to the rest of 3D-RA cases. It should
be noted that the high intensity values found inside the implant made these filters
enhance these regions despite their blob-like local structure. However, the real shape
of the aneurysm was not recovered due to the abrupt discontinuity in the intensity
profile at the boundary of the aneurysm.
The parametric probabilistic filter was not able to converge due to the huge intensity
variance associated to the image.
In contrast, our non-parametric model was still able to correctly enhance vessel
segments. However, as happened with multi-scale filters, the shape of the aneurysm
was distorted due to the effect of the abrupt intensity change on the multi-scale local
structure. In this case, the algorithm also incorrectly enhanced bone tissue.
3.5. Results
68
axial
coronal
sagittal
Frangi et al.
Lin et al.
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.5: Vessel enhancement in MRA. First row shows axial, coronal and sagittal views
of the image. Second row shows volume rendering images of the results from the multi-scale vessel
enhancement filters and the probability for vessel estimated from the GMM. Third row shows volume
rendering images of the probability for vessel estimated from our non-parametric method using the
ellipsoid, prototype and differential invariant representations, respectively.
69
GMM
axial
K-Means
NP-Ellipsoid
NP-Prototype
NP- Invariant
Figure 3.6: Vessel enhancement in MRA. First row shows axial views of the image and the
corresponding K-Means tissue classification (vessel tissue is labelled in white, background tissue in
grey and partial volume tissue in black). Second row shows the probability for vessel estimated from
the parametric and non-parametric models.
3.5. Results
70
axial
coronal
sagittal
Frangi et al.
Lin et al.
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.7: Vessel enhancement in CTA. First row shows axial, coronal and sagittal views
of the image. Second row shows volume rendering images of the results for the multi-scale vessel
enhancement filters and the probability for vessel estimated from the GMM. Third row shows volume
rendering images of the probability for vessel estimated from our non-parametric method using the
ellipsoid, prototype and differential invariant representations, respectively.
71
sagittal
K-Means
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.8: Vessel enhancement in CTA. First row shows sagittal views of the image and the
corresponding K-Means tissue classification (vessel tissue is labelled in grey, background tissue in black
and bone tissue in white). It should be noted that, in this example, K-Means classification is not able
to separate the partial volume tissue. Second row shows the probability for vessel estimated from the
parametric and non-parametric models. Third row shows the corresponding probability for bone.
3.5. Results
72
axial
coronal
sagittal
Frangi et al.
Lin et al.
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.9: Vessel enhancement in 3D-RA (standard example). First row shows axial, coronal
and sagittal views of the image. Second row shows volume rendering images of the results from the
multi-scale vessel enhancement filters and the probability for vessel estimated from the GMM. Third
row shows volume rendering images of the probability for vessel estimated from our non-parametric
method using the ellipsoid, prototype and differential invariant representations, respectively.
73
GMM
axial
K-Means
NP-Ellipsoid
NP-Prototype
NP- Invariant
Figure 3.10: Vessel enhancement in 3D-RA (standard example). First row shows axial views
of the image and the corresponding K-Means tissue classification (vessel tissue is labelled in white,
background tissue in grey and partial volume tissue in black). Second row shows the probability for
vessel estimated from the parametric and non-parametric models.
3.5. Results
74
axial
coronal
sagittal
Frangi et al.
Lin et al.
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.11: Vessel enhancement in 3D-RA (low-dose example). First row shows axial, coronal
and sagittal views of the image. Second row shows volume rendering images of the results from the
multi-scale vessel enhancement filters and the probability for vessel estimated from the GMM. Third
row shows volume rendering images of the probability for vessel estimated from our non-parametric
method using the ellipsoid, prototype and differential invariant representations, respectively.
75
GMM
axial
K-Means
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.12: Vessel enhancement in 3D-RA (low-dose example). First row shows coronal views
of the image and the corresponding K-Means tissue classification (vessel tissue is labelled in white,
background tissue in grey and partial volume tissue in black). Second row shows the probability for
vessel estimated from the parametric and non-parametric models.
3.5. Results
76
axial
coronal
sagittal
Frangi et al.
Lin et al.
GMM
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.13: Vessel enhancement in 3D-RA (adquired after coil implant). First row shows
axial, coronal and sagittal views of the image. Second row shows volume rendering images of the
results from the multi-scale vessel enhancement filters and the probability for vessel estimated from
the GMM. Third row shows volume rendering images of the probability for vessel estimated from
our non-parametric method using the ellipsoid, prototype and differential invariant representations,
respectively.
77
GMM
axial
K-Means
NP-Ellipsoid
NP-Prototype
NP-Invariant
Figure 3.14: Vessel enhancement in 3D-RA (acquired after coil implant). First row shows
coronal views of the image and the corresponding K-Means tissue classification (coil tissue is labelled
in white, vessel tissue in dark grey, background in black and partial volume tissue in light grey). Second
row shows the probability for vessel estimated from the parametric and non-parametric models.
3.6. Conclusions and Perspectives
3.6
78
Conclusions and Perspectives
In this Chapter, we have presented a novel method for probabilistic vessel enhancement.
Instead of using a parametric assumption as usual, we propose a non-parametric model
for probability estimation. This way, no constraint is imposed on statistical estimation,
that is performed by learning the distribution of probabilities associated to the main
tissue types presented in the medical images from a representative sample of examples.
In this work, Parzen windows has been selected for non-parametric estimation because
of its generalization properties and simplicity. The feature space is composed of highorder differential image descriptors in a multi-scale framework. In this work, we have
studied three different spaces of features based on state of the art representations of
the second-order local structure of the image.
The method has been compared to reference vessel enhancement techniques in different examples of angiographic modalities from the Circle of Willis territory that are
usually acquired in clinical practice for the diagnosis of cerebral vascular pathologies
like cerebral aneurysms (MRA, CTA and 3D-RA). The examples included a 3D-RA
image after coil insertion, usually acquired in clinical practice in order to assess how
the blood flow is affected after coil embolization.
Multi-scale filters showed good results in the enhancement of thick arterial structures in MRA and 3D-RA. However, they showed poor results at small vessels detection
and high responses at certain regions belonging to background tissues. In CTA, these
methods showed to enhance the brightest ridges of bone tissues difficulting vessel identification.
Parametric probabilistic enhancement showed good results in the enhancement of
medium and large size arteries. However, this filter showed high responses at background tissues. In CTA and low-dose 3D-RA, it enhanced the whole region of partial
volume voxels surrounding bone tissues. It should be noted that the Gaussian Mixtures
model was not able to estimate the probabilities for tissues in images acquired after
coil implants.
In all the examples, our non-parametric method showed to outperform the other
considered techniques providing acceptable results even in the most challenging examples. For example, our method was able to enhance vessel tissues in a wide range of
diameters in all cases. Moreover, it was able to minimize the error in the estimation of
the probability for vessel in the regions of partial volume voxels at bone tissue in CTA
and low-dose 3D-RA. Furthermore, the non-parametric model was able to estimate
correctly the probability for vessel in images acquired after coil implants. It should
be noted that these results were consistent independently of the feature space used to
describe the local structure of the images. From these results, we think that, after
proper validation, our method may be used in a wide variety of clinical applications
providing a great preprocessing tool helpful in tasks like visualization, segmentation
and centerline extraction.
Perhaps, the main limitation for the applicability of our method in clinical practice
may be found in the training stage of the algorithm that forces our method to be non-
79
automatic. Although the whole image can be used for training in the majority of MRA
and 3D-RA cases, the user needs to supervise the selection of the number of K-Means
classes. In addition, we have found that in some 3D-RA examples and the majority
of CTA images, the user needs to spend a considerable amount of time on finding the
combination of sample images that would provide acceptable tissue labeling for the
generation of a single training set. In some cases we found that a correct labeling
cannot be obtained using automatic algorithms. One possible solution would be to
rely on the generation of a single training set from a representative sample of images
where a correct labeling could be achieved. This alternative training method leads to
a trade off between automaticity and performance that will be explored in the next
Chapter.
Although our algorithm showed to outperform reference existing techniques, we still
identified some problems that need to be overcome. For example, a correct probability
estimation for partial volume voxels between objects is difficult to achieve. Moreover,
although important improvements with respect to parametric estimation have been
achieved in CTA, vessel enhancement and probability estimation at these locations
still constitute challenging open issues.
As future directions, it would be interesting to work in further improvements of the
elements involved in the definition of the machine learning algorithm. Just to name a
few, these improvements may include the exploration of alternative feature spaces, the
definition of more discriminative metrics in the feature space, the inclusion of scaleselection procedures during learning, the study of intrinsic metrics for nearest-neighbors
search, the use of more sophisticated machine learning algorithms like Support Vector
Machines or Neural Networks, and the use of manifold learning methods for improving
estimation.
Chapter 4
Non-parametric Geodesic Active
Regions for cerebral aneurysms
segmentation in 3D-RA and CTA
Abstract
The use of patient-specific 3D models of cerebral aneurysms plays an
important role in clinical practice helping the specialist in the selection of
the most appropriate treatment and in interventional planning. Furthermore, the availability of these models may provide geometric descriptors or
hemodynamic parameters associated to the aneurysm development or the
risk of rupture. In this Chapter, we focus on the automatic generation of
3D models of cerebral aneurysms from segmentation in 3D-RA and CTA.
The complexity of the aneurysm shape and the problems involved when
dealing with these image modalities make segmentation a challenging task
that has been successfully tackled in few works. We propose a method
for segmentation based on Geodesic Active Regions where region-based
information is estimated in a non-parametric framework from high-order
multiscale features. Due to the large variability of the feature space, the most
sensitive part of the algorithm is constituted by the selection of the training
set. Therefore, a protocol for training set construction has been devised. In
addition, the non-parametric model has been selected in order to achieve an
optimal probability estimation in the 3D-RA and CTA images. Once the
supervised part of the method has been carried out, the algorithm allows to
obtain automatic segmentations of images with characteristics similar to the
datasets used for training. The method has been evaluated with respect to
manual segmentations on a representative set of aneurysms obtaining a high
overlap index with respect to the ground truth. Moreover, this technique
has shown to outperform other region-based implicit deformable models.
4.1. Introduction
82
Contents
4.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.2
Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.2.1
Region-based implicit deformable models . . . . . . . . . . . . . . . .
86
4.2.2
Methods for the segmentation of the cerebral vasculature . . . . . . .
87
4.3
Non-parametric Geodesic Active Regions
4.4
Training set construction
4.5
4.6
4.7
4.1
. . . . . . . . . . . . . .
89
. . . . . . . . . . . . . . . . . . . . . . . .
92
4.4.1
Datasets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.4.2
Protocol followed for training sets construction . . . . . . . . . . . . .
93
4.4.3
Training set selection
95
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameter and model selection . . . . . . . . . . . . . . . . . . . . .
4.5.1
Selection of the optimal number of neighbors . . . . . . . . . . . . . .
4.5.2
Model selection
99
99
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Segmentation results and Evaluation
. . . . . . . . . . . . . . . . .
102
4.6.1
Datasets and experimental setting . . . . . . . . . . . . . . . . . . . . 102
4.6.2
Evaluation framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6.3
Evaluation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
110
Introduction
A cerebral aneurysm is an abnormal enlargement of the arteries located in the brain
(Figure 4.1). This pathology tends to appear at or near bifurcations of the arteries
in the Circle of Willis (CoW), with more frequency at the Anterior Communicant
(ACoA), Posterior Communicant (PCoA), Middle Cerebral (MCA), Internal Carotid
(ICA), and basilar arteries [127]. The prevalence of unruptured cerebral aneurysms is
unknown, but prospective studies estimate it to be as high as the 6% of the general
population. Approximately the 30% of patients present multiple aneurysms. From
them, there is a clear female preponderance ranging from the 54 to the 61% of the
cases [267, 260, 263]. Rupture, usually accompanied with subarachnoid aneurysmal
hemorrhage (SAH), is the most serious complication. These events have an incidence
of sudden death of 12.4% and rates of fatality from the 32% to the 67% after the
hemorrhage [120, 119]. Morbidity rates reach the 10.9% due to intra cranial bruise,
subsequent recurrent bleeding, stroke, hydrocephaly and vessel spasm [202]. The risk of
rupture appears to be related to its size and location. With respect to size, aneurysms
smaller than 7 mm have a benign natural history whereas larger aneurysms have greater
83
Chapter 4. Non-parametric Geodesic Active Regions
Figure 4.1: Cerebral aneurysm. Image from the book frontcover The brain aneurysm, V. G. Khurana
and R. F. Spetzler, Barrow Neurological Institute, Phoenix, USA. Reproduced with the permission of
Dr. V. G. Khurana.
risk of bleeding. Aneurysms in certain locations of the brain may have a high risk of
bleeding even with aneurysm sizes less than 7 mm.
The treatment of cerebral aneurysms aims at their complete elimination from the
blood circulation. Either surgery or endovascular coiling techniques can be used in
the treatment of this pathology. In surgical treatment, the aneurysm is reached by a
craniotomy and the surgeon places a metal clip across its neck, as shown in Figure 4.2,
in order to stop blood flow inside the aneurysm. Endovascular coiling is a minimallyinvasive procedure that reaches the treatment area from inside the blood vessel. This
procedure involves the insertion of a catheter into the femoral artery of the patient
that goes through the vascular system up to reaching the aneurysm sac. As shown in
Figure 4.3, platinum micro-coils are threaded through the catheter and deployed into
the aneurysm, partially or totally blocking the blood flow into its dome. The International Subarachnoid Aneurysm Trial (ISAT) has recently provided the evidence that,
if medically possible, all patients presenting ruptured brain aneurysms or unruptured
ones with high risk of rupture should receive the endovascular treatment [71, 176, 29].
However, not all the aneurysms are susceptible to receive an endovascular treatment. The specialist has to make the decision depending on the aneurysm size, geometry, location and position with respect to parent flow vessel [255, 57]. Two dimensional
Digital Substraction Angiography (DSA) is considered the gold standard technique for
the detection, characterization and quantification of brain aneurysms in clinical prac-
4.1. Introduction
84
Figure 4.2: Clip device and scheme of clipping procedure. Once the aneurysm is reached by craniotomy, the clip is placed across the aneurysm neck in order to stop blood flow inside the aneurysm.
Images obtained from http://www.aesculapusa.com and http://www.ubneurosurgery.com.
Figure 4.3: Guglielmi Detachable Coil device and scheme of coil embolization procedure. The
aneurysm sac is reached from a micro-catheter inserted through the femoral artery. The platinum
coils are deployed into the aneurysm blocking the blood flow into its dome. Images obtained from
http://www.endovascular.jp and http://www.ubneurosurgery.com.
tice. However, other less invasive acquisition techniques like Computed Tomography
Angiography (CTA), 3D Rotational Angiography (3D-RA), or Magnetic Resonance
Angiography (MRA) are being increasingly used as complementary methods for these
aims [4, 262, 115]. In order to estimate the dimensions of an aneurysm from these
data, it is customary to perform lineal measurements on maximum intensity projections (MIP) of the original volumetric scan [189]. The MIP provides a projection image
in an angle considered optimal by the neuroradiologist. The optimal viewpoint is usually chosen so that the magnitudes of interest are maximal. Manual measurements are
then carried out on the basis of this 2D image. The selection of the optimal angle
introduces a high degree of subjectivity to the quantification of the aneurysm. Furthermore, manual window leveling is often used to enhance the arterial structures of
interest increasing the subjectivity in the quantification of the aneurysm size. Therefore, measurements of the aneurysm dimensions based on patient specific 3D models of
the aneurysm and the surrounding vascular tree play an important role in improving the
treatment and intervention planning. Moreover, the availability of 3D models of brain
aneurysms may provide geometric descriptors or hemodynamic parameters associated
to the aneurysm development or the risk of rupture [157, 203, 168, 166, 35, 36].
This Chapter is focused on the generation of 3D models of cerebral aneurysms
from segmentation in 3D-RA and CTA. Segmentation in 3D-RA is a difficult task due
to noise, inhomogeneous image gradient, and the presence of vessels of a wide range
85
of sizes. In datasets acquired under low contrast dose, strong image artifacts appear
next to the arteries and part of the bone tissue turns out visible with the same range
of intensities than the thinnest vessels. Figure 4.4 shows some examples of 3D-RA
images in these situations. Segmentation in CTA is even more challenging because
of the presence of bone tissue in the image with intensity values that highly overlap
with intensity inside vessel tissues. As shown in Figure 4.5, these images present high
partial volume effects, specially in locations where the vessel is next or attached to other
vessels, the aneurysm, or the skull. In these locations, it is difficult or even impossible
to visually distinguish among vessel and background or bone tissues, specially, in the
location of the sphenoid bone that surrounds the carotid grooves next to the Turkish
saddle.
In this Chapter, an automatic method for the segmentation of cerebral aneurysms
in 3D-RA and CTA has been devised. The method is based on a Geodesic Active
Regions (GAR) model that couples the information from the edges and the statistics of the regions presented in the image [187]. Region descriptors are defined from
the non-parametric vessel enhancement filter presented in Chapter 3. Therefore, up
to second-order differential image descriptors are used in a multi-scale framework for
non-parametric estimation of the probabilities associated with the main tissue types
present in the medical images. The probabilities associated with each region are estimated using an adaptive Parzen windows computed from a k-Nearest Neighbors (kNN)
rule. Instead of using patient-specific training sets, we rely on the generation of a single
training set from a representative sample of images. Due to the large variability of patterns found in the feature space, the most sensitive part of the algorithm is constituted
by the selection of this training set. Therefore, a protocol for training set generation
has been devised. In addition, the non-parametric model has been selected in order to
achieve an optimal probability estimation for each of the tissue classes present in the
images. Once the supervised part of the method has been carried out, the algorithm
allows to obtain automatic segmentations of images with the same characteristics than
the datasets used for training. The method has been evaluated using manual segmentations and favorably compared with some of the region-based implicit deformable
models that are most related to our work and could compete in performance with our
segmentation algorithm for our specific application.
The method for segmentation of cerebral aneurysms was presented at the international conference MICCAI’03 [113] with application to CTA. Further improvements on
the method including the design of different feature spaces and the incorporation of
customary techniques of pattern recognition for parameter and model selection in order
to increase the generalization performance of the algorithm were presented in MIAR’04
for CTA and SPIE’05 for both CTA and 3D-RA [108, 109]. The most relevant results
were finally published in the journal Medical Image Analysis (MEDIA) [110].
The Chapter is organized as follows. Section 4.2 surveys the state of the art techniques most related to our work. The proposed non-parametric Geodesic Active Regions segmentation method is introduced in Section 4.3. Sections 4.4 and 4.5 present
the details of the protocol for training set generation and model selection. The segmen-
4.2. Related works
86
a
b
c
Figure 4.4: Examples of 3D-RA images. (a) Slice of an image acquired under standard protocol.
(b) Slice of an image acquired under low contrast dose showing vessel artifacts. (c) Slice of an image
acquired under low contrast dose showing bone tissue.
a
b
c
Figure 4.5: Examples of CTA images. (a) Slice showing a vessel located next to the aneurysm
dome. (b) Slice showing a cerebral aneurysm located next to bone tissue. (c) Slice showing part of
the Turkish saddle next to the Internal Carotid Arteries (ICA).
tation results and evaluation study is reported and discussed in Section 4.6. Finally,
some concluding remarks are given in Section 4.7.
4.2
Related works
In this section, we provide a review of the state of the art techniques most related to
our work. In first place, we begin with an overview of general region-based implicit
deformable algorithms. In second place, we review of the state of the art literature
devoted to vascular segmentation with special attention to cerebral vasculature and
brain aneurysms segmentation.
4.2.1
Region-based implicit deformable models
As shown in Chapter 2, since the method of region competition proposed by Zhu and
Yuille [284] there have been several works that include statistical region-based information in the evolution of the deformable model [279, 187, 38, 12, 207, 197, 160, 177].
In these different versions of region-based implicit deformable models, the interface is
deformed according to a PDE that minimizes an energy functional depending on regionbased statistical information. In places with weak gradient, region-based information
drives the evolution of the surface thus avoiding non-desirable effects of gradient-driven
evolution.
Geodesic Active Regions (GAR) combine classical Geodesic Active Contours (GAC)
87
with region-based statistical information defined in terms of the probability associated
with a region. In the case of medical images, each region is assumed to be in correspondence with a tissue or texture. The estimation of the probabilities for each tissue
involves the definition of the feature space that characterizes the image inside the different tissues and the selection of the model for probability estimation. In most attempts,
the estimation of the probabilities is usually based on two main assumptions: image
intensity is the most discriminant tissue descriptor and the statistics of the intensity
distribution can be described using parametric estimators. In particular, the probabilities are usually modeled with a Finite Mixture Model [284, 187, 160] or even simpler
assumptions [279, 38, 12].
In order to overcome with the limitations of parametric models, non-parametric
estimators have been included in GAR for probability estimation in several works simultaneous or subsequently to ours. In 2003, Rousson et al. proposed an implicit
deformable model for unsupervised texture segmentation that uses the structure tensor as texture descriptor in an non-parametric framework [207]. In 2003, Pichon et
al. proposed a generic statistical flow for GAR segmentation that considers the intensity and the gradient of the image as tissue descriptors and Parzen windows as
non-parametric estimator [196, 197]. More recently, Mory et al. have proposed in 2007
a non-parametric model from local intensity distributions [177].
4.2.2
Methods for the segmentation of the cerebral
vasculature
Some of the approaches for 3D vascular segmentation are based on statistical thresholding. These techniques approximate the distribution of probabilities of the intensity
values in the image by Finite Mixture Models (FMM) estimated from the Expectation Maximization algorithm. The FMM allows obtaining a segmentation by means of
an automatic global thresholding. These methods have been applied to the segmentation of the cerebral vascular tree and cerebral aneurysms in Time of Flight (TOF)
and Phase Contrast (PC) Magnetic Resonance Angiography (MRA) [265, 45], and in
3D-RA [85]. Other approaches are based on segmentation from atlases of the cerebral
vasculature [41, 191]. However, most of the recent approaches for 3D vascular segmentation are based on deformable models. The model is represented by a surface that
deforms for recovering the shape of the vascular structure. Depending on the representation of the model, these approaches can be divided into parametric and implicit
deformable models.
Parametric deformable models assume a predefined surface parameterization with
fixed topology. In vascular analysis, cylindrical or line-like shapes parameterized by
the vasculature centerlines are frequently used [211, 80, 266, 139, 280, 56, 17, 84,
256, 165, 147, 272]. In many cases, these models are not able to extract a complex
arterial tree without an important user interaction. Moreover, the tubular constraint
usually prevents the model from representing pathological shapes as stenosis or large
aneurysms.
4.2. Related works
88
Table 4.1: Summary of the most important characteristics of state of the art methods for segmentation of vascular structures.
Authors
Modality and location
Segmentation approach
Details
Sato et al. [211]
abdominal CTA
model not valid
multiscale line model
(1998)
cerebral vasculature in MRA
for brain aneurysms
Wilson et al. [265]
labeling based on
not valid for
(1999)
statistical information
large brain aneurysms
Frangi et al. [80]
model not valid
carotid in MRA
deformable cylinder
(1999)
for brain aneurysms
Krissian et al. [139]
model not valid for
multiscale medial model
(2000)
phantoms
Wink et al. [266]
model not valid
abdominal CTA and MRA
deformable cylinder
(2000)
for brain aneurysms
Lorigo et al. [156]
co-dimension 2
model not valid for
(2000)
implicit deformable model
Yim et al. [280]
deformable model
(2001)
Deschamps et al. [60]
cerebral vasculature in 3D-RA
parametric GAR
include brain aneurysms
(2001)
Aylward et al. [17]
not valid for
medial representation
(2002)
brain aneurysms
Vasilevskiy et al. [248]
flux maximizing flow
(2002)
centerline extraction
model not valid for
van Bemmel et al. [244]
abdominal MRA
(2003)
Bruijne et al. [56]
model not valid for
active shape model
abdominal CTA
(2003)
brain aneurysms
Antiga et al. [5]
model not valid for
carotid MRA
(2003)
brain aneurysms
Chillet et al. [41]
model not valid for
atlas based segmentation
(2003)
brain aneurysms
Chen et al. [39]
abdominal MRA
vessel enhancement filter
model not valid for
(2004)
carotid in MRA
Chung et al. [45]
labeling based on
not valid for
(2004)
statistical information
Fridman et al. [84]
not valid for
deformable cylinder
(2004)
Gan et al. [85]
labeling based on
cerebral vasculature in 3D-RA
(2005)
MIP statistical information
Manniessing et al. [160]
parametric GAR
requires additional
cerebral vasculature in CTA
(2005)
uses bone masking
CT acquisition
Volkau et al. [256]
deformable cylinder
(2005)
Holtzman et al. [118]
Haralick-Canny edge detector
(2006)
Yan et al. [276]
Capillary Active Contours
(2006)
Passat et al. [191]
model not valid for
atlas based segmentation
(2006)
brain aneurysms
McIntosh et al. [165]
deformable organisms
(2006)
for vessel segmentation
Manniessing et al. [162]
topology constrained
achieves segmentation
(2007)
parametric GAR
through skull basis
model not valid for
Li et al. [147]
4D-parametric model
brain aneurysms
(2007)
coronary in cardiac CT
Worz et al. [272]
model not valid for
abdominal CTA and MRA
deformable cylinder
(2007)
brain aneurysms
Law et al. [141]
cerebral vasculature in
multirange filter
(2007)
PC-MRA
Descoteaux et al. [61]
cerebral vasculature in
vessel enhancement filter
(2008)
PC-MRA
flux maximizing flow
89
The use of implicit deformable models within the level set framework has become
very popular in recent years. Their ability to handle with changes of topology and
adapt to the shape of complex structures makes them a very suitable technique for
the automatic segmentation of vascular structures including cerebral aneurysms. In
geometric deformable models solely based on gradient information, the evolution of the
deformable surface strongly depends on the image quality. Due to limited resolution or
artefacts present in medical imagery, the gradient usually presents discontinuities at the
boundaries and inside narrow locations of the objects. The evolving surface suffers from
leakage in such places. Moreover, the curvature and edge constraints prevent the surface
to evolve through narrow and twisted vessels. To deal with these limitations, some
improvements to this model have been proposed in the literature for the segmentation
of vascular structures in MRA, 3D-RA and CTA, that consist of smart initializations of
the model [60, 248, 5, 244], modifications in the energy functional in order to deal with
the specific characteristics found on angiographic images [156, 60, 276, 160, 118, 61],
or hybrid approaches [39].
Table 4.1 summarizes the most remarkable contributions to the segmentation of
vascular structures pointing out its applicability to cerebral aneurysms segmentation.
It should be noted that the great majority of these algorithms have been developed
and tested on MRA data, specially those for the segmentation of the cerebral vasculature. Among them, only a few may deal with brain aneurysms with a large size
range [265, 280, 60, 45, 85, 160, 118, 141]. Some of them are designed to specifically address with vascular segmentation in MRA [265, 280, 45, 141]. The rest are
region-based segmentation methods based on parametric models for probability estimation [60, 85, 160, 118]. Therefore, as shown in Chapter 3, they may not be appropriate
for segmentation of brain aneurysms from imaging modalities like CTA 2 , where bone
tissue presents high intensity values overlapped with vessel tissue intensities, or very
noisy 3D-RA acquired from patients in which an standard amount of contrast injection
is not possible.
4.3
As discussed in Chapter 2 implicit deformable models unify parametric models for curve
and surface evolution and the level set method. These models borrow the idea from
geodesic snakes of evolving an initial curve or surface towards a local minimum of an
energy functional [126]. Geodesic Active Regions (GAR), as introduced by Paragios et
al. [187], incorporates both edge and region-based statistical information into the energy
functional. In this work, we propose a non-parametric GAR model where region-based
2
It should be noted that Manniessing et al. [160] dealt with segmentation in CTA after bone
masking. This required an additional CT acquisition which is not usual in clinical practice. Holtzman
et al. [118] dealt with cerebral aneurysms segmentation in CTA. However, the method is included into
a hierarchical approach where a huge amount of bone tissue should be removed in a subsequent step.
Indeed, the proposed method for bone removal does not guarantee accurate segmentations.
4.3. Non-parametric Geodesic Active Regions
90
information is estimated from the non-parametric vessel enhancement filter presented
in Chapter 3.
In our method, the GAR energy functional is defined from
E(t) = α
Rin (x) dx + α
Rout (x) dx + γ
g(x)dσ
(4.1)
Ωin (t)
Ωout (t)
S(t)
where Ω = Ωin (t) ∪ Ωout (t) ∪ S(t) is the partition of the image domain Ω provided at
time t by the evolving interface S(t), and α and γ control the contribution of the region
and boundary based information, respectively.
Region-based information is defined in terms of the region descriptors
Rin (x) = −log(Pin (x))
and
Rout (x) = −log(Pout (x))
(4.2)
where Pin and Pout are the probabilities for a voxel x to belong to Ωin and Ωout , respectively. In our method, each region is assumed to be in correspondence with a single
tissue. 3D-RA and CTA images usually present the inside tissue, that corresponds
to the aneurysm and vessels, and one or two outside tissues that correspond to the
background and bone. In the case of images with two regions, the probability of the
inner region is computed as
Pin (x) = P (x ∈ Cvessel )
(4.3)
and the probability outside corresponds to
Pout (x) = P (x ∈ Cback ).
(4.4)
In the case of images with three regions, the probability outside is computed, under
the assumption of independence between classes distribution, as
Pout (x) = P (x ∈ Cback ) + P (x ∈ Cbone ).
(4.5)
Edge-based information is based on a function of the gradient that is positive in
homogeneous regions and zero at the edges. In our method,
g(|∇σ I|) =
1
1 + |∇σ I|2
(4.6)
where ∇σ I is the gradient of the image after convolution with a Gaussian filter of
standard deviation σ close to the in-plane resolution. This function helps the evolving
interface stopping at the boundaries of the regions.
91
As region descriptors are time independent, the Euler-Lagrange equation associated
to the minimization of the variational problem is given by
−
→
−
→ −
→
∂S(x; t)
= α(Rin − Rout ) · N + γ(gKmean + ∇g · N ) · N
∂t
(4.7)
where Kmean is the mean curvature of the evolving surface, controls the contribution
−
→
of the curvature to the evolution, and N is its outer unitary normal vector. As our
method is intended for the segmentation of vascular structures, the mean curvature in
Equation 4.7 is replaced with the minimum curvature of the surface, Kmin , as Lorigo
et al. proposed for the segmentation of vascular structures in [156].
The level set method is used to capture the motion of the surface allowing topological changes during evolution and avoiding numerical instabilities. As discussed in
Chapter 2, the level set method consists in embedding the evolving surface in a manifold one dimension higher than S implicitly represented by a function φ. The surface
S can be reconstructed as the level set zero of φ. If the manifold evolves following the
equation
∂φ
+ α(Rin − Rout ) · |∇φ| + γ(gKmin · |∇φ| + ∇g · ∇φ) = 0
∂t
(4.8)
then the evolution of the level set zero of φ is equivalent to the evolution of S driven
by Equation 4.7.
The level sets associated with φ evolve towards a local minimum of the energy
functional in order to maximize the probability for the inner region inside the zero level
set and the probability for the outer region outside having into account the gradient
information at the boundaries also. The minimum curvature term favors the flow inside
narrow vessels while it provides a similar evolution to mean curvature motion inside
cerebral aneurysms. The zero level set of the resulting steady-state solution is a 3D
model of the segmentation.
The rest of implementation details proceed as described in Chapter 2. For the
discretization of the numerical computation domain Ω, the values Δx, Δy, and Δz are
selected to be the in- and out-plane image resolution, respectively. The discretization
of the level set equation is given by
n
n
φn+1
ijk = φijk − Δt( α(Rin − Rout )ijk · |∇φijk |
balloon−term
+ γ(gijk (Kmin )nijk · |∇φnijk | + ∇gijk · ∇φnijk )) (4.9)
dif f usive−term
advection−term
The numerical approximation of the spatial derivatives is performed using a Godunov
scheme for the balloon term, central differences for the diffusive term and, finally, an
upwind scheme for advection.
4.4. Training set construction
92
In addition, the implicit function φ is initialized to be a signed distance function
of the initial surface S(0), computed using fast marching algorithm [216]. The size of
time sampling, Δt, is computed from the CFL condition
Δt · max (α|Rin − Rout | + γ|∇g|) ·
Ω
1
1
1
+
+
Δx Δy Δz
2γ
2γ
2γ
< 1 (4.10)
+
+
+
(Δx)2 (Δy)2 (Δz)2
For efficiency, all the computations are performed in a narrow band of the zero level
set [1]. The level set function φ is periodically reinitialized to be a signed distance
function using the method proposed by [140]. Finally, convergence is assumed to be
reached when the rate of volume change of the segmentations between iterations is
under a tolerance value.
4.4
Training set construction
As described in Chapter 3, the construction of the training sets in our non-parametric
method for probability estimation involves the selection of the image data sets, cropping
in the locations of interest, image pre-processing, tissue labeling, and point sampling.
Ideally, the use of image specific training sets including a sufficiently representative
sample of the features presented in the image would provide the best estimation results.
However, the construction of patient-specific training sets is a semi-automatic operation
that requires considerable user interaction. Moreover, depending on the image modality
and aneurysm location, some of the most representative features may not be included
into the training sets canceling the advantage of working with image-specific data.
For these reasons, we propose to perform the construction of one general training set
including the most representative features observed in the database of patients after
a deep empirical study of the available data. Although some of these stages could be
approached in a different way, we detail in this section the protocol followed for the
selection of the datasets and point sampling that, according to our experience, would
provide satisfactory training sets for our applications.
4.4.1
Datasets
The 3D-RA clinical datasets were acquired at Inova Fairfax Hospital (Fairfax, Virginia, USA) using a Phillips Integris Biplane unit (Philips Medical Systems; Best,
The Netherlands). Rotational angiographies were performed using a 6-second constant
injection of contrast agent and a 180-degree rotation with imaging at 15 frames per
second over 8 seconds for acquisition of 120 images. Each dataset was transferred to
a Philips Integris Workstation and reconstructed on a 128 x 128 x 128 image with a
square field-of-view (FOV) of 54.04 mm yielding a voxel size of 0.42 x 0.42 x 0.42 mm3 .
93
The CTA clinical datasets were provided by Miguel Servet Hospital (Zaragoza,
Spain). The acquisition was performed using an Helical Elscint CT Twin scanner
(Marconi; Haifa, Israel) with 120 kV/300 mA for the amplifier tube, 1.2-mm collimation
with an helical pitch of 1 and slice spacing of 0.65 mm. The images were reconstructed
on a 512 x 512 volume with a square FOV of 20.8 cm yielding an in-plane resolution of
0.4 mm. A total of 140 ml of non ionic contrast fluid was intravenously administrated
(Omnitrast 300 mg; Schering, Berling, Germany) at a rate of 3 ml/s, starting the
scanning 20 seconds after the onset of contrast administration.
At this point I would like to acknowledge to Dr. C.M. Putman from Inova Fairfax
Hospital and to Dr. R. Barrena from Miguel Setvet Hospital for providing the 3D-RA
and CTA datasets, respectively, and for their clinical support during the development
of this part of the Thesis.
4.4.2
Protocol followed for training sets construction
Training Sets in 3D-RA
The protocol for training set construction in 3D-RA started with the selection of a
group of clinical datasets in the most typical locations of the Circle of Willis. The
maximum dome sizes of the aneurysms presented in these datasets covered a wide
range of the sizes existing in the database. Thus, the images selected for learning
included vessel patterns from all the Circle of Willis and aneurysm patterns covering
the most frequent sizes in the data base. In our application, each of the candidate
images was selected to represent a prototype case of cerebral aneurysm located at
the Anterior Communicant (ACoA), Posterior Communicant (PCoA), Middle Cerebral
(MCA), basilar, and Internal Carotid (ICA) arteries, respectively, resulting in a total
of five datasets.
It should be noted that the training candidates need to present homogeneous characteristics consistent with the ones found on test images. In this work, we are presenting
results for a database of standard 3D-RA images. For this reason, images obtained
under low contrast dose, after coil or clip implants, or presenting giant aneurysms were
not considered either for training or evaluation. For example, if the algorithm would be
applied for the segmentation of aneurysms with coil, images presenting coil implants
should be used for training.
The selected images were first cropped to exclude zero-intensity regions while preserving most of the vascular tree. Then, the cropped images were pre-processed using
an anisotropic diffusion filter [25]. The use of anisotropic diffusion was intended here
to reduce noise while preserving image features, thus improving labeling results. The
filter was computed as the solution of the PDE equation
∂I
= div(g · ∇I)
(4.11)
∂t
where g is the edge function defined in Equation 4.6. In our application, the number of
iterations was selected equal to 5 and the stable value for the time step equal to 0.025.
94
At this point, vessel, background, and partial volume tissues were labeled in the
cropped images. During the design of the protocol for training set construction, several algorithms were considered for automatic labeling, as manual thresholding, region
growing [2] or K-Means [65]. Our approach used K-Means as it is fully automatic, parameter independent and yielded excellent segmentation results compared to manual
labeling.
Finally, the training points were randomly selected from vessel and background
tissues, and the corresponding feature vectors were computed and stored. In our application, a total of 2000 points were selected from each tissue and image resulting in
a training set of 20 000 points. Thus, the training points represent a uniform sample
of the local patterns presented in vessels of all locations and a wide range of widths,
aneurysms of a wide range of sizes, and the background.
Training Sets in CTA
Compared to 3D-RA, CTA has the additional challenge of the presence of bone structures in the image whose intensity values highly overlap with the intensities of vessel
and aneurysm, especially in partial volume voxels. Moreover, most of the vessels and
aneurysms in the Circle of Willis are in close proximity to the skull. Therefore, the
variability of the local patterns increases in CTA with respect to 3D-RA depending on
the presence of bone tissues in the image and its proximity to vessel tissues. As in the
3D-RA case, images obtained under low contrast dose, after coil or clip implants, or
presenting giant aneurysms were not considered either for training or evaluation.
The protocol for training set construction in CTA started with the selection of a
group of clinical datasets with cerebral aneurysms. The maximum dome sizes of the
aneurysms and vessel diameters presented in these datasets covered a wide range of
sizes existing in the database. The images were cropped in several locations of interest
including aneurysms, vessel segments from the cerebral vasculature, and bone tissue
next to the middle and posterior circulation of the Circle of Willis. In our application,
a total of six crops were used for training. Among them, four crops included aneurysms
located at the ACoA and MCA arteries. These crops were tight enough to not include
bone tissue. From the rest, two crops included bone tissue surrounding the Circle
of Willis territory and two crops included vessel segments showing a wide range of
diameters.
At this point, K-Means algorithm was considered for automatic labeling. The labels
of vessel tissue were obtained from crops that do not include voxels at the bone tissue,
as the K-Means labeling resulted more accurate in those cases. As in the case of 3DRA, several algorithms as manual thresholding and region growing were considered
as alternative to K-Means for automatic labeling. However, none of these algorithms
provided satisfactory labeling results in the crops involving bone tissue. Finally, vessel,
background, bone, and partial volume tissues were labeled in the cropped images. The
training points were randomly selected from vessel, background and bone tissues, and
the corresponding feature vectors were computed and stored. This way, only correct
95
samples were added to the training set. In our application, 10 000 points were selected
from vessel, background and bone tissues, resulting in a training set of 30 000 points.
4.4.3
Training set selection
In order to assess if the datasets selected for training significantly influenced the final outcome of our non-parametric probability estimation method, different groups of
datasets were considered during the devising of this algorithm. From these datasets,
different training sets were generated following our proposed protocol for training set
construction and their respective estimated probabilities were compared. From the
experiments developed in this study, we are showing two representative ones.
Figures 4.6 and 4.8 show the images selected for training and the corresponding
estimated probabilities for the five 3D-RA datasets used for evaluation in Section 4.6.
In the images, it can be appreciated that all the probabilities resulted quite similar.
The differences in the estimated probabilities were usually located at partial volume
voxels in the transition between vessel and background, and inside vessels with low
contrast dose. These results were consistent with the selection of alternative datasets
for training. Therefore, our proposed protocol for training set construction seems to
provide a robust probability estimation in 3D-RA.
Figures 4.7 and 4.9 show the images selected for training and the corresponding
estimated probabilities for the five representative CTA datasets used for evaluation
in Section 4.6. In the images, it can be appreciated that the non-parametric method
incorrectly estimated the probability for vessel in the transition between bone and
background. These errors were specially remarkable at the carotid grooves, where the
ICA crosses the skull basis, at the Turkish saddle (shown in PCoA example), and at
bone tissue with low calcification (shown in MCA-2). It should be noted that similar
results were also obtained in the CTA example shown in Chapter 3 despite the use of
patient-specific training sets. The most important differences in the estimated probabilities were found at these locations while the estimation obtained with the different
training sets was similar in the locations belonging to vessel tissue. Therefore, our
proposed protocol for training set construction seems to provide a robust probability
estimation in vessel tissues unleashed from bone tissue. Regretfully, the probability
estimation at locations close to bone tissue seems to be strongly dependent on the
selection of the datasets used for training. In the following, we selected Train #1 as
optimal in both 3D-RA and CTA.
96
Train #1
Train #2
ACoA
PCoA
MCA
basilar
ICA
Figure 4.6: Training sets in 3D-RA. Maximum Intensity Projection (MIP) of the 3D-RA images
selected for training. First row corresponds to the datasets selected for the first training set (Train
#1). Second row corresponds to the datasets selected for the second training set (Train #2).
Train #1
Train #2
Figure 4.7: Training sets in CTA. Volume rendering of the CTA images selected for training.
Upper group corresponds to the datasets selected for the first training set (Train #1). Lower group
corresponds to the datasets selected for the second training set (Train #2).
Invariant 2
Invariant 1
Prototype 2
Prototype 1
Ellipsoid 2
Ellipsoid 1
97
ACoA
PCoA
MCA
Basilar
ICA
Figure 4.8: Probability estimation in 3D-RA. Comparison of the probabilities for vessel estimated with Train #1 and #2 in the three different feature spaces considered in the non-parametric
estimation. The figure shows the original slice images and the probabilities associated to the training
sets.
98
Invariant 2
Invariant 1
Prototype 2
Prototype 1
Ellipsoid 2
Ellipsoid 1
ACoA-1
ACoA-2
PCoA
MCA-1
MCA-2
Figure 4.9: Probability estimation in CTA. Comparison of the probabilities for vessel estimated
with Train #1 and #2 in the three different feature spaces considered in the non-parametric estimation.
The figure shows the original slice images and the probabilities associated to the training sets.
99
4.5
4.5.1
Parameter and model selection
Selection of the optimal number of neighbors
The number of neighbors k, is a parameter that controls the complexity of kNN nonparametric estimation model. As k increases, the model becomes more complex, being
able to estimate the distribution of more complicated structures (low bias), but the
generalization error increases and, therefore, the accuracy in the estimation falls (high
error variance). This problem, illustrated in Figure 4.10, is known in the statistical
learning community as the bias/variance dilemma [101]. In between, there exist at least
one value of k providing a model complexity that gives the minimum test generalization
error. The optimal number of neighbors is then chosen among the ones that provide
the maximum model complexity with minimum generalization error.
High Bias
Low Variance
Underfitting
Low Bias
High Variance
Overfitting
optimal model complexity
Generalization error
test sample
training sample
Model complexity
Figure 4.10: Bias/Variance dilemma in pattern recognition. Illustration of test and training generalization error behavior as the model complexity is increased.
Given a set of N test inputs X = (x1 , ..., xN ) with corresponding labels C =
(C(x1 ), ..., C(xN )) independent from the training data, the generalization error associated to the kNN model is defined as
GE(k) =
N
1 LCE (xi , k)
N i=1
(4.12)
where
LCE (x, k) = −2 δ(Ci = C(x)) log(P (x ∈ Ci , k))
(4.13)
i
is the cross-entropy loss function [101], where Ci corresponds to the i-th class, δ represents Dirac’s delta function, and P (x ∈ Ci , k) is the probability estimated from the
kNN rule associated to class Ci with number of neighbors k. If the set of test inputs
is used to train the kNN model, then the expression for GE(k) is identified with the
training set error.
4.5. Parameter and model selection
100
As shown in Figure 4.10, the generalization error cannot be estimated from the
training set error as this error consistently decreases to zero with model complexity.
This is known as overfitting phenomenon. In rich-data situations, the available data
can be distributed into a set for training and a testing set. However, in the majority
of applications, including ours, this is not the case. As alternative, there exist several
techniques that estimate the generalization error either analytically (Bayesian information criterion (BIC), minimum description length (MDL)...) or by an efficient re-use of
the available data (cross validation, bootstrap...) [101]. Among them, cross validation
is the simplest and most widely used method for estimating the generalization error.
The available data is randomly split into M equal-sized parts and the generalization
error of the model trained with the remaining M -1 parts is estimated for the m-th
part.
In our method, we used cross validation for the selection of the optimal number of
neighbors. The generalization error associated with the training sets was estimated in
ten different experiments using m-fold cross validation with m = 10 folds. Figure 4.11
shows the curves of the generalization error in 3D-RA and CTA as the model complexity
increases. In addition, the curves of the computation time spent in the kd-tree search
for a volume of size 100 × 100 × 100 are shown in this figure.
Comparing both sets of cross validation curves, it should be noted that the generalization error in CTA was higher than in 3D-RA. This may be due to the high
variability of local features in CTA decreases the generalization performance of the
learning algorithm. In addition, the differential invariant representation showed the
lowest error in all cases, whereas the ellipsoid representation showed the highest error.
The error for the prototype representation in 3D-RA was similar to the error for the
ellipsoid representation. Interestingly, this error was similar to the optimal differential
invariant representation error in the case of CTA.
Comparing the time spent in the kd-tree search, the algorithm showed to be more
efficient for the differential invariant representation. In contrast, the algorithm showed
the worst efficiency for the ellipsoid representation. These differences in time may
reflect the spatial distribution of the different second-order features in the kd-tree.
In the case of 3D-RA the generalization error shows an asymptotic behavior starting from 20 neighbors for the feature space based on the differential invariant representation, and 30 neighbors for the feature space based on the ellipsoid and prototype
representations. In the case of CTA, the minimal generalization error is reached approximately at 20 neighbors for the ellipsoid representation and at 30 neighbors for
the prototype and differential invariant representations. This error gradually increases
from 40 neighbors. From these results we selected 30 neighbors as an optimal parameter
for our non-parametric estimation method.
101
0.25
Ellipsoid
Prototype
Invariants
0.25
3DRA
0.2
0.2
0.175
0.175
0.15
0.125
0.1
0.075
0.15
0.125
0.1
0.075
0.05
0.05
0.025
0.025
0
0
1
10
20
CTA
0.225
0.225
30
40
50
60
70
80
90
Ellipsoid
Prototype
Invariants
1
100
10
20
30
2500
Ellipsoid
Prototype
Invariant
2500
3DRA
50
60
70
Ellipsoid
Prototype
Invariant
2250
2000
2000
1750
1750
1500
1500
Time (s)
Time (s)
2250
40
80
90
100
Number of neighbors (k)
1250
CTA
1250
1000
1000
750
750
500
500
250
250
0
0
1
10
20
30
40
50
60
70
80
90
100
1
10
20
30
40
50
60
70
80
90
100
Figure 4.11: Upper row, curves of the generalization errors associated to the number of neighbors,
k. Computations from ten different cross-validation experiments. Lower row, curves of the average
computation time spent in the kd-tree search for a volume of size 100 × 100 × 100. Left, curves
corresponding to 3D-RA. Right, curves corresponding to CTA.
4.6. Segmentation results and Evaluation
4.5.2
102
Model selection
Prior to the generation of the segmentation results, we compared the performance of
the non-parametric method for the different feature spaces. From this comparison, we
finally selected the model showing the best results for probability estimation.
Figure 4.12 shows the differences of vessel probabilities for 3D-RA. All the probabilities resulted quite similar with differences located at the partial volume voxels in the
transition between vessel and background. In this case, the computation time spent in
the kd-tree search and the generalization error associated to the learning system were
the criteria that lead us to select the differential invariants as optimal feature space for
our non-parametric segmentation method in 3D-RA.
Figure 4.13 shows the differences of vessel probabilities for CTA. The probabilities resulted quite similar in the locations belonging to vessel tissues. However, the
best probability estimation for vessel in bone tissue locations was achieved by the
non-parametric model associated to the prototype space. This lead us to select the
prototype space as optimal feature space for our non-parametric segmentation method
in CTA.
4.6
Segmentation results and Evaluation
In this experimental section we present the segmentation results of our non-parametric
Geodesic Active Regions in a representative database of cerebral aneurysms in 3D-RA
and CTA. In addition, we provide the evaluation of the method with respect to ground
truth manual segmentations and compare our algorithm with the region-based implicit
deformable models most related to our work that could compete in performance with
our segmentation algorithm for this specific application.
4.6.1
Datasets and experimental setting
In this work, segmentation results are shown on a total of 10 cerebral aneurysms located
at the Circle of Willis and selected from the 3D-RA and CTA databases described in
Section 4.4.1. The datasets were selected in order to cover the most representative
geometries and locations found in our databases of images. Therefore, the results shown
in this experimental section are extensible to datasets with similar geometries and
locations. In the case of 3D-RA, the images were selected to represent a prototype case
of cerebral aneurysms located at the ACoA, PCoA, MCA, basilar, and ICA arteries,
respectively. From them, the MCA example showed part of the bone tissue. In the case
of CTA, two aneurysms were located, at the ACoA. One was located at the PCoA. As
happens with the majority of the aneurysms of this type, the aneurysm was in contact
to bone tissue. The last two aneurysms were located at the MCA territory. From
them, one of the aneurysms was selected to be in contact to bone tissue.
Manual segmentations were performed by one experienced observer twice. The
observer manually traced the contours with a period of 1 month between the trac-
Prot vs Inv
Ell vs Inv
Ell vs Prot
103
ACoA
PCoA
MCA
Basilar
ICA
Prot vs Inv
Ell vs Inv
Ell vs Prot
Figure 4.12: Probability estimation in 3D-RA. Comparison of the probabilities for vessel
estimated from the three different feature spaces considered in the non-parametric estimation. The
figure shows the absolute differences between probabilities for all the possible permutations.
ACoA-1
ACoA-2
PCoA
MCA-1
MCA-2
Figure 4.13: Probability estimation in CTA. Comparison of the probabilities for vessel estimated
estimated from the three different feature spaces considered in the non-parametric estimation. The
figure shows the absolute differences between probabilities for all the possible permutations.
104
ings using SNAP application [283]. To reduce intra-expert variability, the vascular
structures with diameter less than 1 mm (∼ 3 voxels) were excluded, and an average
segmentation was derived by calculating the mean shape of the manual segmentations
using a shape based interpolator [204]. This average segmentation was regarded as the
ground-truth in our quantitative study.
In the comparative analysis, we considered K-Means and the two region-based implicit deformable models proposed in the literature for the segmentation of cerebral
aneurysms in 3D-RA and CTA.
• K-Means clustering (KM) was considered in the comparison as it was used for
automatic labeling in the training stage of our algorithm. K-Means is an unsupervised classification method based on the minimization of the variance inside classes. In our implementation, the images were first preprocessed using an
anisotropic diffusion filter to improve classification results [25]. Three clusters
(vessel, background and partial volume) were considered in the case of 3D-RA,
and five clusters (vessel, background, bone, air and partial volume) in the case
of CTA.
• Parametric Geodesic Active Regions (P-GAR), was successfully used for the segmentation of cerebral aneurysms in 3D-RA [60]. In our implementation, the
probabilities of the region descriptors were assumed to be Gaussian, estimated
from a Gaussian Mixture Model (GMM). As we found that the Expectation Maximization (EM) algorithm presented convergence problems in our datasets, we
estimated the GMM parameters from the segmentations achieved by K-Means.
• Active contours without edges (ACWE) minimizes the variance inside and outside
the surface in evolution. In this model, the gradient information is dropped.
Thus, the energy functional depends only on region information. This method
was used for the segmentation of CTA in [118]. In our implementation, if the
image presents two tissues, a single surface was considered in the evolution [38].
In the case of three tissues, a multiphase version of this algorithm was used [253].
The same initialization was used for the three deformable models consisting in the
signed distance transform of N seed points randomly selected inside the ground-truth.
The common parameters, collected in Table 4.2, were set equal to the same values for
all the methods. The number of iterations, nit , was selected to be the needed to reach
convergence. Time step Δt was computed according to the Courant-Friedrichs-Levy
(CFL) condition inside a narrow band of amplitude w.
4.6.2
Evaluation framework
In our quantitative study, two different measures of similarity were considered to evaluate the accuracy of a segmentation by comparing the set of segmented voxels (S) with
the set of ground-truth voxels (G). We compared the volumetric overlap between two
105
Table 4.2: Parameters involved in the considered model based techniques (non-parametric Geodesic Active Regions (NP-GAR), parametric Geodesic Active Regions (P-GAR) and Active Contours
Without Edges (ACWE)) and values used in the evaluation experiments.
Parameter
Description
Model
Value
N
number of seed points
NP-GAR, P-GAR, ACWE
100
region based energy scaling NP-GAR, P-GAR, ACWE
1.0
ζ
η
gradient based energy scaling
NP-GAR, P-GAR
1.0
curvature scaling
NP-GAR, P-GAR, ACWE
0.25
nit
number of iterations
Δt
time step
NP-GAR, P-GAR, ACWE
CFL condition
w
narrow band amplitude
NP-GAR, P-GAR, ACWE
6 voxels
NP-GAR, P-GAR, ACWE until convergence
segmentations using the Dice Similarity Coefficient (DSC). In addition, we evaluated
the subvoxel accuracy between two segmentations using statistics derived from the
Euclidean distance between surfaces.
DSC is a special case of the Kappa statistic commonly used in reliability analysis
for multiple applications. DSC is defined as
DSC(S, G) = 2 ·
|S ∩ G|
|S| + |G|
(4.14)
where |X| denotes the cardinality of the set X. DSC ranges from 0, if the objects do not
overlap, to 1, if the overlap between the two segmentations is maximum. This coefficient
has been extensively used for validation of segmentation algorithms in different medical
image applications [285, 88, 287, 197, 85].
In order to compare the distances between the ground-truth and the surfaces from
model-based segmentations, we used the distance between the boundary points of
the ground-truth and the surface representation of the model. This distance can
be considered as a random variable, D, which describes the discrepancy between
the ground-truth G and the surface S. The statistics derived from this random
variable, provide a quantitative interpretation about the performance of the evaluated model based technique. In this work, we considered the probability of error,
P E = P (D > 0), the mean of errors, μD>0 = mean(D|D > 0), the standard deviation of errors, σD>0 = stdev(D|D > 0) and the error distance of the worst f % voxels,
Df = f − quantile (D).
4.6.3
106
Evaluation results
Figures 4.14 and 4.15 show the segmentations used in the evaluation study. In the case
of the 3D-RA datasets, our non-parametric method (NP-GAR) showed to provide the
best performance through the evaluation datasets. Segmentation with K-Means (KM)
was usually not able to recover some of the vessels of interest as can be appreciated
in the basilar example. In general, the performance of K-Means in the rest of the
vasculature was good, although the algorithm failed in the segmentation of the arteries
at the middle cerebral territory. The quality of the segmentations provided by the
parametric model (P-GAR) was, in general, lower than the quality achieved by the
other models, specially in PCoA, MCA and basilar examples. The model without
edges (ACWE) showed to provide segmentations thicker than the ground-truth.
In the case of CTA datasets, our non-parametric model (NP-GAR) again showed to
provide the segmentations with the best quality in all cases. In the images with bone
tissue next to the aneurysm (PCoA and MCA-2), all the models were prone to lead
the front evolution towards bone to background partial volume areas thus segmenting
parts of bone tissue attached to the aneurysm body. This problem was less dramatic
in our method as, in most of the cases, the attached piece of bone was usually smaller,
and could be easily removed with surface editing without affecting the overall quality
of the segmentation. K-Means (KM) usually included vessel and bone tissue in a
single cluster (MCA-2) and vessel with low intensity and background tissue in the
same cluster (PCoA). In the images with aneurysms not in contact with bone tissue
(ACoA-1, ACoA-2 and MCA-1), K-Means, the non-parametric (NP-GAR) and the
parametric (P-GAR) models showed to perform similar (except for the segmentation
of the thin vessel in MCA-1) and the model without edges (ACWE) provided thicker
segmentations.
Figure 4.16 and Table 4.3, present the DSC values for the segmentations considered
in the evaluation study when compared to the ground-truth. In addition, Table 4.4
shows the results of the statistics based on the error distance D between manual and
model based segmentations. Both tables present the mean and the standard deviation
of the statistical values computed for each method over the 3D-RA and CTA data sets.
In the 3D-RA evaluation set, the best average overlap index was achieved by our
non-parametric method (NP-GAR), with an average DSC of 90.05%. In Figure 4.16,
it is shown that all the segmentations provided by our technique produced DSC values
greater than the 70%, which indicates an excellent agreement with the ground-truth in
all cases [285]. K-Means (KM) showed a low overlap index due to wrong segmentation
of vessels in the middle cerebral territory (DSC = 46.74%). In cases involving small
vessels (basilar), or high partial volume effects (PCoA and MCA), the parametric model
(P-GAR) usually achieved the worst overlap indexes (DSC average = 69.03%). In all
cases, the method without edges provided segmentations thicker than the ground truth
with average DSC equal to 77.39%.
Table 4.4 shows that our non-parametric method (NP-GAR) achieved the lowest
probability of error (P E = 17.88%) while the parametric model (P-GAR) and the
ICA
basilar
MCA
PCoA
ACoA
107
Manual
NP-GAR
KM
P-GAR
ACWE
Figure 4.14: Comparison of the 3D-RA segmented vascular models with the reference techniques
considered in the evaluation study (ACoA, PCoA, MCA, basilar, and ICA). The columns show the
results from the manual, our non-parametric (NP-GAR), K-Means (KM), parametric (P-GAR) and
Active Contour Without Edges (ACWE) methods, respectively.
108
MCA-2
MCA-1
PCoA
ACoA-2
ACoA-1
Manual
NP-GAR
KM
P-GAR
ACWE
Figure 4.15: Comparison of the CTA segmented vascular models with the reference techniques
considered in the evaluation study (ACoA-1, ACoA-2, PCoA, MCA-1, MCA2). The columns show
the results from the manual, our non-parametric (NP-GAR), K-Means (KM), parametric (P-GAR),
and Active Contour Without Edges (ACWE) methods, respectively.
109
1
3DRA
0.9
1
NP-GAR
KM
P-GAR
ACWE
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ACoA
PCoA
MCA
basilar
CTA
0.9
DSC
DSC
NP-GAR
KM
P-GAR
ACWE
0
ICA
ACoA-1
ACoA-2
PCoA
MCA-1
MCA-2
Figure 4.16: DSC values for the methods considered in the evaluation study: our non-parametric
Geodesic Active Regions (NP-GAR), K-Means (KM), Parametric Geodesic Active Regions (P-GAR),
and Active Contours Without Edges (ACWE). The left figure presents the results for the 3D-RA and
the right figure for the CTA datasets, respectively.
Table 4.3: Mean and standard deviation of the DSC values for the methods considered in the
evaluation study: Non-parametric Geodesic Active Regions (NP-GAR), K-Means (KM), Parametric
Geodesic Active Regions (P-GAR), and Active Contours Without Edges (ACWE).
NP-GAR
3D-RA
CTA
KM
90.05 ± 4.18
P-GAR
ACWE
80.73 ± 19.72 69.03 ± 28.19
77.39 ± 5.27
71.69 ± 13.72 67.54 ± 24.62 60.50 ± 27.03
63.63 ± 9.63
Table 4.4: Statistics for the study of the similarity between manual and model-based segmentations:
Non-parametric Geodesic Active Regions (NP-GAR), Parametric Geodesic Active Regions (P-GAR),
and Active Contours Without Edges (ACWE). The first row for each model represents the mean and
the second row, the standard deviation of the quantities over the datasets (measured in mm).
3D-RA
PE(%)
μD>0 [mm]
σD>0 [mm]
D0.95 [mm]
D0.99 [mm]
NP-GAR
17.88 ± 6.80
0.42 ± 0.12
0.49 ± 0.41
0.85 ± 0.21
2.99 ± 3.02
P-GAR
41.93 ± 31.35
0.48 ± 0.12
0.45 ± 0.41
0.97 ± 0.27
2.71 ± 3.07
ACWE
36.64 ± 7.02
0.61 ± 0.10
0.47 ± 0.32
1.06 ± 0.13
2.99 ± 2.71
PE(%)
μD>0 [mm]
σD>0 [mm]
D0.95 [mm]
D0.99 [mm]
NP-GAR
42.63 ± 17.39
0.40 ± 0.09
0.30 ± 0.05
0.93 ± 0.15
1.34 ± 0.32
P-GAR
52.58 ± 26.28
1.40 ± 2.07
1.54 ± 2.57
4.83 ± 8.23
6.27 ± 9.69
ACWE
52.75 ± 10.26
0.55 ± 0.10
0.39 ± 0.23
1.37 ± 0.74
1.86 ± 1.26
CTA
110
model without edges (ACWE) presented the highest probability of error (41.93 and
36.64%, respectively). The lowest mean distance error corresponded to our method,
that was close to the in-plane resolution (0.40 mm). Besides, in our model, the 95%
of the points had a distance to the ground truth less or equal than twice the in-plane
voxel size (0.85 mm). On the other hand, the value of D0.95 for the parametric model
(P-GAR) and the model without edges (ACWE) was equal to 0.97 and 1.06 mm,
respectively.
In the CTA evaluation set, the quality of the segmentations resulted strongly dependent on the presence of bone tissue in the dataset that is evaluated, as shown
in Figure 4.16. The DSC coefficient indicated an excellent overlap, with the same
trends appreciated in the 3D-RA evaluation set, if the image presented just vessel
and background tissues (ACoA-1 and MCA-1), or the bone tissue was not located in
vessel proximities (ACoA-2). However, in the cases with bone tissue located next to
vessel tissue (PCoA and MCA-2), the DSC coefficient was lower than the 70%. In
the cases without bone proximity, the best overlap index was achieved by K-Means
method (KM), although, our non-parametric model (NP-GAR) achieved the best average overlap index (DSC average = 71.69%), as in the cases with bone proximity
(PCoA, MCA-2), our non-parametric method improved considerably the performance
of K-Means.
In Table 4.4, it can be appreciated that the probability of error (PE) was usually
higher than in 3D-RA. Our non-parametric method (NP-GAR) still showed the lowest
probability of error (PE = 40.77%). In the parametric model (P-GAR) and the model
without edges (ACWE), the probability of error was greater than the 50%. The mean
distance error and the standard deviation are comparable to 3D-RA in our method
(NP-GAR) and the model without edges (ACWE). The lowest mean distance error
corresponded to our method (NP-GAR) resulting close to the in-plane resolution. The
parametric model (P-GAR) showed a mean distance error next to 4 voxels (1.40 mm)
with a standard deviation of 1.54 mm. In our model (NP-GAR), the 95% of the points
have a distance to the ground truth less than 0.90 mm while in the model without
edges and the parametric model D0.95 equals to 1.37 and 4.83 mm, respectively.
4.7
In this Chapter, we have devised an automatic method for segmentation of cerebral vascular structures with application to the generation of 3D models of cerebral aneurysms
in 3D-RA and CTA. The method consists in an implicit deformable model that evolves
minimizing an energy functional that incorporates region-based information estimated
from the probabilistic vessel enhancement filter presented in Chapter 3.
In most related works, the estimation of the probabilities for region description
assumes that the distribution of the feature space can be described using parametric
models. More in particular, the probabilities are usually computed from the intensity
distribution using a Gaussian Mixture Model (GMM). In our application, vascular
111
tissue usually occupies a very small proportion compared to the background and bone
tissue image volume. Moreover, depending on the aneurysm size and location, this
proportion hugely varies from one dataset to another. For these reasons, the Gaussian
model can lead to inaccurate probability estimations and introduce severe errors in the
segmentation, as shown in the results section. As alternative, we propose the use of a
non-parametric method for probability estimation. This way, no constraint is imposed
over the statistical estimation, that is performed from a significative sample of points
in the feature space.
The feature space is composed of up to second-order differential image descriptors
in a multi-scale framework, appropriate for the description of the local image structure
and the discrimination of ridges from other geometric features. The use of high-order
information provide a richer description of the different tissues in medical imagery
than the description solely provided by the intensity distributions. In this work, we
explored the performance of two different feature spaces specifically oriented to the
discrimination of tissues involving tubular structures, and the feature space including
the second-order irreducible differential invariants. We found that for 3D-RA the three
feature spaces provided a similar performance although the computation time needed
in kd-tree search was notably reduced for the algorithm associated to the differential
invariants feature space. In the case of CTA, the performance of the three feature
spaces at vessel tissue was similar. However, the non-parametric method showed to
erroneously classify bone tissue with high values for vessel probability. This error was
minimized for the model associated to the prototype space. Therefore, the differential
invariants in the case of 3D-RA and the prototype space in the case of CTA were
selected in our non-parametric GAR segmentations.
Due to the large variability of patterns found in the feature space, the most sensitive
part of the algorithm is constituted by the selection of the training set. In this work, a
protocol for the selection of the datasets and point sampling for training set generation
has been proposed. The process used for training set generation determines the decision
boundary of our non-parametric method. It remains an open issue how to construct
the training set with the highest generalization for our application, due to the high
variability of patterns existing in the feature spaces.
In the 3D-RA datasets we have shown that the decision boundary includes the
region of partial volume features and vessels with low contrast dose inside. In the case
of CTA datasets, the decision boundary includes partial volume features, vessels with
low contrast dose inside, aneurysm with high contrast dose inside, and bone tissue with
low calcification. In both cases, segmentations located at vessel to background partial
volume voxels are robust due to Geodesic Active Regions include both statistical and
gradient based information in the evolution of the model. However, depending on
the samples used for training in CTA, the segmentations could perform quite different
in the segmentation of thin vessels or bone tissues. Nevertheless, these arteries are
usually of low interest in both clinical and technical applications like computational
fluid dynamics (CFD) or shape characterization. Moreover, in the great majority of
cases, the bone tissue can be removed with surface editing tools.
112
In this work, the proposed methodology was thoroughly evaluated against manual segmentations as ground-truth and compared to other techniques. The proposed
method was applied to a total of 10 clinical datasets of cerebral aneurysms representing
the most typical geometries observed in our 3D-RA and CTA databases in the most
frequent locations. First, a qualitative evaluation of the segmentation results was visually assessed. For the quantitative analysis, both an overlap coefficient (DSC) and the
statistics over the distance (D) from the ground-truth to the surface representation of
the segmentations were computed. With our technique (NP-GAR), the average DSC
indexes were equal to 90.05 and 71.69% in 3D-RA and CTA, respectively. The mean
distance error was close to the in-plane resolution (0.42 and 0.40 mm, respectively)
and the 95% of the points of the model had a distance to the ground truth less than
0.85 and 0.93 mm, respectively. In comparison, the average DSC indexes for the active
contours without edges model (ACWE) were equal to 77.39 and 63.63%, respectively
and the average distance to the ground truth were equal to 0.61 and 0.55 mm, respectively. The quantities computed from the parametric geodesic active contours (P-GAR)
indicated a much lower performance. Thus, both quantitatively and qualitatively, our
technique has proven to yield more accurate results when compared to these other
competing techniques.
Therefore, our method has shown to be a suitable technique for the automatic
segmentation of cerebral aneurysms in 3D-RA once the non-parametric method has
been properly trained. In the case of CTA, our algorithm has shown to outperform
reference existing techniques. However, as we mentioned in Chapter 3 this work has
identified some problems that need to be overcome. In particular, the segmentation of
vessels and cerebral aneurysms in contact to bone tissue constitutes a challenging open
issue that has been tackled in few works by using postprocessing, surface editing tools
or combining complementary information from other image modalities.
As future directions, it would be interesting to work in further improvements of
the segmentation algorithm. Just to name a few, these improvements may include the
combination of non-parametric Geodesic Active Regions with the algorithms proposed
in the literature for thin vessels segmentation shown in Chapter 2 or the incorporation
of prior shape information in the deformable model. In addition, due to the generality
of the proposed framework, our method could be applied to the segmentation of other
organs and/or medical imaging modalities by adapting the method to the suitable data.
Finally, our method for segmentation could be incorporated in clinical practice for the
quantification of the clinical parameters needed for treatment selection and surgical
planning. In addition, the segmentations could be used in the generation of 3D models
suitable for Computational Fluid Dynamics simulation or geometry characterization.
Part II
Chapter 5
Diffeomorphisms in Computational
Anatomy
Abstract
Computational Anatomy is a recently emerging discipline that aims for the
study of shape variability from anatomical images. Anatomical information
is encoded by the spatial transformations existing between the images and a
template selected as anatomical reference. Statistics on spaces of transformations allow modeling the anatomical variability existing across a population
of samples. In the absence of a more justified physical model for inter-subject
variability some particular manifolds of diffeomorphisms provide a suitable
mathematical setting for the analysis of the anatomical variability. In this
Chapter, the fundamental aspects of the differential and algebraic structure of
infinite dimensional Riemannian manifolds of diffeomorphisms are thoroughly
studied. In addition, the variational problems for diffeomorphic registration
proposed in Computational Anatomy in closest relationship with the methods
developed in this Thesis are reviewed. We end with the study of the efficient
second-order methods proposed in the literature simultaneously to our work
that circumvent the high computational complexity and memory requirements
of the classical large deformation paradigm for diffeomorphic registration.
5.1. Introduction
116
Contents
5.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
5.2
Riemannian manifolds of diffeomorphisms . . . . . . . . . . . . . .
119
5.3
5.4
5.1
5.2.1
Differentiable structure on infinite dimensional manifolds of diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.2
The manifold of H s -diffeomorphisms
5.2.3
The ”Lie” group of H s -diffeomorphisms . . . . . . . . . . . . . . . . . 124
Diffeomorphic Registration Methods
. . . . . . . . . . . . . . . . . . 121
. . . . . . . . . . . . . . . . .
127
5.3.1
Large Deformation Kinematics for diffeomorphic registration . . . . . 128
5.3.2
Large Deformation Diffeomorphic Metric Mapping (LDDMM)
5.3.3
Numerical aspects of the LDDMM method . . . . . . . . . . . . . . . 130
5.3.4
LDDMM from Jacobi Fields
5.3.5
Efficient algorithms for diffeomorphic registration
. . . . 129
. . . . . . . . . . . . . . . . . . . . . . . 134
. . . . . . . . . . . 134
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
Introduction
The interest for understanding shape differences in related anatomies can be traced
back to the beginnings of modern science. More concretely, in the treatise On growth
and form (1917), D’Arcy Thompson provided one possible vision on how to approach
the study of anatomical differences. In the chapter titled The Comparison of Related
Forms he explored the degree to which differences in form of related animals could
be described by means of relatively simple mathematical transformations showing up
the role of physical and mechanical laws in determining the shape and structure of
anatomical structures in life organisms:
”In a very large part of morphology, our essential task lies in the
comparison of related forms rather in the precise definition of each;
and the deformation of a complicated figure may be a phenomenon
easy of comprehension, though the figure itself may have to be left
unanalyzed and undefined. This process of comparison, of recognizing
in one form a definite permutation or deformation of another, apart
altogether from a precise and adequate understanding of the original
‘type’ or standard of comparison, lies within the immediate province
of mathematics and finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of
Coordinates, on which is based the Theory of Transformations.”
117
Chapter 5. Diffeomorphisms in Computational Anatomy
Computational Anatomy aims for the study of shape variability in anatomical structures following D’Arcy Thompson vision. Anatomical information is encoded by the
spatial transformations existing between the images and a template selected as anatomical reference [96, 97, 67, 170]. The statistical analysis of these transformations allows
modeling the anatomical variability existing across a population [234, 171, 97, 236,
123, 243, 170, 172]. Different models of anatomical variability have been successfully
used in order to identify anatomical differences between healthy and diseased individuals or improve the diagnosis of pathologies [233, 258, 52, 257, 201]. Moreover,
models of growth have been built for the assessment of the anatomical change over
time [232, 19, 170, 87, 55]. Although Computational Anatomy has traditionally focused on the study of neuroanatomy, there exist recent efforts for the extension to the
study of variability in all organ systems [20, 104].
In Computational Anatomy, mathematical models of anatomies are defined via orbits associated to the action of transformation groups on a representative image of the
anatomy. The group action allows to generate a whole family of new images belonging
to the same anatomy than the reference image. This family of images, introduced by
Grenander et al. in [96], is called deformable template. As groups of transformations
can be also seen as Riemannian manifolds, the differentiable Riemannian structure allows measuring distances between the elements in the group. Hence, the deformable
template can be endowed with a metric distance by means of the group action. This
allows to translate the study of anatomies from images to their associated transformations.
Finite dimensional Lie groups of rigid or affine transformations have been used for
characterizing the orbits of object configurations in Computer Vision applications [128].
However, the use of non-rigid transformations with a large number of degrees of freedom
becomes necessary for dealing with the complexity of biological shapes like the ones
found on human neuroanatomy. In the absence of a more justified physical model
for inter-subject anatomical variability and under the reasonable assumption that the
deformation model responsible of organ growth is related to smooth and invertible
transformations, some particular groups of diffeomorphisms (i.e. differentiable maps
with differentiable inverse) provide a suitable setting for the deformable template model
in the study of both anatomical variability and growth.
Diffeomorphic registration is formulated as the problem of finding the path of diffeomorphisms with minimal energy that smoothly transforms the source into the target
image. The origins of such problem can be found on a physical context, more concretely, in Fluid Dynamics, as the deformation of the image domain can be simulated
from the deformation of a viscous fluid subject to external forces action [6, 116]. In
Computational Anatomy, diffeomorphic registration is approached in the large deformation setting as a variational problem defined from the minimization of an energy
functional involving the following elements
• An image matching metric that measures the similarity between the images after
registration
5.1. Introduction
118
• The characterization of the diffeomorphic transformations
• A regularization constraint to favor stable numerical solutions
• An optimization technique to search for the optimal transformation in the space
of valid diffeomorphisms.
The Large Deformation Diffeomorphic Metric Mapping method (LDDMM) is considered the reference paradigm for diffeomorphic registration in Computational Anatomy [67, 239, 23]. In this registration scenario, the image matching metric is usually
selected as the sum of squared intensity differences. Transformations are represented
as end points of paths of diffeomorphisms parameterized by time-varying vector field
flows defined on the tangent space of a convenient Riemannian manifold of diffeomorphisms. Regularization is imposed on the energy associated to the length of the path,
and classical gradient descent is used for numerical optimization.
In the last years, some variations in the definition of the elements of the LDDMM
variational problem have been proposed providing different algorithms for diffeomorphic registration:
Variations on image matching energy
Since Christensen et al. [42] different authors have proposed inverse consistent modifications of the LDDMM variational problem incorporating a source to target matching
symmetry during registration [123, 14, 22]. These algorithms have shown to improve
forward and backward correspondences improving the suitability of the transformations for statistical analysis. In addition, Avants et al. have introduced landmarkbased diffeomorphic registration and cross-correlation in these inverse consistent frameworks [15, 16]. Lorenzen et al. have used information theory measures in the image
matching term for multimodal diffeomorphic registration [154]. Cao et al. have proposed a variational problem for vector valued images in order to deal with registration
of diffusion tensor MRI [31].
Variations on diffeomorphisms characterization and regularization
Younes et al. have proposed to restrict the domain of transformations to diffeomorphisms belonging to paths that fulfill the momentum conservation [282]. In this case,
regularization is imposed on the norm of the initial vector fields associated to geodesic
paths. Ashburner et al. have proposed to restrict the domain of transformations to
diffeomorphisms belonging to paths parameterized by stationary vector field flows [11].
These paths are identified with the one-parameter subgroups of a convenient group of
diffeomorphisms. Regularization is imposed on the norm of the infinitesimal generator
of the one-parameter subgroup. Simultaneously, the stationary parameterization has
been included by Vercauteren et al. in a variant of Demons algorithm for diffeomorphic
registration [251].
119
Variations on optimization
Efficient second-order methods have been combined with the use of the stationary
parameterization of diffeomorphisms in order to improve robustness and efficiency during optimization. Ashburner et al. have proposed a Levenberg-Maquardt optimization
strategy in stationary-LDDMM diffeomorphic registration [11]. Vercauteren et al. have
proposed an efficient second-order optimization strategy in diffeomorphic Demons [251].
Variations on the deformable template model
Although this Thesis deals with classical Grenander’s deformable template model where
the group action is defined on dense images, it should be noted that Grenander’s model
is extensible to representations of anatomical shapes like landmarks, curves and surfaces by simply defining a convenient group action on these entities. Diffeomorphic
registration between these shape representations has been also approached in the LDDMM framework. In the case of landmarks, three different variational formulations
have been proposed in the literature [124, 28, 164]. Diffeomorphic registration of curves
and surfaces has been approached within the theory of currents in [90, 242].
This Chapter is focused on the study of diffeomorphisms in Computational Anatomy. In Section 5.2 we study those fundamental aspects of Riemannian geometry
related to infinite dimensional Riemannian manifolds and groups of diffeomorphisms
involved in Computational Anatomy. In addition, Section 5.3 overviews diffeomorphic
registration algorithms proposed in Computational Anatomy in closest relationship
with the methods developed in this Thesis. Special attention is paid to the theoretic
and implementation details found on the LDDMM method and recent efficient registration algorithms.
5.2
Riemannian manifolds of diffeomorphisms
This section provides an overview of the fundamental aspects of Riemannian geometry
regarding to infinite dimensional manifolds and groups of diffeomorphisms used in
Computational Anatomy. This review has been done after compiling and carefully
analyzing the results shown on the works by some of the authors that have extensively
contributed to this new and fascinating field of pure mathematics. More details can
be found in the works of differential calculus on infinite dimensions and mathematical
physics by Arnold, Omori, Kriegl, Schmid, Glockner and collaborators [6, 183, 136,
214, 91] and in the works related to Computational Anatomy [170, 116, 49, 172, 7].
5.2. Riemannian manifolds of diffeomorphisms
5.2.1
120
Differentiable structure on infinite dimensional
manifolds of diffeomorphisms
The study of infinite dimensional manifolds is more complicated than the case of finite
dimensions. A finite dimensional differentiable manifold is an abstract mathematical
space that is locally homeomorphic 3 , and therefore topologically equivalent, to an Euclidean space, Rn . In consequence, the local properties of the manifold can be studied
in terms of linear spaces translating the methods of calculus in linear spaces to differentiable manifolds. On the other hand, infinite dimensional differentiable manifolds are
locally homeomorphic to infinite dimensional metric vector spaces showing a complex
differentiable structure.
Manifolds of diffeomorphisms are defined from the set
Dif f (Ω) := {ϕ : Ω → Ω, ϕ and ϕ−1 smooth homeomorphism}
(5.1)
together with the group operations
• Composition
μ : Dif f (Ω) × Dif f (Ω) → Dif f (Ω), μ(ϕ1 , ϕ2 ) = ϕ2 ◦ ϕ1
• Inverse
ν : Dif f (Ω) → Dif f (Ω), ν(ϕ) = ϕ−1
where the composition operation μ provides two different group homomorphisms
• Left composition
Lϕ : Dif f (Ω) → Dif f (Ω), Lϕ (ρ) = ρ ◦ ϕ
• Right composition
Rϕ : Dif f (Ω) → Dif f (Ω), Rϕ (ρ) = ϕ ◦ ρ
and Ω is a compact simply connected differentiable manifold. Dealing with the noncompact case is also possible, although it results much more complicated and is out of
the scope of application domain of Computational Anatomy.
The structure of differentiable manifold in Dif f (Ω) is defined by endowing with
a Riemannian metric to the space of smooth vector fields in Ω, X (Ω). This structure provides a local homeomorphism from every element ϕ in the manifold to the
tangent space Tϕ (Dif f (Ω)) at that element, which is isomorphic to the algebra of
right-invariant vector fields in Dif f (Ω), Xr (Dif f (Ω)). The right-composition Rϕ provides the canonical isomorphism between all the tangent spaces and this algebra. As
3
Homeomorphisms are defined in Topology as continuous bijective mapping with continuous inverse.
121
in the finite dimensional case, the tangent space represents the closest approximation
of the manifold by a vector space on a neighborhood of ϕ. The elements related to
the differentiable structure (charts, tangent bundle, differentiable curves, vector fields,
differential maps...) can be defined analogously to the finite dimensional case [62].
Depending on the degree of differentiability of the set of diffeomorphisms (i.e. the
meaning of the term ”smooth” in Equation 5.1), different degrees of smoothness can
be identified in the corresponding vector fields and therefore, different structures of
differentiable manifold can be defined in Dif f (Ω) [214]. For example,
• The set of C ∞ -diffeomorphisms Dif f ∞ (Ω) together with the space of C ∞ -vector
fields in Ω is a Frechet space.
• The set of C k -diffeomorphisms Dif f k (Ω) with the space of C k -vector fields is a
Banach space (k < ∞).
• The set of Sobolev H s -diffeomorphisms Dif f s (Ω) with the space of H s -vector
fields is a Hilbert space (s > 12 dim(Ω), dim(Ω) 1).
Dealing with C ∞ -diffeomorphisms is problematic as these spaces lack of easy generalizations of the inverse and implicit function theorems. Fortunatelly, suitable extensions
for both theorems are available for both C k and H s -diffeomorphisms. This makes these
spaces appropriate candidates for physical, and therefore, Computational Anatomy applications. However, the existence of a complete scalar product makes Hilbert spaces
preferable. Thus, the properties of finite dimensional vector spaces can be naturally
extended to these infinite dimensional spaces. Furthermore, the theorems of existence
and uniqueness of PDE solutions do hold for Hilbert spaces.
The Hilbert differentiable structure was originally studied in Physics in the context
of Continuum Mechanics [68, 6, 116]. The analogies existing between the motion of
a continuum system and diffeomorphic registration allowed to translate the setting
for working with Dif f s (Ω) from Physics to Computational Anatomy [6, 239]. In the
following, we will focus on the study of Dif f s (Ω) in relation to these disciplines.
5.2.2
The manifold of H s -diffeomorphisms
Characterization of diffeomorphisms
Diffeomorphisms are characterized as transformations belonging to a smooth path on
Dif f s (Ω)
φ : [0, 1] → Dif f s (Ω), t → φ(t)
(5.2)
parameterized as the solution of the transport equation
φ̇(t) = v(t, φ(t))
(5.3)
Tφ(t)(Diff(Ω ))
122
φ (1) = ϕ
v(t, φ(t))
v(0,φ (0))
φ (t)
Diff( Ω )
φ (0) = id
Figure 5.1: Diagram of diffeomorphism characterization. Path in Dif f s (Ω) starting at φ(0) = id
parameterized from time-varying vector field flow v(t, φ(t)).
with initial condition φ(0) = id (identity element), where
v : [0, 1] → T (Dif f s (Ω)), t → v(t, φ(t)) ∈ Tφ(t) (Dif f s (Ω))
(5.4)
is a time-varying flow curve of smooth vector fields in the tangent bundle constituted by
the directional derivatives associated to the path at each point (Figure 5.1). Therefore,
diffeomorphisms can be computed as solutions of non-stationary ODEs. The Hilbert
differentiable structure guarantees that the solution to the transport equation is a path
of diffeomorphisms in Dif f s (Ω).
Riemannian metric. Geodesics.
The Riemannian metric in Dif f s (Ω) is constructed from the scalar product defined at
the identity element, id. Thus, the scalar product of v, w ∈ Tid (Dif f s (Ω)) is defined
from
v, wTid (Dif f s (Ω)) = Lv, LwL2
(5.5)
where L is a linear invertible differentiable operator related to the physical deformation
model imposed on Ω. From this definition, the Riemannian metric is extended to the
whole tangent bundle by right-translation. Thus, given v, w ∈ Tϕ (Dif f s (Ω))
v, wTϕ (Dif f s (Ω)) = (dRϕ−1 )ϕ v, (dRϕ−1 )ϕ wTid (Dif f s (Ω))
(5.6)
where (dRϕ−1 )ϕ denotes the differential map of Rϕ−1 at Tϕ (Dif f s (Ω)). From this
construction, it follows that this metric is invariant under right composition (rightinvariant). However, invariance under left composition is not preserved.
123
The distance d : Dif f s (Ω) × Dif f s (Ω) → R+ associated to this Riemannian metric
is given by
d(ϕ1 , ϕ2 ) = min{L(φ) | φ smooth path between id and ϕ2 ◦ ϕ−1
1 }
(5.7)
where L(φ) corresponds to the length of the path φ
1
φ̇(t)V (t) dt =
L(φ) =
0
1
(dRφ(t)−1 )φ(t) v(t, φ(t))V dt =
1
v(t)V dt
(5.8)
0
0
where V (t) and V denote the tangent spaces Tφ(t) (Dif f s (Ω)) and Tid (Dif f s (Ω)), respectively and · V (t) is the norm associated to the scalar product ·, ·V (t) . The energy
functional
E(φ) =
0
1
φ̇(t)2V (t) dt
=
0
1
v(t)2V dt
(5.9)
represents the kinetic energy of an hypothetical particle moving on the manifold following that path. Paths minimizing this energy, and therefore minimizing their length,
are called geodesic curves [62]. In this case, geodesic paths are associated to a rightinvariant metric and, therefore, they constitute right-geodesics. The length of the
geodesic is identified with the right-invariant distance existing between the extremal
points of the path. In the case of geodesic paths starting at the identity element, the
energy of the geodesic provides a right-invariant measure of the amount of deformation
associated to the diffeomorphism φ(1).
At this point, it is important to remark that Dif f s (Ω) is geodesically complete [238].
This means that the manifold has no singular point that can be reached in a finite time.
Therefore, given two points on the manifold, there always exists at least one minimizing
geodesic between them (Hopf-Rinow-DeRham theorem).
The Riemannian exponential and logarithm maps
Given ϕ ∈ Dif f s (Ω) and v0 ∈ Tϕ (Dif f s (Ω)), there exists a unique right-geodesic
φ(t) defined in some interval (−, ) such that φ(0) = ϕ and φ̇(0) = v0 [68, 62]. The
Riemannian exponential map at point ϕ, expϕ : Tϕ (Dif f s (Ω)) → Dif f s (Ω), is defined
as the diffeomorphism reached by the geodesic φ at time 1, i.e. expϕ (v0 ) = φ(1). In
the following, without loss of generality we are considering geodesics starting at the
identity element, φ(0) = id and v0 ∈ V .
In Fluid Dynamics, the configuration of right-geodesic paths is governed by the
Euler-Poincare equation for diffeomorphisms (EPDiff equation) [49]. Thus, given φ(t)
right-geodesic such that φ̇(t) = v(t, φ(t)) then
124
∂(L† Lv(t))
+ v(t) · ∇(L† Lv(t)) + ∇T v(t) · (L† Lv(t)) + (L† Lv(t)) · div(v(t)) = 0 (5.10)
∂t
The EPDiff equation is equivalent to the conservation of the momentum L† Lv(t)
through right-geodesic paths [167]
L† Lv(t) = Dφ−1 (t)T · (L† Lv(0)) ◦ φ−1 (t) · det(Dφ−1 (t))
(5.11)
This result was also reached by Miller et al. from the extension of the Lagrangian
momentum conservation in classical mechanics to diffeomorphisms [172]. However, the
existence of such conservation laws goes back to Noether’s theorem in 1918 [181].
From the momentum conservation, the right-geodesic characterized by a given v0 ∈
V is defined as the solution of the transport equation φ̇(t) = v(t, φ(t)) where v(t)
is the unique non-stationary flow of vector fields in V that fulfills the momentum
conservation equation for v(0) = v0 . This procedure is popularly known as geodesic
shooting as, given a point in the manifold and a direction, the points on the geodesic
can be computed by shooting the initial point on the given direction a determined
amount of time [116, 21]. By definition, the Riemannian exponential map is identified
with the point shooted at time t = 1.
The Riemannian exponential is locally defined on a neighborhood of the origin
0 ∈ V . In this neighborhood, the exponential map is a local diffeomorphism [137].
The inverse is called Riemannian logarithm map. The existence and uniqueness is
guaranteed from the fact that, although in infinite dimensional spaces there is no inverse
function theorem, the Nash-Moser theorem provides a generalization of the inverse
function theorem in Banach spaces. Unfortunately, no algorithm for the computation
of v(0) for a given diffeomorphism has been provided yet. Moreover, even the feasibility
of such algorithm is still unknown.
The lack of left-invariance of the Riemannian metric together with the absence of a
method for the computation of the Riemannian logarithm map pose some theoretical
and practical limitations for statistical calculus on Riemannian manifolds of diffeomorphisms that will be detailed in Chapter 8. As alternative, some authors have remarked
the possibility of relying on the algebraic properties of the ”Lie” group structure defined
on Dif f s (Ω) [7, 8].
5.2.3
The ”Lie” group of H s -diffeomorphisms
Algebraic structure
The elements in Dif f s (Ω) constitute an algebraic group together with the operations
• Composition
μ : Dif f s+k (Ω) × Dif f s (Ω) → Dif f s (Ω), μ(ϕ1 , ϕ2 ) = ϕ2 ◦ ϕ1
125
• Inverse
ν : Dif f s (Ω) → Dif f s (Ω), ν(ϕ) = ϕ−1
and the identity element id. Given an element ϕ ∈ Dif f s (Ω), the composition operation μ provides two different group homomorphisms
• Left composition
Lϕ : Dif f s+k (Ω) → Dif f s (Ω), Lϕ (ρ) = ρ ◦ ϕ
• Right composition
Rϕ : Dif f s (Ω) → Dif f s (Ω), Rϕ (ρ) = ϕ ◦ ρ
The inverse and the right composition are C ∞ -mappings with the differentiable structure of Riemannian manifold. However, the left composition is only a C k -mapping and,
in consequence, the group composition is not C ∞ . Therefore, although the definition
of the group operations are quite similar to the finite dimensional case, strictly speaking, Dif f s (Ω) is not a Lie group in the same sense as in finite dimensions. However,
this infinite dimensional group is an Inverse Limit of Hilbert (ILH) Lie group and the
properties of ILH-groups allow to formally work with a Lie group structure analogously
to the finite dimensional case [183].
The Lie algebra associated to Dif f s (Ω) corresponds to the space of right-invariant
vector fields on Dif f s (Ω), Xr (Dif f s (Ω)). The Lie bracket is defined from the Lie
derivative of vector fields [v, w] = Lv (w) − Lw (v), where Lv (w) = ∂w/∂v. It should
be noted that this operator is not closed as smoothness is lost through differentiation.
Therefore, Xr (Dif f s (Ω)) is not a Lie algebra in the finite dimensional sense. However,
as happened with the group, this algebra is an Inverse Limit of Hilbert (ILH) Lie
algebra and the properties of ILH-algebras allow to formally work with the Lie algebra
structure analogously to the finite dimensional case [183].
One-parameter subgroups
One-parameter subgroups are defined as algebraic structures on Dif f s (Ω) represented
by a family of diffeomorphisms (φ(s))s∈R+ where φ is a Lie group homomorphism,
φ : R+ → Dif f s (Ω), and therefore, the group operations between R+ and the subgroup
are preserved. Thus, φ(s + t) = φ(s) ◦ φ(t) and φ(−s) = φ(s)−1 or equivalently,
φ(0) = id. As R+ is conmutative, group elements belonging to the same one-parameter
subgroup conmute. One-parameter subgroups can be seen as differentiable paths of
diffeomorphisms. The derivative at zero w = φ̇(0) is called the infinitesimal generator
of the subgroup and parameterizes the path via the transport equation φ̇(t) = w(φ(t))
with initial condition φ(0) = id. With this parameterization diffeomorphisms can be
computed as solutions of stationary or autonomous ODEs.
126
The group Exponential and Logarithm maps
The paths of diffeomorphisms that can be parameterized using stationary vector field
flows are exactly the one-parameter subgroups. In this case, the solution of the transport equation at time 1 is identified with the group exponential map, Exp : V →
Dif f s (Ω), Exp(w) = φ(1) where φ̇(t) = w(φ(t)).
The group logarithm map is defined in a neighborhood of the identity as the unique
vector field w ∈ V such that Exp(w) = ϕ. In the finite dimensional case, the existence
and uniqueness of the group logarithm comes from the differentiability of the exponential map in the domain of definition and the fact that the differential of the exponential
map equals to the identity in this domain. In the group of diffeomorphisms,
φ(t) − id
Exp(tw) − Exp(0)
= lim
=w
t→0
t→0
t
t
lim
(5.12)
where φ is the one-parameter subgroup with infinitesimal generator w. Therefore, under
the assumption that the group exponential is differentiable, its differential at zero is
the identity map and the existence and uniqueness of the logarithm holds. Moreover,
algorithms for the computation of the group exponential and logarithm maps have
been provided by Arsigny et al. [8]. As these algorithms will be used throughout the
rest of this Thesis, we detail here the details for its implementation.
The algorithm for the computation of the group exponential consists in a stable
implementation of autonomous ODEs solutions. It is based on the fact that, given w ∈
V and N ∈ N such that w/2N is small enough 4 , then the first order approximation of
the exponential map Exp(w/2N ) ≈ x+w/2N can be used to compute group exponential
map Exp(w) from N recursive compositions of this approximation. Thus, denoting with
ϕ1 = Exp(w/2N ) ◦ Exp(w/2N ) the group exponential map is provided by the formula
ϕN ≈ Exp(w) where ϕi = ϕi−1 ◦ ϕi−1 , i = 1, ..., N
(5.13)
This algorithm resembles the Scaling and Squaring method for matrices [114]. Hence, in
this Thesis it will be referred to the Scaling and Squaring method for the computation
of the exponential in diffeomorphism groups.
The algorithm for the computation of the group logarithm map is formulated from
the generalization of the Inverse Scaling and Squaring method (ISS) from matrices to
−N
diffeomorphisms. It is based on the identity Log(ϕ) = 2N · Log(ϕ2 ), ϕ ∈ Dif f s (Ω).
−N
If N is such that Log(ϕ2 ) is close to the identity, this expression can be approxi−N
−N
can be computed from N recursive
mated by ϕ2 − id. The diffeomorphism ϕ2
taking of squared roots. The square root of a transformation ϕ is approached from the
4
In this Thesis N = 10 is chosen as appropriate value
127
1
E(T ) = T ◦ T − ϕ22
2
(5.14)
with initial condition 12 (ϕ − id) and energy gradient given by ∇E(T ) = (DT )t ◦ T · (T ◦
T − ϕ).
Baker-Campbell-Hausdorff formula
The Baker-Campbell-Hausdorff (BCH) formula is a fundamental relationship existing
between the group exponential and logarithm maps in general Lie groups. This formula
provides a formal expression for the computation of u = Log(Exp(v) · Exp(w)) in terms
of Lie brackets [., .]
u = v + w + 1/2[v, w] + 1/12[v, [v, w]] − 1/12[w, [v, w]] − 1/24[w, [v, [v, w]]]
(5.15)
+O((v + w)5 )
Intuitively, the BCH formula measures how much Log(Exp(v) · Exp(w)) differs from
v + w due to the non-commutativity of the product operation in the group. This
formula provides useful approximations in Lie calculus.
As a final remark, it should be noted that set of diffeomorphisms obtained with
the stationary parameterization does not span all diffeomorphisms in Dif f s (Ω). In
fact, the group exponential map is not onto [94]. Unfortunately, this means that
the group exponential map is not differentiable. Hence, working with the stationary
parameterization poses some theoretical limitations. Arsigny et al. pointed out that
this limitation could be due to these spaces may provide a too large structure and
that a possible solution may be considering to deal with smaller infinite dimensional
spaces (for example those called locally exponential) where the usual properties of the
exponential in finite dimensions still hold [7, 91]. It remains a question of active research
to identify the adequate spaces of diffeomorphisms in which the group exponential and
logarithm maps could be used in a rigorous mathematical way and to investigate if
these spaces are appropriate for Computational Anatomy applications.
5.3
Diffeomorphic Registration Methods
In this section we describe the algorithms for diffeomorphic registration most related to
the methods developed in this Thesis. We start with the large deformation kinematic
model that introduced large diffeomorphic transformations in registration. Then, the
Large Deformation Diffeomorphic Metric mapping and the most remarkable implementation details are thoroughly discussed. Finally, we focus on the methods proposed in
the literature for efficient diffeomorphic registration.
5.3. Diffeomorphic Registration Methods
5.3.1
128
Large Deformation Kinematics for diffeomorphic
registration
Early diffeomorphic registration approaches are based on the viscous fluid registration
method of Christensen et al. [44]. In this framework, diffeomorphisms are represented
by a finite composition of transformations
ϕ(x) = ϕ1 (x) ◦ ϕ2 (x) ◦ ... ◦ ϕN (x)
(5.16)
where ϕi (x) is represented by its corresponding displacement field ui (x), ϕi (x) = x +
ui (x). Displacement fields u(x) are parameterized by flows of time varying vector fields
v(x, t) defined on Ω. The parameterization is described by the PDE
v(x, t) =
∂u(x, t)
+ ∇u(x, t) · v(x, t).
∂t
(5.17)
This equation provides the time rate of change experienced by an element of material
at point x and time t. The term ∇u(x, t) · v(x, t) is responsible of the kinematic
nonlinearities of the displacement field allowing particles in the Eulerian domain to
have curved trajectories.
The deformation of the material associated to the displacement field u is governed
by a PDE derived from fundamental conservation laws of continuum mechanics on
deformable bodies. In particular, the conservation of the momentum with a mass
source of density ρ and rate of mass change η
ρ
dv
+ vη − ∇ · T − b = 0
dt
(5.18)
provides the relationship between the body forces b and the resulting material deformation. The deformation model imposed on Ω comes from the specification of the Cauchy
stress tensor T. In this fluid registration framework, the Navier-Poisson-Newtonian
fluid model is used. Thus, the momentum conservation equation results into a viscous
fluid PDE. In practice, a simplified model is used, leading to the viscous fluid PDE
μ∇2 v + (λ + μ)∇(∇ · v) + b(u) = 0
(5.19)
where the term μ∇2 v is related to the constant-volume viscous flow of the template
image and the term (λ + μ)∇(∇ · v) controls the rate of growth or shrinkage of local
regions in the deforming template thus regularizing the Jacobian.
In image registration, the body force b(u) follows from the minimization of the
energy functional
1
E(u(x, t)) = I0 (x − u(x, t)) − I1 2L2
2
(5.20)
129
yielding
b(u(x, t)) = −(I0 (x − u(x, t)) − I1 (x))∇I0 (x − u(x, t)) · |det(Du(x, t))|
(5.21)
The solution of the viscous fluid registration problem involves iteratively solving
the PDEs in Equations 5.17 and 5.19 and recomputing the body force from Equation 5.21. Due to bad numerical conditioning, the transformation becomes singular for
large displacements. To overcome this problem, the algorithm generates a new template image by applying the current deformation to the former template every time
the determinant of the Jacobian displacement becomes lower than a given threshold,
det(Du(x, t)) < τ , and the registration is restarted for a new ϕi . This procedure is
popularly known as regridding. Finally, the resulting transformation is computed from
the composition of the displacement fields associated to the sequence of propagated
templates (Equation 5.16).
One of the limitations of this method is that there are no explicit constraints that
limit transformations to be diffeomorphic. The folding of the grid over itself can frequently occur destroying the smoothness of the transformation. However, this method
still provides the basic formulation that inspired the latter large deformation methods.
5.3.2
Large Deformation Diffeomorphic Metric Mapping
(LDDMM)
The Large Deformation Diffeomorphic Metric Mapping (LDDMM) model for diffeomorphic registration overcomes the limitations of Christensen et al.’s viscous fluid model
ensuring that the transformations are diffeomorphic [67, 239]. The transformation from
a template image I0 to a target I1 is represented by the end point ϕ = φ(1) of a path
of diffeomorphisms φ(t) resulting from the minimization of the energy functional
E(ϕ) = Ereg (φ) +
1
Eimg (I0 ◦ ϕ−1 , I1 )
σ2
(5.22)
The term Ereg (φ) imposes a regularization on the path energy. The term Eimg (I0 ◦
ϕ−1 , I1 ) measures the similarity between the images after registration. Although it is
usually selected to be the sum of squared intensity differences, I0 ◦ ϕ−1 − I1 2L2 , this
term can be replaced by any other energy proposed in usual registration techniques.
The weighting factor 1/σ 2 balances the energy contribution between regularization and
matching.
The space of transformations in which the admissible solutions of the variational
problem lie depends on the parameterization chosen to represent paths of diffeomorphisms. In the original LDDMM framework, paths of diffeomorphisms φ(t) are parameterized using time-varying vector field flows v : [0, 1] → T (Dif f s (Ω)), t → v(t) ∈ V
130
satisfying the transport equation φ̇(t) = v(t, φ(t)). Therefore, diffeomorphic registration is obtained computing the flow of vector fields resulting from the minimization of
the energy functional
1
E(v) =
0
v(t)2V dt +
1
I0 ◦ ϕ−1 − I1 2L2
σ2
(5.23)
and obtaining the solution at time t = 1 of the transport equation associated to v(t).
Dupuis et al. pointed out that the selection of the metric defined in the tangent space
V is crucial for the well posedness of the variational problem [67]. He suggested a
metric from the scalar product u, wV = Lu, LwL2 where operator L was derived
from the Navier-Stokes PDE introduced in Christensen et al.’s registration method
(Equation 5.19) under the assumption of incompressible flow (i.e. ∇ · v = 0). This lead
to L = γId − α∇2 . Vector field flows that generate diffeomorphisms do verify that
0
1
v(t)2V < ∞
(5.24)
Under this condition, it was demonstrated that the solution of the variational problem
exists. This solution provides a path of diffeomorphisms with minimal energy and
maximum image matching at t = 1.
5.3.3
Numerical aspects of the LDDMM method
Euler-Lagrange equation for LDDMM
The numerical solution associated to the minimization of the LDDMM variational
problem is computed using customary techniques of optimization on Hilbert spaces.
Optimization is commonly approached using standard gradient descent [23], where
the Euler-Lagrange equation associated to the energy functional E(v) corresponds to
∇v E = 0. Thus, the gradient descent update scheme is given by
v n+1 (t) = v n (t) − (∇v E(t))n
(5.25)
The gradient operator ∇v E can be computed from the Gâteaux derivatives of the
energy functional along a small perturbation h of the velocity field flow v as the Gâteaux
variation of the energy functional E(v) along h ∈ L2 ([0, 1], V ) is related to its Frechet
derivative ∇v E(v) by
E(v + εh) − E(v)
=
∂h E(v) = lim
ε→0
ε
1
∇v E(t), h(t)V dt
0
For simplicity, the computation of ∂h E(v) is divided into
(5.26)
131
∂h Ereg (v) = ∂h
0
1
v(t)2V dt
1
v(t), h(t)V dt
=2
(5.27)
0
and
∂h Eimg (v) = ∂h
=
1
I0 ◦ φ(1)−1 − I1 2L2
σ2
(5.28)
2
I0 ◦ φ(1)−1 − I1 , ∇(I0 ◦ φ(1)−1 ) · ∂h φ(1)−1 L2
σ2
where the Gâteaux derivative of the path φ(1)−1 along the perturbation h is obtained
from (Lemma 2.1 in [23])
−1
φ−1
v+εh (1) − φv (1)
= lim
(5.29)
ε→0
ε
1
(D(φ(t) ◦ φ(1)−1 ))−1 · [h(t) ◦ (φ(t) ◦ φ(1)−1 )]dt
= −(Dφ(1)−1 )
−1
∂h φ(1)
0
where the subscript v in φv is used to explicitly denote the dependence of φ on the
associated velocity field v. Inserting this expression into Equation 5.28, making the
change of variables x = φ(t) ◦ φ(1)−1 and isolating the vector field flow h in the second
component of the scalar product we get
∂h Eimg (v) =
2
− 2
σ
1
|det(D(φ(1) ◦ φ(t)−1 ))| · (I0 ◦ φ(t)−1 − I1 ◦ φ(1) ◦ φ(t)−1 )·
0
∇(I0 ◦ φ(t)−1 ), h(t)L2 dt (5.30)
which in space V yields
2
∂h Eimg (v) = − 2
σ
2
− 2
σ
0
1
0
1
L−1 |det(D(φ(1) ◦ φ(t)−1 ))| · (I0 ◦ φ(t)−1 − I1 ◦ φ(1) ◦ φ(t)−1 )
!
·∇(I0 ◦ φ(t)−1 ) , L−1 (h(t))V dt =
(5.31)
(L† L)−1 |det(D(φ(1) ◦ φ(t)−1 ))|(I0 ◦ φ(t)−1 − I1 ◦ φ(1) ◦ φ(t)−1 )
!
·∇(I0 ◦ φ(t)−1 ) , h(t)V dt
(5.32)
132
Finally, the Euler-Lagrange equation associated to the minimization of the LDDMM
variational problem is obtained from the combination of the results in Equations 5.27
and 5.32
†
−1
∇v E(v)(t) = 2v(t) − (L L)
2
(I0 ◦ φ(t)−1 − I1 ◦ φ(1) ◦ φ(t)−1 )·
2
σ
(5.33)
|det(D(φ(1) ◦ φ(t) ))| · ∇I0 (φ(t) )
−1
−1
Discretization
The update of the numerical scheme of the LDDMM variational problem is computed
on a regular sample of the computation domain Ω. Time samples are denoted by tk .
The numerical scheme for gradient descent optimization is given by
n+1
n
(tk ) = vijk
(tk ) − (∇v Eijk (tk ))n
vijk
(5.34)
n
where vijk
(tk ) corresponds to the velocity discretization in the computation domain
at time tk and iteration n and (∇v Eijk (tk ))n approximates the energy gradient at the
same spatial point and time.
The discretization of the energy gradient involves the computation of the diffeomorphism φijk (tk )−1 and φijk (1) ◦ φijk (tk )−1 associated to vijk (tk ), the image deformation
I0 ◦φijk (tk )−1 and I1 ◦φijk (1)◦φijk (tk )−1 , the gradient ∇(I0 ◦φijk (tk )−1 ) and the Jacobian
D(φijk (1) ◦ φijk (tk )−1 ) in the computation domain. The computation of image deformation is performed by linear interpolation of the original image in the points given by
the diffeomorphism, the image gradient and the Jacobian matrix are computed using
central differences for the computation of the derivatives.
The computation of the path φ from v is performed by solving the transport equation φ̇(t) = v(t, φ(t)). The expression of this equation in the Eulerian frame of reference corresponds to an advection type hyperbolic PDE. Therefore, standard schemes
as those described in Chapter 2 should be used for obtaining a numerical solution.
Forward Euler time approximation in the Eulerian coordinate system constitutes the
most simple method for solving the transport equation. However, the size of time
discretization is bounded by the CFL condition that forces the selection of small time
steps in order to obtain numerically stable solutions.
As alternative, Beg et al. proposed to use semi-Lagrangian schemes [225, 224].
These numerical methods combine the advantages of Lagrangian with Eulerian approaches for the solution of this class of PDEs. The solution at current time step in a
grid point in the Lagrangian coordinate system is computed following the characteristic
line at that point and using the velocity of the point that will end on this regular point
133
at the next time-step. This numerical scheme allows the choice of larger time-steps
than those provided by the CFL condition in the Eulerian scheme. Therefore, the
numerical solution to the transport equation is computed from the iterative scheme
φtk (y) = φtk−1 (y − αtk )
(5.35)
with initial condition φ(t−1 ) = id where αtk represents the direction of the characteristic
computed from the solution of the iterative scheme
= Δt vtk−1 (y − 0.5 · αtnk )
αtn+1
k
(5.36)
with initial condition αt0k = 0.
Stability
The gradient descent update scheme used for optimization can be included into the
general formulation v k+1 = v k − d(v k ) where d(v k ) is the search direction at iteration
k and controls the step size made along this direction. During optimization, the
selection of the parameter is critical as it can determine the convergence of the
algorithm. For example, if is selected to be too big, the step made along the search
direction may not provide a step minimizing the energy. Otherwise, if is selected to be
too small the algorithm may get trapped into a local minimum far from an acceptable
solution. To our knowledge, LDDMM literature lacks of information about the linesearch procedures used to guarantee the stability during optimization. This issue will
be tackled in Chapters 6 and 7.
Computation of L and (L† L)−1 operators action
The action of operator L = γId − α∇2 on v ∈ V involves the computation of the action
∂2
∂2
∂2
of the Laplacian operator ∇2 = ∂x
2 + ∂x2 + ∂x2 on each vector field component vi that
can be implemented using central finite differences on a periodic domain
Lvi = −α
vi (x + Δx, y, z) − 2vi (x, y, z) + vi (x − Δx, y, z)
Δx2
vi (x, y + Δy, z) − 2vi (x, y, z) + vi (x, y − Δx, z)
+
Δy 2
vi (x, y, z + Δz) − 2vi (x, y, z) + vi (x, y, z − Δz)
+
Δz 2
+ γ vi (x, y, z) (5.37)
134
where Δx, Δy, and Δz yield a discretization of the computation domain Ω in the unit
cube. The operator L is self-adjoint, this means that L = L† . Therefore, the action of
the operator (L† L) can be implemented computing the discretized version of L2 .
The action of the operator (L† L)−1 cannot be expressed with a simple analytic
expression. Therefore, it is implemented through the Fourier domain. Thus, given
a vector field v in Ω, (L† L)−1 v is approached computing the action of (L† L)−1 on
each vector component. Let vi (x, y, z) be the i-th component of v and Vi (k1 , k2 , k3 )
the corresponding Fourier transform. Then, (L† L)−1 vi in the Fourier domain is given
by A−2 (k1 , k2 , k3 ) · Vi (k1 , k2 , k3 ), where A corresponds to the Fourier transform associated to the operator L. Finally, the inverse Fourier transform provides the action of
(L† L)−1 vi on the spatial domain.
5.3.4
LDDMM from Jacobi Fields
LDDMM from Jacobi Fields is based on the fact that the non-stationary vector field
flow v that parameterizes a geodesic path of diffeomorphisms fullfils the momentum
conservation equation (given in Equation 5.11) [282]. This constraint allows to express
the variational problem associated to LDDMM just in terms of the initial velocity field
v0 = v(0) as follows
E(v0 ) = v(0)2V +
1
I0 ◦ ϕ−1 − I1 2L2
2
σ
(5.38)
The solution to the minimization of this variational problem is approached using
gradient descent optimization on Hilbert spaces as in the original LDDMM. Hence,
the Euler-Lagrange equation associated to the minimization of the energy functional
is given by
∗
∇v0 E(v0 ) = 2v(0) − Q (1)
2
−1
−1
−1
(I0 ◦ ϕ − I1 ) · |det(Dϕ )| · ∇(I0 ◦ ϕ )
σ2
(5.39)
where the operator Q∗ (1) corresponds to the dual of the evolution of the Jacobi field
associated to the Gâteaux derivative of φ(t) with respect to v0 at time t = 1.
The solution to the variational problem provides a geodesic path in Dif f s (Ω) with
minimal energy and maximum image matching at t = 1. This method has lower
computational requirements than original LDDMM. However, the dependence of φ(t)
in v0 is quite complex. This complexity is reflected in the expression of Q∗ (1). Hence
in practice, the minimization of E(v) using classical LDDMM is still preferred for
Computational Anatomy applications.
5.3.5
Efficient algorithms for diffeomorphic registration
Recently, two different fast algorithms for diffeomorphic registration have been proposed in order to alleviate the computational complexity of original LDDMM registra-
135
tion in Computational Anatomy applications. Both algorithms rely on the stationary
parameterization of diffeomorphisms that allows a considerable reduction of memory
and time requirements [8]. Moreover, optimization is approached using second-order
techniques introducing efficiency and robustness in the optimization proccess. It should
be noted that these algorithms were introduced in the literature simultaneously to the
methods developed in this Thesis. A deep theoretical and experimental comparison
(presented in Chapter 7) is provided in this Thesis. In this Section we describe the
most remarkable theoretical and implementational aspects of both methods.
Diffeomorphic Anatomical Registration using Exponentiated Lie algebra
(DARTEL)
DARTEL was introduced by Ashburner et al. as an efficient second-order optimization
method in the LDDMM registration paradigm [11]. The method uses the stationary parameterization of diffeomorphisms and, to our knowledge, it can be considered
the first effort to incorporate second-order methods for optimization in the LDDMM
framework.
In DARTEL, Newton’s method is implemented using Levenberg – Marquardt approximation. Thus, optimization is approached with the iterative scheme
wk+1 = wk − · (λI + Hw E(wk ))−1 · ∇w E(wk )
(5.40)
where the gradient ∇w E(w) and the Hessian Hw E(w) are computed from Frechet
differentials defined in the space of L2 -functions
(∇w E(w))L2 = 2 (L† L)w − (I0 ◦ Exp(w)−1 − I1 ) · ∇(I0 ◦ Exp(w)−1 ) (5.41)
(5.42)
(Hw E(w))L2 = 2 (L† L) + ∇(I0 ◦ Exp(w)−1 )T · ∇(I0 ◦ Exp(w)−1 )
The computation of the expression (L† L) in Hw E(w)−1 has to be approached using
the matrix representation of the momentum operator [175]. As a consequence, the
algorithm results into a large dimensional matrix inversion problem with huge memory
requirements. This makes the algorithm impractical on standard machines for 3D
applications, as the storage of the full inverse Hessian matrix is not possible because
of memory limitations.
In order to circumvent with this limitation, DARTEL computes the expression
(λI + Hw E(w))−1 · ∇w E(w) by solving the sparse system of equations
(λI + Hw (E(w)))h = ∇w E(w)
(5.43)
For L = γId−α∇2 , the matrix λI +Hw (E(w)) is diagonally dominant and Gauss-Seidel
method for solving linear systems converges. Therefore, the matrix can be decomposed
136
into E + R where E is an easily invertible matrix and R is the reminder and h can be
approximated using the iterative algorithm
hn+1 = E −1 (∇w E(w) − R · hn )
(5.44)
In order to achieve a fast convergence, this iterative scheme is implemented using
multigrid approaches [199].
Diffeomorphic Demons
Diffeomorphic Demons was introduced by Vercauteren et al. as an extension of Demons
algorithm for computing diffeomorphic transformations [251]. Originally proposed by
Thirion et al. in [231], Demons registration computes the transformation of the image domain from the action of forces similarly to Maxwell proposal for solving Gibbs
paradox in thermodynamics. Demons forces push the grid points of the computation
domain according to local characteristics of the image inspired on the optical flow
equations. The resulting displacements are regularized by a simple Gaussian smoothing. Although this heuristic formulation was not well understood, the efficiency of
Demons registration made this algorithm quite popular for applications. Later on, it
was stated the close relationship between Demons forces and second-order optimization
schemes associated to the least squares minimization problem [194] that evolved into
a well-posed variational formulation [27].
Thus, in Demons variational problem, the transformation that deforms the source
I0 into the target I1 is computed from the minimization of the energy functional
E(ϕ, ψ) =
1
2
σsim
I0 ◦ ϕ − I1 2L2 +
1
1
d(ϕ, ψ)2 + 2 Ereg (ψ)
2
σx
σreg
(5.45)
where the terms d(ϕ, ψ) and Ereg (ψ) are introduced in order to cast Demons variational
formulation as a well-posed minimization problem. Both terms can be derived from
the introduction of the hidden variable ψ in the registration process. This variable
represents an estimation of the transformation existing between the source and the
target image on the search space. The regularization criterion Ereg (ψ) poses a prior on
the smoothness of this transformation. Instead of requiring an exact correspondence
between ϕ and ψ some error is allowed at each image point regularized by the distance
d existing between both transformations. The weighting factors σreg and σsim balance
the energy contribution between regularization and matching and σx accounts for the
spatial uncertainty on the correspondences.
In diffeomorphic Demons, transformations ϕ and ψ are considered to belong to some
space of diffeomorphisms. The regularization term is considered to be the harmonic
energy associated to the Jacobian of the transformation
Ereg (ψ) = Dψ − I2fro
(5.46)
137
and the distance d is given by
d(ϕ, ψ) = ϕ − ψL2
(5.47)
The solution of the minimization of this variational problem is approached decoupling
the estimation of ϕ and ψ in each iteration. Thus, for a given ψ, the transformation ϕ
is estimated from the minimization of
E(ϕ) =
1
2
σsim
I0 ◦ ϕ − I1 2L2 +
1
d(ϕ, ψ)2
2
σx
(5.48)
and for the estimated ϕ, the regularization on ψ is imposed by a simple Gaussian
smoothing [27] 5 .
In order to compute the minimization of the decoupled energy functional E(ϕ) the
relationship between ϕ and ψ has to be previously stated. This is done by extending
the additive scheme (ϕ = ψ + u where u is a displacement vector field) from Demons
algorithm to its diffeomorphic variant which leads to
ϕ = ψ ◦ Exp(u)
(5.49)
where u is such that Exp(u) belongs to the space of diffeomorphisms where Demons
update can be performed. Then, the decoupled energy functional can be written as
E(ψ ◦ Exp(u)) =
1
2
σsim
I0 ◦ (ψ ◦ Exp(u)) − I1 2L2 +
1
u2L2
σx2
(5.50)
The minimization is approached using second-order optimization techniques by extending Newton’s method on finite dimensional Lie groups to diffeomorphisms [249] leading
to the intrinsic iterative scheme
ψ k+1 = ψ k ◦ Exp( · uk )
(5.51)
uk = −Hu E(ψ k )−1 · ∇u E(ψ k )
(5.52)
where
The gradient and Hessian expressions depend on the second-order Taylor approximation of the energy functional. Usually, the energy is approximated using
5
In fact, the harmonic regularization has shown to be equivalent to Gaussian smoothing
138
E(ψ ◦ Exp(u)) = E(ψ) + ∇u E(ψ) · u +
1 T
u · Hu E(ψ) · u + O(u3L2 )
2
(5.53)
which yields6
∇u E(ψ) =
1
2
σsim
(I0 ◦ ψ − I1 ) · ∇(I0 ◦ ψ)
(5.54)
and
Hu E(ψ) ≈
1
2
σsim
∇(I0 ◦ ψ)T · ∇(I0 ◦ ψ) +
1
σx2
(5.55)
Finally, the Sherman-Morrison formula (a.k.a. matrix inversion lemma) together with
the selection of σsim = |I0 ◦ ψ − I1 | as the local estimation of the image noise provide
the familiar expression of the update for Demons-like algorithms
u=
I0 ◦ ψ − I1
∇(I0 ◦
ψ)22
+
σ2
sim
σx2
· ∇(I0 ◦ ψ)
(5.56)
It should be noted that the Hessian of the image term involved in the second order
derivatives of the computations of Hu E(ψ) is neglected leading to a Gauss-Newton’slike optimization where the accuracy descends from O(u3L2 ) to O(u2L2 ). Although
the rate of converge is slower than in Newton’s method, Gauss-Newton avoids any
convergence problem that may arise when dealing with ill-conditioned Hessians. As alternative, it has been recently pointed out the possibility of introducing a second-order
minimization method that does not need the computation of this Hessian matrix while
preserves O(u3L2 ) accuracy [158]. The efficient second-order minimization (ESM)
uses information related to the solution of the problem in order to improve the search
direction of second-order optimization methods.
Thus, ESM method proceeds computing the Taylor expansion of the energy gradient
∇u E(ψ ◦ Exp(u)) = ∇u E(ψ) + uT · Hu E(ψ) + O(u2L2 )
(5.57)
and inserting it into the usual second-order Taylor approximation of the energy (Equation 5.53)
1
E(ψ ◦ Exp(u)) = E(ψ) + (∇u E(ψ ◦ Exp(u)) + ∇u E(ψ)) · u + O(u3L2 )
2
6
∇u E(ψ) = limu→0
1
σ2
sim
(I0 ◦ (ψ ◦ Exp(u)) − I1 )∇(I0 ◦ (ψ ◦ Exp(u))) +
1
2 u
σx
(5.58)
139
The computation of ∇u E(ψ ◦ Exp(u)) is approached considering that the update ψ ◦
Exp(u) provides the optimal transformation. Therefore, when the images are aligned
with the optimal transformation, the warped image and its gradient are close to the
target image and its gradient, respectively, and ∇u E(ψ ◦ Exp(u)) can be approximated
by ∇I1 . This leads to the ESM update step
u=
I0 ◦ ψ − I1
∇(I0 ◦ ψ) + ∇I1 22 +
σ2
sim
σx2
· (∇(I0 ◦ ψ) + ∇I1 )
(5.59)
that corresponds to the familiar expression for the ad hoc symmetrization of Demons
forces proposed by Thirion et al. [231].
5.4
Summary
In this Chapter we have studied the fundamental aspects of Riemannian geometry related to the manifolds of diffeomorphisms involved in Computational Anatomy. These
are infinite dimensional spaces that show a much more complex differential structure
than the finite dimensional case. Computational Anatomy deals with Sobolev diffeomorphisms where thanks to the Hilbert differentiable structure many of the results valid
for finite dimensions can be extended to infinite dimensions. This Hilbert structure
was originally studied in the physical context of Continuum Mechanics. The analogies
existing between the models of deformation of a continuum system and the models
of anatomical variability and growth allowed to translate the results from Physics to
Computational Anatomy.
Calculus on the Riemannian manifold of diffeomorphisms Dif f s (Ω) is based on
the parameterization of the paths of diffeomorphisms and metric definition. Paths of
diffeomorphisms are parameterized from the solution of the transport equation associated to a time-varying smooth vector field flow defined on the computation domain.
The metric is defined from the action of a linear invertible differentiable operator on
the elements of the tangent space. This metric is invariant with respect to right multiplication but lacks of left-invariance. Right-geodesics can be characterized from the
solutions of the Euler-Poincare equation for diffeomorphisms which provides an effective algorithm for the computation of the Riemannian exponential map. Although
the existence of the Riemannian logarithm map can be guaranteed from Nash-Moser
generalization of the inverse function theorem, no algorithm for its computation has
been provided yet. The lack of a left-invariant Riemannian metric together with the
absence of a method for the computation of the Riemannian logarithm map pose some
theoretical and practical limitations for statistical calculus on Riemannian manifolds
of diffeomorphisms that will be discussed in Chapter 8.
As alternative, it was pointed out the possibility of relying on the algebraic properties of the group structure of Dif f s (Ω) as it was previously done with some particular
5.4. Summary
140
Lie groups of matrices [8, 9, 7]. The first theoretical limitation is that no Lie group
structure compatible with the differential structure of Riemannian manifold can be
established on Dif f s (Ω). Fortunatelly, an algebraic definition close to Lie groups has
been recently defined on Dif f s (Ω). This allows to deal with Dif f s (Ω) as in the
finite dimensional case. One-parameter subgroups are identified with paths of diffeomorphisms parameterized from the solution of the transport equation associated
to constant-time flow of smooth vector fields. The group exponential map provides
elements on these subgroups. The group logarithm of an element in Dif f s (Ω) is identified with the infinitesimal generator of the one-parameter subgroup. In this case,
there have been proposed algorithms for the computation of both the group exponential and logarithm maps from extending efficient algorithms for the computation of the
group exponential and logarithm in groups of matrices [8].
However, the most important limitation posed by relying on the group structure in
Dif f s (Ω) is that the group exponential map is not onto and, therefore, it is not a diffeomorphism at the identity [94]. This means that there may exist some aspects of the
anatomical variability in a population that could be better encoded by right-geodesics
than by one-parameter subgroups. However, there may also exist the possibility that
the space Dif f s (Ω) would be too complex for dealing with Computational Anatomy
applications and alternative groups of diffeomorphisms would provide an appropriate
framework for encoding the whole anatomical variability of a population in a rigorous
algebraic way may be feasible. At this point, we need to wait for advances in the theory of infinite dimensional Riemannian manifolds and perform thorough experimental
validations that would corroborate one of these hypothesis.
Diffeomorphic registration is used in Computational Anatomy to compute the transformations encoding the anatomical variability existing between different images. Diffeomorphic registration algorithms arose in a physical context described from a variational formulation solved by PDEs of Continuum Mechanics. From them, the LDDMM
method and its variants are considered the reference paradigm for image registration in
Computational Anatomy. These algorithms are based on the non-stationary parameterization of paths of diffeomorphisms and the output transformations are well suited
for calculus on the Riemannian manifold of diffeomorphisms Dif f s (Ω). However, the
high computational complexity represent the main practical limitations.
Recently, efficient algorithms for diffeomorphic registration have been proposed in
order to overcome with LDDMM limitations. These algorithms rely on the group
structure defined in Dif f s (Ω) based on the stationary parameterization of paths of
diffeomorphisms. Therefore, the output transformations are restricted to elements belonging to one-parameter subgroups or finite composition of group exponentials and
are well suited for calculus on the group of diffeomorphisms. In particular, these algorithms seem to be better positioned than non-stationary LDDMM for the computation of statistics on diffeomorphisms with desirable invariance properties. In addition,
optimization is approached using second-order techniques introducing efficiency and
robustness in the optimization process.
In conclusion, infinite dimensional Riemannian manifolds of diffeomorphisms pro-
141
vide a rigorous mathematical setting where Computational Anatomy applications can
be developed. The LDDMM method for diffeomorphic registration has shown well
fitted in this rigorous framework although with limited applicability in Computational
Anatomy in practice due to its huge computational requirements. This has motivated
in this Thesis the design of alternative algorithms that rely on the algebraic structure
of Dif f s (Ω) that would compete in performance and efficiency with reference LDDMM
while fitting into an alternative valid framework for the computation of statistics on
diffeomorphisms.
Chapter 6
LDDMM from one-parameter
subgroups of diffeomorphisms
Abstract
In diffeomorphic registration the space of solutions depends on the parameterization used to represent paths of diffeomorphisms. In the Large Deformation
Diffeomorphic Metric Mapping (LDDMM) paradigm, transformations are
represented as end points of paths parameterized by time-varying flows of
smooth vector fields defined on the tangent space of a convenient Riemannian manifold of diffeomorphisms, computed as solution of non-stationary
ODEs associated to these flows. With this characterization, optimization in
LDDMM is performed on the space of flows of smooth vector fields resulting
into a time and memory consuming algorithm. Recently, an alternative
parameterization of paths of diffeomorphisms based on constant-time vector
field flows has been proposed in the literature. With this parameterization,
diffeomorphisms can be computed as solutions of stationary ODEs. In
this Chapter, we propose to include the stationary parameterization for
diffeomorphic registration in the LDDMM framework. We formulate the
variational problem related to the registration scenario and derive the
associated Euler-Lagrange equations. Moreover, the performance of the
non-stationary vs the stationary parameterizations in real and simulated
3D-MRI brain datasets is evaluated. Compared to non-stationary LDDMM,
stationary-LDDMM has shown to provide similar performance in terms
of image matching and similar local differences between the diffeomorphic
transformations while drastically reducing memory and time requirements.
6.1. Introduction
144
Contents
6.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
6.2
Stationary-LDDMM for diffeomorphic registration . . . . . . . . .
145
6.3
6.4
6.1
6.2.1
Euler-Lagrange equation for stationary-LDDMM . . . . . . . . . . . . 146
6.2.2
Numerical implementation
Results
. . . . . . . . . . . . . . . . . . . . . . . . 148
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
6.3.1
6.3.2
Regularization parameters selection . . . . . . . . . . . . . . . . . . . 150
6.3.3
Evaluation framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3.4
Registration results in real datasets
6.3.5
Simulated datasets
6.3.6
Registration results in simulated datasets . . . . . . . . . . . . . . . . 164
6.3.7
Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
. . . . . . . . . . . . . . . . . . . 153
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
182
Introduction
As we have seen in Chapter 5, diffeomorphic registration is approached in Computational Anatomy as a variational problem where the space of solutions depends on
the parameterization used to represent paths of diffeomorphisms. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) paradigm, transformations are
represented as end points of paths of diffeomorphisms parameterized by time-varying
flows of smooth vector fields defined on the tangent space of a convenient Riemannian
manifold of diffeomorphisms, computed as solution of non-stationary ODEs associated
to these flows. With this characterization of diffeomorphisms, optimization is performed on the space of valid flows of vector fields. The computational requirements of
this algorithm linearly grow with the size of time sampling. Hence, LDDMM diffeomorphic registration results into a time and memory demanding algorithm that cannot
be afforded in certain applications.
In order to alleviate the computational requirements of LDDMM, Younes et al.
have recently proposed to parameterize paths of diffeomorphisms from time-varying
flows of vector fields that fulfill the momentum conservation equation [282]. Thus, the
variational problem associated to LDDMM is restricted to the space of valid initial
momenta allowing great time and memory savings. However, the dependence of the
diffeomorphisms on the initial velocity fields results quite complex and the expensive
although simpler optimization of the variational problem in classical LDDMM is still
preferred for Computational Anatomy applications.
145
Chapter 6. LDDMM from one-parameter subgroups of diffeomorphisms
Recently, Arsigny et al. have proposed an alternative parameterization of paths
of diffeomorphisms that uses constant-time flows of smooth vector fields [8]. This
parameterization is closely related to the group structure of diffeomorphisms as the
paths that can be parameterized using stationary vector fields are exactly identified
with the one-parameter subgroups. With this parameterization, diffeomorphisms can
be represented from the group exponential map of the infinitesimal generator associated
to the one-parameter subgroup, computed as solution of stationary ODEs.
In this Chapter, the stationary parameterization is included for diffeomorphic registration in the variational problem studied in the LDDMM framework. This restricts
transformations to belong to paths identified with one-parameter subgroups. Thus,
the variational problem associated to LDDMM is restricted to the space of valid oneparameter subgroup generators allowing great time and memory savings. Inspired on
Younes et al. proposal, we have formulated the variational problem related to the
registration scenario and derived the Euler-Lagrange equation associated to the minimization of the energy functional for gradient descent optimization. Moreover, we
have studied the influence of the parameterization on the performance of LDDMM
registration in real and simulated 3D-MRI brain datasets.
The method for stationary-LDDMM diffeomorphic registration was presented at
the international workshop MMBIA’07 held in conjunction with ICCV’07 conference
obtaining the best paper award mention [105]. An improved version of the manuscript
has been recently accepted with minor revisions at the International Journal on Computer Vision (IJCV). The stationary parameterization has been also considered for
diffeomorphic registration by other authors in [11] and [251] at the same time than
our conference results. The main differences among the three methods are thoroughly
analyzed in Chapter 7.
The rest of the chapter is divided as follows. In Section 6.2 we introduce the method
for stationary-LDDMM diffeomorphic registration, and detail the implementation details. The experiments in real and simulated datasets are presented in Section 6.3.
Finally, Section 6.4 presents the most remarkable concluding remarks.
6.2
Stationary-LDDMM for diffeomorphic
registration
In the LDDMM framework, diffeomorphic registration from a template I0 to a target
image I1 is represented by the end point ϕ = φ(1) of a path of diffeomorphisms φ(t)
resulting from the minimization of the energy functional
E(ϕ) = Ereg (φ) +
1
I0 ◦ ϕ−1 − I1 2L2
σ2
(6.1)
The term Ereg (φ) imposes a regularization constraint to favor stable numerical solutions. The image metric measures the matching between the images after registration.
6.2. Stationary-LDDMM for diffeomorphic registration
146
The weighting factor 1/σ 2 balances the energy contribution between regularization and
matching. The space of valid solutions depends on the parameterization chosen to represent the paths of diffeomorphisms. In this section we introduce stationary-LDDMM,
where diffeomorphisms are characterized from stationary vector field flows.
In stationary-LDDMM, paths of diffeomorphisms are identified with one-parameter
subgroups spanned by infinitesimal generators w ∈ V . The energy of such paths in the
Riemannian manifold is given by
1
w(t)2V dt = w2V
(6.2)
E(φ) =
0
Hence, the diffeomorphism that connects I0 and I1 is computed from the infinitesimal
generator of the one-parameter subgroup resulting from the minimization of the energy
functional
E(w) = w2V +
1
I0 ◦ Exp(w)−1 − I1 2L2
σ2
(6.3)
and obtaining the solution at time t = 1 of the stationary ODE associated to w.
As in the non-stationary case, the minimization of this variational problem is approached using the gradient descent update
wn+1 = wn − (∇w E(w))n
(6.4)
where the energy gradient is given by
†
−1
∇w E(w) = 2w − (L L)
2
−1
−1
(I0 ◦ Exp(w) − I1 ) · ∇(I0 ◦ Exp(w) )
σ2
(6.5)
At the steady-state, the resulting w provides the infinitesimal generator of a oneparameter subgroup of diffeomorphisms with minimal energy and maximum image
matching at t = 1.
6.2.1
Euler-Lagrange equation for stationary-LDDMM
We now present the computations to obtain the Euler-Lagrange equation associated
to the minimization of the energy functional E(w). In general convex vector spaces,
the Euler-Lagrange equation associated to a Frechet differentiable energy functional
F (t, v(t), v̇(t)) is given by
d ∂F
∂F
−
=0
(6.6)
∂v
dt ∂ v̇
147
In the stationary case, the dependence of the energy E(w) on ẇ is constant. Therefore,
the corresponding Euler-Lagrange equation is obtained from ∂E/∂w = ∇w E(w) = 0.
In Frechet spaces like V , the gradient operator relates the Frechet derivative and
the Gâteaux derivative (whenever both derivatives exist) by
∂h E(w) = ∇w E(w), hV
(6.7)
The Gâteaux derivative of the energy functional E(w) along h ∈ V is defined as the
variation of E(w) under the perturbation of w in the direction of h
E(w + εh) − E(w)
ε→0
ε
∂h E(w) = lim
(6.8)
For simplicity, we divide the computation of ∂h E(w) into ∂h E1 (w) and ∂h E2 (w), where
E1 (w) = w2V
1
I0 ◦ Exp(w)−1 − I1 2L2
2σ 2
Straightforward computations provide
E2 (w) =
∂h E1 (w) = 2w, hV
(6.9)
(6.10)
(6.11)
The chain rule allows to compute the variation of E2 (w) in the direction of h
∂h E2 (w) =
1
I0 ◦ Exp(w)−1 − I1 , ∇I0 ◦ Exp(w)−1 · ∂h Exp(w)−1 L2
2
σ
(6.12)
The Gâteaux derivative of the exponential map is obtained using the approximation ∂h Exp(w) ≈ D(Exp(w)) h. The Gâteaux derivative of the inverse exponential
map is obtained from the fact that Exp(w)−1 = Exp(−w). Thus, ∂h Exp(w)−1 ≈
−D(Exp(w)−1 ) h. It should be mentioned that this approximation is valid only on
a neighborhood of 0V in the fiber Tid (Dif f (Ω)). Therefore, it may not be suitable
for dealing with too large diffeomorphisms. However, as pointed out in the experimental section, it may provide acceptable enough results for diffeomorphic registration
of anatomical images. In addition, including an expression of ∂h Exp(w) valid in the
whole tangent bundle T (Dif f (Ω)) in the computations would considerably increase the
computational complexity of the algorithm invalidating its advantages with respect to
non-stationary LDDMM.
Hence,
6.2. Stationary-LDDMM for diffeomorphic registration
∂h E2 (w) = −
1
(L† L)−1 ((I0 ◦ Exp(w)−1 − I1 ) · ∇(I0 ◦ Exp(w)−1 )), hV
σ2
148
(6.13)
Finally, the energy gradient (Equation 6.5) can be obtained combining the results in
Equations 6.11 and 6.13 into Equation 6.7.
6.2.2
In this work, the numerical implementation for finding the solution of stationary LDDMM variational problem proceeds as in non-stationary LDDMM [23]. Thus, the algorithm based on the non-stationary parameterization initializes with iteration k = 0,
v(t) = 0V and φ(t) = id ∀t whereas the algorithm based on the stationary parameterization initializes with w = 0V , and ϕ = id. Each iteration in the gradient descent
consists of the steps collected in Table 6.1.
The selection of the optimal step size is approached using a backtracking inexact
line-search strategy starting from an initial guess 0 . In each iteration the step size is selected to be the first parameter that provides a sufficient decrease in the energy
according to Armijo’s condition [180]. In the case of non-stationary LDDMM Armijo’s
condition is given by
E(vk (t) − k ∇v(t) E(vk (t))) ≤ E(vk (t)) + c1 k ∇v(t) E(vk (t))2L2 ∀t
(6.14)
whereas in the case of stationary LDDMM
E(wk − k ∇w E(wk )) ≤ E(wk ) + c1 k ∇w E(wk )2L2
(6.15)
where c1 is selected 7 equal to 1e−4 .
The computation of the diffeomorphisms in non-stationary and stationary LDDMM is performed by solving the corresponding transport equations using the semiLagrangian numerical scheme described in Chapter 5, Section 5.3.3. It should be noted
that the Scaling and Squaring method was proposed as a stable implementation of the
Euler method in the stationary case [8]. However, in practice both algorithms showed
to provide similar results. Convergence is considered if the value of in the search
strategy is too small or the rate of change in the energy is less than a tolerance value.
7
This value has been empirically chosen as optimal in a wide range of applications. We have seen
that this selection is also optimal for this application.
149
Table 6.1: Algorithm for non-stationary and stationary diffeomorphic registration.
(1) Compute the energy gradient from the Euler-Lagrange Equation:
∇v E(vk )(t) and ∇w E(wk ), respectively
(2) Perform line search for (3) Gradient descent update:
vk (t) = vk−1 (t) − ∇v E(vk−1 )(t) and wk = wk−1 − ∇w E(wk−1 ), respectively
(4) Compute the inverse path of diffeomorphimsms: φ−1 (t) and ϕ = Exp(−wk ), respectively
(5) Compute the transformed images: I0 ◦ φ−1 (t) and I0 ◦ ϕ−1 , respectively
(7) Check for convergence criterion
6.3
Results
The aim of this experimental section is to study the influence of the parameterization
on the performance of LDDMM registration. Our experiments have been carried out
on the non-stationary and stationary LDDMM variational formulations. Registration
results have been both qualitatively and quantitatively compared in terms of image
similarity after registration and local differences between transformations in a database
of real anatomical MRI brain images. Moreover, the performance of the registration
algorithms has been evaluated in a database of simulated MRI brain images computed
from the deformation of a template using simulated diffeomorphic transformations.
Figure 6.1 shows a sagittal view of the brain for anatomical reference.
6.3.1
A set of 15 T1-MRI images were used in our registration experiments. The images
were acquired at Clinic Barcelona Hospital, using a General Electric Signa Horizon
CV 1.5 Tesla scan. I would like to acknowledge to Dr. C. Junque for providing these
datasets. As preprocessing steps, the images were first resampled yielding volumes of
size 128 × 128 × 110 with a spatial resolution of 2.0 × 2.0 × 2.0 mm. Next, the skull
was removed from the images using the tools available at BrainSuite 2 software [63]
(http://brainsuite.use.edu). Finally, the image intensity was normalized using a
histogram matching algorithm and all the images were aligned to a common coordinate system using a similarity transformation with the algorithms available in the
Insight Toolkit (http://www.itk.org). Figures 6.2 and 6.3 show sample slices from
the database of images after preprocessing.
Registration results were obtained selecting one image from this database of patients
as reference Iref and registering the remaining images I1 , ..., IN using non-stationary
and stationary LDDMM algorithms. The non-stationary path of diffeomorphisms was
sampled into T = 10 intervals. In order to make both algorithms comparable, the same
implementation criteria were adopted in their common stages. Both algorithms were
6.3. Results
150
Figure 6.1: Brain anatomy. Sagittal view of the brain from a cadaver section showing some of the
anatomical regions and subcortical structures that are referenced in this Thesis. Image courtesy of
the Digital Anatomist Project, University of Washington, USA.
stopped when the criteria used for convergence were reached or after a maximum of
100 iterations, thus achieving a trade off between total execution time and performance
improving.
6.3.2
Regularization parameters selection
The selection of the regularization parameters is crucial in any deformable registration
algorithm. Strong regularization constraints hinder large deformations and provide
a poor image matching after registration. In contrast, parameters leading to weak
regularization do not constrain the deformation model. In this case, although transformations are large enough to provide a high image matching, unrealistic transformations
can be often obtained. Therefore, parameter selection should be performed as a previous step to any registration experiment. The criteria for parameter selection depend on
the specific application. In this work, we aimed at finding a tradeoff that provided the
best intensity match with minimum deformation on the diffeomorphic transformation.
The Relative Sum of Squared Differences (RSSD) between the images before and after
registration
RSSD =
2
Ii ◦ ϕ−1
i − Iref L2
Iref − Ii 2L2
(6.16)
was used to quantify image similarity while deformation was assessed using the Jacobian minimum, Jmin . The RSSD metric is equal to 1 before registration and decreases
151
with the degree of deformation of the source image towards the target. At convergence, the RSSD measure summarizes the amount of error in the registration due
to photometric variations between the images and inaccurate matching consequence
of diffeomorphic regularization constraints. Therefore, among multiple regularization
parameters those providing diffeomorphic transformations with minimum RSSD and
maximum Jmin (above zero) constituted our optimal selection.
In LDDMM regularization parameters γ and α determine the shape of the kernels
associated to the linear operators L = γ − α∇2 and (L† L)−1 in Fourier domain. Variations of parameter γ produce a shift of kernel values. Therefore, the selection of γ
influences on the magnitude of the velocity field w. Without loss of generality, this
value can be selected equal to 1.0. On the other hand, parameter α determines the
height of the kernel. Therefore, the selection of α is crucial on the smoothness of the
velocity fields v(t) and w. The lower values of α the higher frequency components are
conserved on the action of (L† L)−1 thus allowing larger deformations. As α goes to
0, the linear operators become close to the identity leading to negligible regularization
and non-diffeomorphic solutions.
In this work, we studied the influence of the regularization parameter α and the
weighting factor 1/σ 2 on registration results. Table 6.2 shows the average and standard
deviation of the metrics for parameter selection obtained for different values of these
parameters. As expected, the best image matching was achieved for low values of
α. For fixed α, values for 1/σ 2 since 1.0e3 provided the same average RSSD. Non
diffeomorphic results were obtained from some α0 between 0.001 and 0.0001. The
best average RSSD achieved with diffeomorphic transformations from non-stationary
LDDMM was 13.40% whereas for stationary LDDMM it was 14.04%. Hence, optimal
parameters for both non-stationary and stationary LDDMM were selected as α = 0.001
and 1/σ 2 = 1.0e4.
6.3.3
Evaluation framework
Image matching after registration was assessed from the residual image differences after
registration, Ii ◦ϕ−1
i −Iref and the intensity variance associated to these residual images.
The intensity variance is defined as
IV =
N
N
1 1 2
(Ii ◦ ϕ−1
−
I)
where
I
=
(Ii ◦ ϕ−1
i
i )
N − 1 i=1
N i=1
(6.17)
and N is the number of images in the database. This measure was proposed to compare registration performance of different registration algorithms in terms of image
matching in the non-rigid image registration evaluation project (NIREP) [43]. The
use of the image variance is motivated by the fact that if the similarity of the images
after registration is high, the average I conserves most of the sharp details found on
Iref . Otherwise, this average results into a blurred image. The intensity variance can
6.3. Results
152
Table 6.2: Average and standard deviation of the RSSD (%) (upper row) and Jmin (lower row) for
different values of the regularization parameters α and 1/σ 2 . Metric values associated to the selected
parameters are outlined in boldface. Non-diffeomorphic results are outlined in red.
Non-stationary LDDMM
HH α
1.0
1/σ 2 HH
1.0
1.0e2
1.0e3
1.0e4
1.0e5
1.0e2
1.0e3
1.0e4
1.0e5
0.0050
0.0025
0.0010
0.0001
93.00 ± 3.77 80.99 ± 10.20 78.71 ± 2.46 78.59 ± 2.55 69.86 ± 2.67 48.93 ± 1.87
0.59 ± 0.20
0.84 ± 0.02
0.84 ± 0.02
0.83 ± 0.01
0.74 ± 0.04
0.33 ± 0.04
92.59 ± 3.85 37.79 ± 4.88 29.29 ± 3.58 21.95 ± 2.51 14.90 ± 1.59 8.84 ± 0.91
0.54 ± 0.18
0.65 ± 0.04
0.51 ± 0.07
0.34 ± 0.10
0.15 ± 0.06 -0.10 ± 0.26
92.13 ± 4.24 26.11 ± 2.83 21.99 ± 2.34 17.95 ± 1.84 13.40 ± 1.28 7.94 ± 0.78
0.55 ± 0.18
0.43 ± 0.11
0.33 ± 0.13
0.24 ± 0.10
0.12 ± 0.06 -0.50 ± 0.71
91.62 ± 4.92 26.11 ± 2.83 21.99 ± 2.34 17.95 ± 1.84 13.40 ± 1.28 7.94 ± 0.78
0.55 ± 0.17
0.43 ± 0.11
0.33 ± 0.13
0.24 ± 0.10
0.12 ± 0.06 -0.50 ± 0.71
91.62 ± 4.92 26.11 ± 2.83 21.99 ± 2.34 17.95 ± 1.84 13.40 ± 1.28 7.94 ± 0.78
0.55 ± 0.17
Stationary-LDDMM
H
H α
1.0
1/σ 2 HH
1.0
0.01
0.43 ± 0.11
0.01
0.33 ± 0.13
0.0050
0.24 ± 0.10
0.0025
0.12 ± 0.06 -0.50 ± 0.71
0.0010
0.0001
93.93 ± 3.91 78.71 ± 2.35 78.68 ± 2.46 78.58 ± 2.56 73.84 ± 5.22 48.74 ± 3.13
0.70 ± 0.14
0.93 ± 0.01
0.91 ± 0.01
0.88 ± 0.02
0.79 ± 0.06
0.34 ± 0.04
93.76 ± 4.02 37.66 ± 4.85 29.26 ± 3.55 22.34 ± 2.57 16.41 ± 1.77 13.92 ± 1.36
0.67 ± 0.12
0.66 ± 0.05
0.52 ± 0.07
0.36 ± 0.09
0.21 ± 0.06
0.02 ± 0.01
93.58 ± 4.33 25.87 ± 2.78 21.75 ± 2.29 17.85 ± 1.83 14.04 ± 1.39 13.55 ± 1.36
0.67 ± 0.12
0.43 ± 0.11
0.33 ± 0.12
0.24 ± 0.10
0.14 ± 0.05 -0.01 ± 0.03
93.02 ± 5.32 25.87 ± 2.78 21.75 ± 2.29 17.85 ± 1.83 14.04 ± 1.39 13.54 ± 1.36
0.67 ± 0.12
0.43 ± 0.11
0.33 ± 0.12
0.24 ± 0.10
0.14 ± 0.05 -0.03 ± 0.08
93.02 ± 5.32 25.87 ± 2.78 21.75 ± 2.29 17.85 ± 1.83 14.04 ± 1.39 13.54 ± 1.36
0.67 ± 0.12
0.43 ± 0.11
0.33 ± 0.12
0.24 ± 0.10
0.14 ± 0.05 -0.03 ± 0.09
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be used, therefore, to quantify the sharpness of the average. Thus, given different
registration algorithms, the best image matching performance is provided by the one
showing the lowest intensity variance.
In general, non-rigid registration is a highly ill-posed problem. Therefore, two
different registration methods may provide similar image matching with qualitatively
different transformations. In this work we evaluated the local and global similarities
existing between the obtained transformations. The local differences between the diffeomorphic transformations were quantified from the distance between the associated
Jacobian matrices (J = Dϕ). We used a distance defined on the group of symmetric
positive definite matrices Sym+ (3) applied to the strain matrix S = (J T · J)1/2
−1/2
−1/2
dAI (S1 , S2 ) = trace(log(S1 S2 S1 )2 )
(6.18)
This was motivated by the fact that Riemannian metrics defined on Sym+ (3) are
increasingly used in morphometric studies [10, 144]. In contrast to those metrics based
on the logarithm of the Jacobian determinant [143]
dlog (J1 , J2 ) =
(log(det(J1 )) − log(det(J2 )))2
(6.19)
Riemannian metrics use the whole strain matrix information showing a greater discriminative power between transformations. For example, there exist an infinite number of
different Jacobian matrices with the same determinant (dlog = 0) but associated to different strain forces (dAI 0). In the most simple case of Jacobian matrices expressing
local expansion or contraction without√torsion, straightforward computations show that
dlog = | log(RV C)| and dAI (S1 , S2 ) = 3| log(RV C)|, where RV C denotes the relative
volume change existing between J1 and J2 . Therefore, both metrics provide exactly
the same information in these simple cases while dAI constitutes a more informative
measurement in more general cases. Furthermore, dAI is a natural metric defined in
the usual group structure of Sym+ (3) and it is invariant under affine transformations.
Complementing this local metric, we also measured the global differences between
the diffeomorphic transformations from the distances existing between corresponding
grid points
dSSD (ϕ1 (x, y, z), ϕ2 (x, y, z)) =
(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2
(6.20)
where ϕ1 (x, y, z) = (x1 , y1 , z1 ) and ϕ2 (x, y, z) = (x2 , y2 , z2 ).
6.3.4
Registration results in real datasets
Image matching assessment
Figures 6.4 and 6.5 show the residual image differences after registration. In addition,
Figure 6.6 shows the intensity variance associated to these residual images. In general,
6.3. Results
154
both non-stationary and stationary LDDMM showed a high image matching. However,
the performance of both algorithms diminished at grey matter area. This may be due
to the differences of intensity between the images are higher in these locations of the
brain despite intensity normalization. Another reason could be that the regularization
constraints imposed on diffeomorphic registration makes the transformation not able
to warp between structures with such high geometrical variability.
Differences between diffeomorphic transformations
Figure 6.7 shows the values of dAI computed between corresponding diffeomorphic
transformations provided by non-stationary and stationary LDDMM. In addition, Figure 6.8 shows the average of dAI measured on the whole population of patients together
with the average of dSSD . In general, values of dAI showed that the Jacobian matrices
at corresponding grid points were similar. Most of dAI values resulted below 0.5 which
means a RVC less than 1.33 in the absence of torsion. The most remarkable differences
were located at the brain stem, the boundary between the cortex and background at
the occipital lobe and the lateral ventricles. The average dSSD showed that the distance
between corresponding grid points for both families of transformations was under voxel
size except for these locations.
Finally, complementing these results, we display in Figure 6.9 the diffeomorphisms
corresponding to a representative example. In addition, Figures 6.10 and 6.11 show
the vector fields v(0) and w encoding the non-stationary and stationary parameterizations of the path of diffeomorphisms associated to this representative example and the
corresponding non-stationary flow of vector fields v(t).
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Figure 6.2: Real datasets (patients 1-7). Top row, sagittal, coronal and axial views of the image
selected as reference, Iref . Left column group, views of the datasets used in this experimental section,
Ii . Right column group, corresponding intensity differences, Ii − Iref .
6.3. Results
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Figure 6.3: Real datasets (patients 8-14). Top row, sagittal, coronal and axial views of the image
selected as reference, Iref . Left column group, views of the datasets used in this experimental section,
Ii . Right column group, corresponding intensity differences, Ii − Iref .
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non-stationary LDDMM results
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Figure 6.4: Registration in real datasets (patients 1-7). Illustration of sagittal, coronal and
i − Iref . Left column group
corresponds to the results obtained with the non-stationary parameterization. Right column group
corresponds to the results obtained with the stationary parameterization.
6.3. Results
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Figure 6.5: Registration in real datasets (patients 8-14). Illustration of sagittal, coronal and
i − Iref . Left column group
corresponds to the results obtained with the non-stationary parameterization. Right column group
corresponds to the results obtained with the stationary parameterization.
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Figure 6.6: Registration in real datasets. Illustration of sagittal, coronal and axial views of the
intensity variance associated to the populations of transformed images. First row shows the results for
non-stationary LDDMM (max(IV ) = 396.28). Second row shows the results for stationary-LDDMM
(max(IV ) = 350.28).
6.3. Results
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dAI , non-stationary vs stationary LDDMM results
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Figure 6.7: Registration in real datasets. Illustration of sagittal, coronal and axial views of the
local differences existing between the transformations obtained with non-stationary and stationary
LDDMM registration algorithms measured in terms of the dAI metric. Left column group shows the
results from patieents 1 to 7. Right column group shows the results from patients 8 to 14.
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dAI , non-stationary vs stationary LDDMM results
1
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1
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0
Figure 6.8: Registration in real datasets. First row, illustration of sagittal, coronal and axial
views of the average of the metric dAI between non-stationary and stationary-LDDMM transformations through the database of patients. Second row, average distance dSSD between corresponding
grid points.
6.3. Results
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Figure 6.9: Registration in real datasets (patient #10). First row, superimposed 2D views of
the transformations obtained with non-stationary (white grids) and stationary (blue grids) LDDMM
registration projected onto corresponding sagittal, coronal and axial planes. Second row, values of
distance dAI between both transformations in these planes.
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Figure 6.10: Registration in real datasets (patient #10). Upper row, illustration of sagittal views
of the transformations obtained with non-stationary (left) and stationary (right) LDDMM registration
algorithms superimposed on the saggital slice image, I10 . Lower row, generators of the corresponding
path parameterizations v(0) and w, respectivelly. Both grids and glyphs are colored with respect to
displacement and vector magnitude, respectively.
Figure 6.11: Registration in real datasets (patient #10). Illustration of the flow of velocity
fields corresponding to the non-stationary path parameterization, v(t), t = 1, ..., 9. Glyphs are colored
with respect to its magnitude.
6.3. Results
6.3.5
164
Simulated datasets
The population of simulated brain anatomical images was composed by two sets of
14 images. Each set was generated from the composition of Iref and diffeomorphic
transformations parameterized using simulated non-stationary and stationary flows of
vector fields, respectivelly.
Stationary vector fields were randomly simulated from the distribution learned from
the sample of vector fields w1 , ..., wN resulting from registration with stationary LDDMM in real datasets with the methods described in Chapter 8. To this end, Principal
Component Analysis (PCA) was performed on the covariance matrix associated to
the residuals wi − w where w is the mean associated to the sample of vector fields.
New instances of stationary vector fields were generated from the most representative modes
K of variation u1 , ..., uK and their corresponding eigenvalues λ1 , ..., λK as
wnew = i=1 αi λi ui . Parameters αi were randomly selected from a normal distribution of zero-mean and standard deviation one. Simulated diffeomorphisms were
obtained computing the corresponding group exponential map. The minumum Jacobian determinant achieved by the resulting simulated transformations was equal to
0.03. Figures 6.14 and 6.15 show sample slices of this simulated database of images.
Non-stationary flows of vector fields were generated from the distribution learned
from the sample of non-stationary flows of vector fields v1 (t), ..., vN (t) resulting from
the registration with non-stationary LDDMM in real datasets. As the space of timevarying vector fields is non linear, the computation of PCA was applied to the residuals
of the initial vector fields v1 (0), ..., vN (0) instead. This way, new instances of initial
vector fields were generated as explained in the stationary case. The non-stationary
vectors associated to the simulated initial vector fields were generated via geodesic
shooting [172]. Simulated diffeomorphisms were obtained solving the corresponding
transport equation. In this case, the minimum Jacobian determinant achieved by the
resulting simulated transformations was 0.04. Figures 6.12 and 6.13 show sample slices
of this simulated database of images.
6.3.6
Registration results in simulated datasets
Figures 6.16 and 6.17 show the residual image differences associated to the datasets simulated from non-stationary diffeomorphisms. Figures 6.18 and 6.19 show the residual
image differences after registration associated the datasets simulated from stationary
diffeomorphims, Ii ◦ϕ−1
i −Iref . Figures 6.20 and 6.21 show the intensity variance associated to these residual images. Table 6.3 shows the RSSD and Jmin values corresponding
to registration results.
Both non-stationary and stationary LDDMM showed to improve image matching
results obtained in real datasets experiments as photometric differences between the
samples and the reference image are negligible in these cases. In fact, they showed
an almost exact image matching regardless the parameterization used to generate the
165
simulated datasets.
Table 6.3: Average and standard deviation of the RSSD (%) and Jmin values associated to registration experiments in simulated datasets.
non-stationary datasets
stationary datasets
non-stationary LDDMM stationary-LDDMM
non-stationary LDDMM stationary-LDDMM
RSSD
3.11 ± 1.57
3.81 ± 1.95
RSSD
3.80 ± 0.58
3.81 ± 0.60
Jmin
0.10 ± 0.07
0.10 ± 0.06
Jmin
0.10 ± 0.04
0.10 ± 0.06
Figures 6.22 and 6.23 show the values of dAI computed between the diffeomorphic transformations provided by non-stationary and stationary LDDMM and the corresponding
non-stationary ground truth transformations. Figures 6.24 and 6.25 show the values
of dAI between these transformations and the corresponding stationary ground-truth
transformations. In addition, Figures 6.26 and 6.27 show the average of distances dAI
and the average distance between corresponding grid points, dSSD .
Values of dAI showed the great similarity of the Jacobian matrices of the registration
results with respect to the ground truth transformations. Most of dAI values resulted
below 0.1 which means a RVC with respect to the ground truth less than 1.05 in the
absence of torsion. These results were consistent regardless the parameterization used
to generate the ground truth transformations. The average distance existing between
corresponding grid points, dSSD , resulted in the majority of cases under voxel size. The
most remarkable differences were located at the brain stem and cerebellum
Finally, complementing these results, we display in Figures 6.28 and 6.29 the diffeomorphisms corresponding to representative examples superimposed to the correspnding
ground truth.
6.3. Results
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Figure 6.12: Non-stationary simulated datasets (patients 1-7). Top row, sagittal, coronal and
axial views of the image selected as reference, Iref . Left column group, views of the datasets used in
this experimental section, Ii . Right column group, corresponding intensity differences Ii − Iref .
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Figure 6.13: Non-stationary simulated datasets (patients 8-14). Top row, sagittal, coronal and
axial views of the image selected as reference, Iref . Left column group, views of the datasets used in
this experimental section, Ii . Right column group, corresponding intensity differences Ii − Iref .
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Figure 6.14: Stationary simulated datasets (patients 1-7). Top row, sagittal, coronal and axial
views of the image selected as reference, Iref . Left column group, views of the datasets used in this
experimental section, Ii . Right column group, corresponding intensity differences, Ii − Iref .
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Figure 6.15: Stationary simulated datasets (patients 8-14). Top row, sagittal, coronal and axial
views of the image selected as reference, Iref . Left column group, views of the datasets used in this
experimental section, Ii . Right column group, corresponding intensity differences, Ii − Iref .
6.3. Results
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Figure 6.16: Registration in non-stationary simulated datasets (patients 1-7). Illustration
of sagittal, coronal and axial views of the difference images between the reference and the deformed
images obtained with LDDMM registration algorithms. Left column group corresponds to the registration results obtained with the non- stationary parameterization. Right column group corresponds
to the results obtained with the stationary parameterization.
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Figure 6.17: Registration in non-stationary simulated datasets (patients 8-14). Illustration
of sagittal, coronal and axial views of the difference images between the reference and the deformed
6.3. Results
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Figure 6.18: Registration in stationary simulated datasets (patients 1-7). Illustration of sagittal, coronal and axial views of the difference images between the reference and the deformed images
obtained with LDDMM registration algorithms. Left column group corresponds to the registration
results obtained with the non- stationary parameterization. Right column group corresponds to the
results obtained with the stationary parameterization.
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Figure 6.19: Registration in stationary simulated datasets (patients 8-14). Illustration of
sagittal, coronal and axial views of the difference images between the reference and the deformed
6.3. Results
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Figure 6.20: Registration in non-stationary simulated datasets. Illustration of sagittal,
coronal and axial views of the intensity variance associated to the populations of transformed images.
First row shows the results for non-stationary LDDMM (max(IV )= 84.42). Second row shows the
results for stationary-LDDMM (max(IV ) = 72.07).
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Figure 6.21: Registration in stationary simulated datasets. Illustration of sagittal, coronal
and axial views of the intensity variance associated to the populations of transformed images. First
row shows the results for non-stationary LDDMM (max(IV ) = 50.42). Second row shows the results
for stationary-LDDMM (max(IV ) = 20.27).
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non-stationary LDDMM results vs ground truth
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1.5
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1.5
coronal and axial views of the local differences existing between the ground truth transformations and
the results of non-stationary LDDMM algorithm measured in terms of the dAI metric. Left column
group shows the results from patieents 1 to 7. Right column group shows the results from patients 8
to 14.
6.3. Results
176
stationary-LDDMM results vs ground truth
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1.5
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1.5
coronal and axial views of the local differences existing between the ground truth transformations and
the results of stationary-LDDMM algorithm measured in terms of the dAI metric. Left column group
shows the results from patieents 1 to 7. Right column group shows the results from patients 8 to 14.
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non-stationary LDDMM results vs ground truth
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and axial views of the local differences existing between the ground truth transformations and the
results of non-stationary LDDMM algorithm measured in terms of the dAI metric. Left column group
6.3. Results
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stationary-LDDMM results vs ground truth
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and axial views of the local differences existing between the ground truth transformations and the
results of stationary-LDDMM algorithm measured in terms of the dAI metric. Left column group
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Figure 6.26: Registration in non-stationary simulated datasets. First two rows, illustration of
saggital, coronal and axial views of the average of the metric dAI between ground truth non-stationary
transformations and the results obtained from non-stationary and stationary-LDDMM, respectivelly.
Last two rows, average distance between corresponding grid points.
6.3. Results
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Figure 6.27: Registration in stationary simulated datasets. First two rows, illustration of
saggital, coronal and axial views of the average of the metric dAI between ground truth stationary
transformations and the results obtained from non-stationary and stationary LDDMM, respectivelly.
Last two rows, average distance between corresponding grid points.
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non-stationary LDDMM registration vs ground truth
stationary-LDDMM registration vs ground truth
Figure 6.28: Non-stationary simulated datasets (patient #1). Illustration of sagittal, coronal and axial views of the 2D projections of the transformations obtained with non-stationary and
stationary LDDMM registration (blue grid) superimposed to the ground truth transformation (white
grid). First row shows the results corresponding to non-stationary LDDMM registration. Second row
shows the results corresponding to stationary-LDDMM registration.
non-stationary LDDMM registration vs ground truth
stationary-LDDMM registration vs ground truth
Figure 6.29: Stationary simulated datasets (patient #7). Illustration of sagittal, coronal and
axial views of the 2D projections of the transformations obtained with non-stationary and stationary
LDDMM registration (blue grid) superimposed to the ground truth transformation (white grid). First
row shows the results corresponding to non-stationary LDDMM registration. Second row shows the
results corresponding to stationary-LDDMM registration.
6.3.7
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Efficiency
On one hand, optimization in non-stationary LDDMM requires the storage in memory
of the time-varying flow of vector fields v(t), the energy gradients ∇Ev (v)(t), the flow
of deformed images I0 ◦ φ(t)−1 and I1 ◦ φ(1) ◦ φ(t)−1 , and the Jacobian determinants
det(Dφ(1) ◦ φ(t)−1 )). If the path of diffeomorphisms is sampled into T pieces, memory
and time requirements in this algorithm linearly increase with this parameter. On the
other hand, optimization in stationary LDDMM just requires the storage in memory
of the vector field w, the energy gradient ∇Ew (w) and the deformed image I0 ◦ ϕ−1 .
As example, in a volume of size 128 × 128 × 110 registration using the non-stationary
parameterization required up to 790 MB while stationary parameterization required
about 460 MB. Time requirements for a single iteration took up to 65.04 seconds using
the non-stationary parameterization whereas the stationary parameterization took 6.01
seconds in a 64-bits machine of 2.33 GHZ.
6.4
In this Chapter, we have presented a method for diffeomorphic registration based on the
LDDMM paradigm spreadly used in Computational Anatomy. Traditional LDDMM
methods compute the transformation connecting two anatomical images from a path
of diffeomorphisms parameterized by a time-varying flow of vector fields. In contrast,
our method restricts transformations to belong to paths identified with one-parameter
subgroups. These paths are parameterized by constant-time flows of vector fields.
In the experimental Section, we have studied the influence of the parameterization
on the performance of LDDMM registration. Registration results with non-stationary
and stationary LDDMM were both qualitatively and quantitatively compared in a
database of anatomical MRI brain images. In addition, the accuracy of both algorithms was evaluated in simulated datasets. The metrics used for comparison and
evaluation aimed at the assessment of image matching quality and similarity between
transformations.
Previously to the generation of registration results, a parameter selection study was
performed. Registration results were sensitive to the selection of parameters 1/σ 2 and
α. The minimum RSSD that was achieved with diffeomorphic transformations was
approximately equal to 13% for non-stationary diffeomorphisms and 14% for stationary
diffeomorphisms.
Both non-stationary and stationary LDDMM algorithms showed to be equivalent in
terms of the image matching and the local differences between corresponding transformations. In addition, both algorithms provided transformations similar to the ground
truth regardless the parameterization used to generate the simulated data. In addition, results obtained with the metric dAI showed that both algorithms could provide
comparable results in morphometric studies based on Jacobian metrics.
Regarding time and memory requirements, stationary LDDMM showed to provide
a considerable reduction of the computational requirements for registration. Therefore,
183
our algorithm may provide an alternative efficient method for computing diffeomorphic
registration in state of the art Computational Anatomy applications (for example, in
the computation of anatomical atlases from group-wise diffeomorphic registration [123]
or temporal regression [55]). Moreover, stationary LDDMM allows to generate elements
belonging to one-parameter subgroups of diffeomorphisms where the group exponential
and logarithm maps can be computed using algebraic techniques. Therefore, it may be
suitable to be used in a framework for statistical analysis on groups of diffeomorphisms.
This issue will be explored in Chapter 8.
However, as a consequence of the non surjectivity of the group exponential map
pointed out in Chapter 5, there exist points arbitrarily close to the identity that cannot be parameterized by stationary vector fields. This means that there may exist two
images where the non-stationary parameterization would provide much better registration performance than the stationary parameterization. However, the experiments
reported in this work showed that, at least for MRI anatomical brain images, one can
find elements from both parameterizations that provide similar and acceptable registration results. This may support the hypothesis provided in Chapter 5 which states
that the space Dif f s (Ω) may be too big for dealing with Computational Anatomy
applications.
As future directions, it would be interesting to further explore the equivalence
of both non-stationary and stationary parameterization in a wider range of datasets,
different anatomies, and Computational Anatomy applications. For example, a good
starting point would be to investigate whether both algorithms would provide the
same classification results in morphometry, statistical modeling or group discrimination
studies.
Chapter 7
Comparing algorithms for efficient
diffeomorphic registration
Abstract
During the development of this Thesis, the stationary parameterization of diffeomorphisms has been being increasingly used for diffeomorphic registration.
In particular, two different registration algorithms have been simultaneously
proposed for Computational Anatomy applications: stationary-LDDMM and
diffeomorphic Demons. Stationary-LDDMM has been formulated into the
solid LDDMM theoretical framework. As shown in Chapter 6, this algorithm
is able to provide similar results to non-stationary LDDMM alleviating the
computational complexity. Diffeomorphic Demons has been formulated as an
extension of Demons algorithm for computing diffeomorphic transformations.
It has been proposed as an efficient diffeomorphic registration algorithm
where optimization is approached using second-order schemes. Although
both methods have arisen from different backgrounds, they consider non-rigid
registration as a diffusion process. Moreover, they can be fitted into similar
variational formulations with the same image matching metric and close
characterizations for the diffeomorphic transformations. This Chapter aims
at providing a comparison between both algorithms in a common framework.
To this end, we have first developed an efficient second order method
for optimization within stationary-LDDMM. Then, we have discussed the
differences in the elements of both registration scenarios and compared the
performance of the registration results. In addition, we have studied the
possible advantages of both algorithms for dealing with the computation of
statistics on diffeomorphisms.
7.1. Introduction
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Contents
7.1
Introduction
7.2
Gauss-Newton optimization in stationary-LDDMM
7.2.1
7.3
7.4
7.5
7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
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188
. . . . . . . . . . . . . . . . . . . . . . . . 190
Stationary-LDDMM vs Diffeomorphic Demons . . . . . . . . . . .
191
7.3.1
General variational formulation
7.3.2
Characterization of diffeomorphic transformations . . . . . . . . . . . 191
7.3.3
Image similarity metric . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.3.4
Regularization energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.3.5
Optimization scheme
Results
. . . . . . . . . . . . . . . . . . . . . 191
. . . . . . . . . . . . . . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
196
7.4.1
7.4.2
Regularization parameters selection . . . . . . . . . . . . . . . . . . . 197
7.4.3
Registration results
7.4.4
Suitability for the computation of statistics . . . . . . . . . . . . . . . 200
7.4.5
Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
213
Introduction
The Large Deformation Diffeomorphic Metric Mapping (LDDMM) is considered the
reference paradigm for diffeomorphic registration in Computational Anatomy [67, 23].
As shown in Chapter 5, the majority of the LDDMM variations are focused on the definition of the image matching energy with the non-stationary characterization of the
diffeomorphic transformations. Much less attention has been paid to the optimization
strategy where classical gradient descent method is often used combined with more or
less sophisticated line search techniques [23, 282]. Despite to the solid foundations of
the mathematical framework, the slow rate of convergence of gradient descent optimization together with the high computational complexity inherent to the non-stationary
parameterization make this methodology not much attractive for clinical applications
where more efficient registration algorithms are usually preferred.
In Chapter 6 it was shown that the use of the stationary parameterization of diffeomorphisms for LDDMM registration provides comparable results in terms of image
matching and differences between the diffeomorphic transformations while drastically
reducing memory and time requirements. Indeed, the use of efficient second-order optimization techniques may improve the rate of convergence during optimization making
stationary-LDDMM suitable for clinical applications requiring fast registration.
187
Chapter 7. Comparing algorithms for efficient diffeomorphic registration
Recently, Ashburner et al. have proposed a version of Gauss-Newton’s method for
optimization in stationary-LDDMM [11]. The computations of the Gâteaux derivatives of the objective function are performed in the space of L2 -functions. Hence, the
action of the linear operator involved in the regularization term has to be formulated
using its matrix representation [175]. In consequence, the algorithm results into a
high dimensional matrix inversion problem and memory requirements for diffeomorphic registration hinder their execution in standard machines. Although, as shown in
Chapter 5, matrix inversion could be efficiently approached by solving a sparse system of equations using well known multi-grid techniques [199], this solution involves
an iterative numerical algorithm that depends on the definition of the linear operator.
Moreover, this optimization scheme does not provide an straightforward extension from
the original LDDMM gradient descent implementation.
Thus, the first contribution of this Chapter is to propose a version of GaussNewton’s method for optimization in stationary-LDDMM as alternative to Ashburner
et al.’s implementation. The computations of the Gâteaux derivatives of the objective
function are performed in the tangent space of the Riemannian manifold of diffeomorphisms. This way, the action of the linear operator involved in the regularization term
can be directly formulated using convolution. As a result, the proposed optimization
scheme results into three-dimensional operations for each point in the computation
domain that can be executed in standard machines. Moreover, it is independent of the
definition of the linear operator and it can be considered a direct extension of the original LDDMM gradient descent. Furthermore, the compactness of the linear operator
associated to the metric defined in V provides a convincing argument for supporting
optimization on this space rather than on L2 [23].
As shown in Chapter 5, alternative efficient second order methods for diffeomorphic
registration have been proposed from the extension of Demons algorithm for computing
diffeomorphic transformations [251]. As alternative to LDDMM formulation, diffeomorphic Demons performs optimization on the group of diffeomorphisms. Moreover,
the stationary parameterization is used for diffeomorphic transformation characterization.
In essence, stationary-LDDMM was formulated into the theoretical complexity of
the non-stationary LDDMM framework although it resulted into a much more efficient
algorithm while providing similar registration results. Diffeomorphic Demons was intended as an extension of original Demons algorithm suitable for practical applications
due to its efficiency and the quality of registration results. Although both methods
have arisen from different backgrounds, they consider non-rigid registration as a diffusion process [175]. They can be fitted into a similar variational formulation with
the same metric for image similarity and close characterizations for the diffeomorphic
transformations. Moreover, both algorithms provide efficient second-order implementations of diffeomorphic registration intended to spread the use of diffeomorphisms in
Computational Anatomy applications requiring efficient registration algorithms.
The second contribution of this Chapter is to provide a comparison between both
algorithms in a common framework. To this end, the elements of the registration
7.2. Gauss-Newton optimization in stationary-LDDMM
188
scenario (transformation characterization, image metric, regularization and optimization schemes) have been discussed for both methods. In the experimental section we
have compared the performance of registration results. Moreover, we have studied the
potential of both algorithms for dealing with the computation of statistics on diffeomorphisms. We have focused on two different desirable aspects related to the computation
of statistics, namely, the inverse consistency associated to the registration algorithms,
and the smoothness of the elements in the tangent space needed to perform the statistics. As will be shown in the next Chapter, these constitute essential issues for further
statistical analysis.
The method for Gauss-Newton optimization in stationary-LDDMM was presented
at the international conference ISBI’08 [111]. The comparison between stationaryLDDMM and diffeomorphic Demons was presented at the international workshop
MFCA’08 that was held in conjunction with MICCAI’08 conference [112].
The rest of the Chapter is divided as follows. In Section 7.2 we introduce the
method for Gauss-Newton optimization in stationary-LDDMM. Section 7.3 discusses
the most remarkable theoretical differences existing between stationary-LDDMM and
diffeomorphic Demons. Section 7.4 presents the comparison results. Finally, Section 7.5
provides the most remarkable conclusions and perspectives.
7.2
Gauss-Newton optimization in
stationary-LDDMM
In this Section we present the computations to obtain Gauss-Newton’s optimization
scheme associated to the minimization of the stationary-LDDMM energy functional
E(w) = w2V +
1
I0 ◦ Exp(w)−1 − I1 2L2
σ2
(7.1)
In order to perform the computations, we must assume that E(w) is a twice Frechet
differentiable energy functional defined in the tangent space V , and V is a convex vector
space. It should be noted that the computations for the inverse consistent version of
stationary LDDMM can be obtained in a straightforwad manner.
Among second-order optimization techniques, Newton’s method is the most popular
one. In Newton’s method, the minimization of E(w) is approached with the iterative
scheme
wk+1 = wk − · Hw E(wk )−1 · ∇w E(wk )
(7.2)
where the search direction exploits not only the gradient but also the Hessian structure
of the energy thus providing a faster rate of convergence than methods just based on
the gradient. Gradient and Hessian expressions depend on the second-order Taylor
189
approximation of the energy functional. In this work we deal with the approximation
provided by
1
E(w + h) = E(w) + ∇w E(w), hV + h, Hw E(w)hV + O(h3V )
2
(7.3)
where ∇w E(w) and Hw E(w) are the Frechet differentials defined in V .
As shown in Chapter 6, the Gâteaux derivative of an energy functional E(w) along
h ∈ V is defined as its variation under the perturbation of w in the direction of h,
∂h E(w) where
E(w + h) − E(w)
→0
∂h E(w) = lim
(7.4)
In addition, the second-order Gâteaux derivative, ∂hh E(w) can be computed recursively
from the first order derivative. In Frechet spaces, the gradient operator relates Frechet
differentials and Gâteaux derivatives (whenever both derivatives exist) by
∂h E(w) = ∇w E(w), hV
(7.5)
Moreover, the Hessian operator relates second-order Frechet differentials and secondorder Gâteaux derivatives by
∂hh E(w) = h, Hw E(w) hV
(7.6)
The computation of Newton’s equations for the energy functional defined in Equation 7.1 can be derived using Gâteaux derivatives. For simplicity, we divide E(w) =
E1 (w) + σ12 E2 (w) where E1 (w) = w2V and E2 (w) = I0 ◦ Exp(w)−1 − I1 2L2 , and
compute ∂h Ei (w) and ∂hh Ei (w), i = 1, 2. Straightforward computations provide the
derivatives related to E1
∂h E1 (w) = 2w, hV
and ∂hh E1 (w) = 2h, hV
(7.7)
The first and second-order variations of E2 (w) in the direction of h are computed using
the chain rule in L2 and projecting the results into the space V using the inverse of
operator L† L. In the computations, the Gâteaux derivative of the exponential map is
computed using a first order approximation as in Chapter 6. In order to simplify the
notation, let denote with r = I0 ◦ Exp(w)−1 − I1 the residual function associated to
E2 (w), J = ∇(I0 ◦ Exp(w)−1 ) the gradient and H = Hess(I0 ◦ Exp(w)−1 ) the Hessian
matrix associated to the transformed template. With this notation,
7.2. Gauss-Newton optimization in stationary-LDDMM
∂h E2 (w) = −2r J, hL2 = −2(L† L)−1 (r J), hV
∂hh E2 (w) = 2h, (J T · J + r H) · hL2
= 2h, (L† L)−2 (J T · J + r H) · hV
190
(7.8)
Finally, Newton’s equations can be obtained from the combination of Equations 7.2
to 7.8
w
k+1
−1
2 † −2 T
=w − ·
2 IR3 + 2 (L L) (J · J + r H)
·
σ
2 † −1
2 w − 2 (L L) (r J)
σ
k
(7.9)
In practice, the computation of the Hessian term Hw E(w) often leads to numerical
problems due to the term J T · J + r H is not always guaranteed to be positive definite during optimization. Gauss-Newton’s method is often used as a simplification of
Newton’s method that overcomes this limitation using a linear approximation of this
term. Thus, Gauss-Newton’s method for the minimization of E(w) is approached with
the iterative scheme
w
k+1
−1
2 † −2 T
=w − ·
2 IR3 + 2 (L L) (J · J)
·
σ
2 † −1
2 w − 2 (L L) (r J)
σ
k
(7.10)
It should be noted that this formula may be extended to the efficient second-order
method (ESM) proposed by Malis et al. [158] just by replacing the Jacobian term
J = ∇(I0 ◦ Exp(w)−1 ) by JESM = ∇(I0 ◦ Exp(w)−1 ) + ∇I1 .
7.2.1
In this work, the algorithm for finding the solution of efficient second-order stationaryLDDMM initializes with iteration k = 0, w = 0V and ϕ = id. The selection of the
optimal step size is approached using a backtracking inexact line-search strategy starting from an initial guess 0 . In the case of Gauss-Newton optimization, the parameter
0 is selected equal to 1 in order to guarantee a super-linear rate of convergence [180].
In each iteration, the step size is selected to be the first parameter that provides a
sufficient decrease in the energy according to Armijo’s condition (in this case, scalar
products are given in L2 space)
191
E(wk − k Hw E(wk )−1 · ∇w E(wk )) ≤
E(wk ) + c1 k (Hw E(wk )−1 · ∇w E(wk ))T · ∇w E(wk ) (7.11)
The computation of the solution to the stationary transport equation is performed
using the Scaling and Squaring method [8] as in [251]. The computation of the action
of operators (L† L)−1 and (L† L)−2 are implemented through Fourier domain for each
component of the energy gradient and Hessian, respectively.
Finally, convergence in each resolution level is considered if the value of in the
search strategy is too small or the absolute rate of convergence in the energy is less
than a tolerance value.
7.3
Stationary-LDDMM vs Diffeomorphic Demons
In this Section we proceed to define the common variational framework for stationaryLDDMM and diffeomorphic Demons and discuss the fundamental differences that can
be found in the elements of both registration scenarios.
7.3.1
General variational formulation
In Computational Anatomy, diffeomorphic registration is defined as a variational problem involving the characterization of the diffeomorphic transformations, an image metric to measure the similarity between the images after registration, a regularization
constraint to favor stable numerical solutions, and an optimization technique to search
for the optimal transformation in the space of valid diffeomorphisms. The transformation that deforms the source I0 into the target I1 is defined as the result from the
E(ϕ) =
1
2
σsim
2
1/σsim
Esim (I0 , I1 , ϕ) +
1
Ereg (ϕ)
2
σreg
(7.12)
2
where the weighting factors
and 1/σreg
balance the energy contribution between
image similarity and regularization. In this common variational formulation, it is possible to fit both non-stationary LDDMM [23] and the efficient second-order methods that
have recently arisen in the literature: stationary-LDDMM [11, 112] and diffeomorphic
Demons [251].
7.3.2
Characterization of diffeomorphic transformations
Stationary-LDDMM
In the LDDMM framework [239, 67], transformations are assumed to belong to a group
of diffeomorphisms endowed with a Hilbert differentiable structure of Riemannian man-
7.3. Stationary-LDDMM vs Diffeomorphic Demons
192
ifold, Dif f s (Ω). The tangent space at the identity V , is a set of second-order Sobolev
class vector fields in Ω. The Riemannian metric is defined from the scalar product
v, wV = Lv, LwL2 where L is a linear invertible differentiable operator. Diffeomorphic transformations are represented by the end point ϕ = φ(1) of paths of diffeomorphisms φ(t) parameterized by time-varying flows v(t) of vector fields from the solution
of the transport equation φ̇(t) = v(t, φ(t)). The Sobolev structure
in V guarantees the
1
existence of diffeomorphic solutions for these equations if 0 v(t)2V dt < ∞.
Stationary LDDMM is embedded into the solid LDDMM theoretical framework
although paths of diffeomorphisms are parameterized by constant-time flows of vector
fields. This stationary parameterization is closely related to the group structure defined
in Dif f s (Ω) as the paths starting at the identity parameterized using stationary vector
fields are exactly the one-parameter subgroups. Diffeomorphisms belonging to oneparameter subgroups can be computed from the group exponential map Exp : V →
Dif f s (Ω)
ϕ = Exp(w)
(7.13)
where w constitutes the infinitesimal generator of the subgroup [8]. Thus, stationary
LDDMM restricts transformations to diffeomorphisms belonging to one-parameter subgroups. As pointed out in Chapter 5, it has been shown that the set of diffeomorphisms
obtained with the stationary parameterization do not comprise all diffeomorphisms in
Dif f s (Ω) [94]. Nevertheless, we showed in Chapter 6 that the stationary parameterization is able to provide a performance similar to the more general non-stationary
parameterization on the registration of MRI brain anatomical images.
In diffeomorphic Demons, transformations are assumed to belong to the group of diffeomorphisms Dif f (Ω). In contrast to the LDDMM framework, no Riemannian structure
is explicitly considered in Dif f (Ω). Diffeomorphic transformations are represented as
the composition of
ϕ = ψ ◦ Exp(u)
(7.14)
where ψ is an element in Dif f (Ω) and u is a vector field in Ω belonging to a convenient
space of vector fields that would guarantee the existence of exponential map and that
the composition ψ ◦ Exp(u) remains in Dif f (Ω). To our knowledge, this space has not
been characterized with the same accuracy than in the LDDMM framework.
This representation of diffeomorphisms restricts transformations to any element in
Dif f (Ω) that can be obtained by finite composition of exponentials of smooth vector
fields ϕ = Exp(u1 ) ◦ ... ◦ Exp(uN ). This representation resembles the original proposal
found in Christensen et al.’s large deformation kinematics registration [44].
193
7.3.3
Image similarity metric
Stationary-LDDMM
In stationary LDDMM the image similarity energy is defined from
Esim (I0 , I1 , ϕ) = I0 ◦ Exp(w)−1 − I1 2L2
(7.15)
This term could be replaced by some other energies proposed in non-stationary LDDMM (as mutual information or cross correlation, among others [154, 16]). In stationary LDDMM, Exp(−w) and Exp(w) are exact inverse transformations. Therefore, the
inverse consistent version of the image matching energy for stationary LDDMM simply
corresponds to
Esim (I0 , I1 , ϕ) = I0 ◦ Exp(w)−1 − I1 2L2 + I1 ◦ Exp(w) − I0 2L2
(7.16)
Diffeomorphic Demons is associated to the minimization of the sum of squared differences
Esim (I0 , I1 , ϕ) = I0 ◦ ψ ◦ Exp(u) − I1 2L2
(7.17)
This term is inherent to Demons’s algorithm and, therefore, it cannot be replaced by
any other. The inverse consistent version of the image matching energy corresponds to
Esim (I0 , I1 , ϕ) = I0 ◦ ψ ◦ Exp(u) − I1 2L2 + I1 ◦ ζ ◦ Exp(w) − I0 2L2
(7.18)
subject to (ψ ◦Exp(u))−1 = ζ ◦Exp(w). In this case, minimization involves the solution
of a constrained optimization problem leading to a more complex algorithm for general
expressions of ψ and ζ.
7.3.4
Regularization energy
Stationary-LDDMM
In stationary LDDMM the regularization term is defined as the norm in V of the
infinitesimal generator w associated to the diffeomorphism ϕ,
Ereg (ϕ) = w2V = Lw2L2
(7.19)
The regularization term favors solutions to belong to one-parameter subgroups with
small energy preventing transformations to be non-diffeomorphic. The regularization
term depends on the selection of the operator L that is usually related to the physical
deformation model imposed on Ω. However, it remains an open question how to choose
the best model in non-rigid registration algorithms [175, 11]. In this Thesis we use the
7.3. Stationary-LDDMM vs Diffeomorphic Demons
194
diffusive model L = Id − α∇2 . This selection restricts w to lie on a space of Sobolev
class two. Thus, smoothness is required on first and second-order derivatives for valid
solutions.
In Demons framework regularization is externally imposed using Gaussian smoothing
on ϕ and u. This way, the physical deformation model imposed on Ω is roughly
equivalent to the combination of a diffusive and a fluid model [194]. It has been
shown that the effect of this Gaussian smoothing is equivalent to using the harmonic
regularization
Ereg (ϕ) = Dϕ − I2fro
(7.20)
in Equation 7.12 [27]. It should be noted that just smoothness in first derivatives is
required for valid solutions.
7.3.5
Optimization scheme
Stationary-LDDMM
In stationary LDDMM, optimization is performed on the tangent space associated
to Dif f s (Ω) (optimization on Hilbert spaces). Although classical gradient descent is
usually used for numerical optimization [23, 105], more efficient and robust second-order
techniques have been recently proposed by Ashburner et al. in [11] and Hernandez et
al. in [112]. Both methods are based on Newton’s iterative scheme
wk+1 = wk − · Hw E(wk )−1 · ∇w E(wk )
(7.21)
although they differ on the space where Gâteaux derivatives are computed and the
simplification of the Hessian term used to overcome the numerical problems posed by
Newton’s method in cases with bad conditioned Hessians.
In Ashburner et al.’s method, Gâteaux derivatives are computed on the space of L2
integrable functions and Levenberg Marquardt Newton’s simplification is used. The
expressions for the gradient and the Hessian are given by
(∇w E(w))L2 = 2 (L† L)w −
(Hw E(w))L2 = 2 L† L +
2
(I0 ◦ Exp(w)−1 − I1 ) · ∇(I0 ◦ Exp(w)−1 )(7.22)
σ2
2
∇(I0 ◦ Exp(w)−1 )T · ∇(I0 ◦ Exp(w)−1 )
σ2
(7.23)
With this approach, the action of the linear operator L† L has to be formulated using the matrix representation of the convolution. As a consequence, the algorithm
results into a high dimensional matrix inversion problem involving huge computational
requirements. Although inversion is approached by solving a sparse system of linear
195
equations combining Gauss-Seidel with multigrid techniques [199], memory requirements for diffeomorphic registration hinder the execution in standard machines.
In our method, Gâteaux derivatives are computed in the space V using a GaussNewton simplification, which leads to
2 † −1
(L L) ((I0 ◦ Exp(w)−1 − I1 ) · ∇(I0 ◦ Exp(w)−1 ))
(7.24)
σ2
2
+ 2 (L† L)−2 (∇(I0 ◦ Exp(w)−1 )T · ∇(I0 ◦ Exp(w)−1 ))
(7.25)
σ
(∇w E(w))V = 2 w −
(Hw E(w))V = 2 IR3
With this approach, the action of the operators (L† L)−1 and (L† L)−2 can be formulated
using convolution and the update of Equation 7.21 can be computed using pointwise
operations with smaller memory requirements.
Apart from the computational requirements, Beg et al. provided an additional
argument supporting optimization on space V rather than on L2 [23]. The linear
operator K = (L† L)−1 is a compact operator in V . Using results from F. Riesz’s
spectral theory of compact operators, there exists an orthonormal basis (n )n∈N in L2
with corresponding singular values (λn )n∈N such that
K=
(λn ·, n L2 ) · n
(7.26)
n∈N
and λn → 0 as n → ∞ due to operator compactness. The expansion of the gradient
expressions in this basis yields 8
(∇w E(w))L2
1
=
2w, n L2 + −b, n L2 · n
λn
n∈N
(∇w E(w))V =
(2w, n L2 + λn −b, n L2 ) · n
(7.27)
(7.28)
n∈N
where b = (I0 ◦ Exp(w)−1 − I1 ) · ∇(I0 ◦ Exp(w)−1 ). Therefore, whereas the action of the
linear operator (L† L)−1 in Equation 7.24 remains bounded, the action of (L† L) Equation 7.22 results into a high frequency components amplification leading to numerical
instabilities in the computation.
In diffeomorphic Demons optimization is performed on the group of diffeomorphisms
Dif f (Ω) (optimization on Lie groups) using the iterative scheme
8
Analogous conclusions can be inferred from expanding the bilinear form associated to L2 − and
V − Hessian expressions in this basis.
7.4. Results
196
I0 ◦ ϕk − I1
· (∇(I0 ◦ ϕk ))
∇(I0 ◦ ϕk )22 + (I0 ◦ ϕk − I1 )2 /τ 2
= ϕk ◦ Exp( · uk+1 )
uk+1 =
(7.29)
ϕk+1
(7.30)
where second-order techniques are used for the computation of uk [251].
Regularization is performed at the end of each iteration by smoothing the updated uk
and ϕk using Gaussian filters of standard deviation σu and σϕ , respectively. Moreover,
the term (I0 ◦ ϕk − I1 )2 /τ 2 also contributes to the regularization by enforcing the numerical stability of the optimization scheme and controlling the maximum update step
length. This term can be seen as a Levenberg-Maquardt approximation of Newton’s
method. Leaving aside the common variational formulation provided in this work,
an identical optimization scheme can be obtained from a variational formulation (described in Chapter 5) resulting from the introduction of a hidden variable that controls
the correspondences between ϕ and the true transformation [27].
As an alternative to this usual Gauss-Newton optimization, the efficient secondorder scheme introduced in [158] was used in [251]. This led to replace the term
∇(I0 ◦ ϕ) in Equation 7.29 by its symmetric version ∇(I0 ◦ ϕ) + ∇I1 . This was shown
to improve the rate of convergence with respect to the original Gauss-Newton scheme.
7.4
Results
In this experimental section we compare the performance of stationary-LDDMM and
diffeomorphic Demons in the registration of MRI brain datasets. We present results on
the popular Internet Brain Segmentation Repository (IBSR) database. Registration
results are compared using the metrics for image similarity and differences between
transformations presented in Chapter 6. In addition, we study the inverse consistency
error and the smoothness of the transformations, towards the assessment of the suitability of both algorithms for the computation of statistics on populations.
7.4.1
A set of 15 T1-MRI images from the IBSR database were used for comparing the performance of stationary-LDDMM and diffeomorphic Demons registration algorithms.
Image size was 256×256×128 with a voxel size of 0.94×0.94×1.5. The images were acquired at Massachusetts General Hospital and are freely available at http://www.cma.
mgh.harvard.edu/ibsr/data.html. This database incorporates manually-guided expert segmentations of white and grey matters (WM and GM), the cerebrospinal fluid
(CSF) and the most relevant subcortical structures. This database has been extensively
used for the evaluation and comparison of registration and segmentation algorithms (see
references in http://www.cma.mgh.harvard.edu/ibsr/PubsUsingIBSR.html).
197
As preprocessing steps, the images were skull stripped using the manual segmentations. Next, the image intensity was normalized using a histogram matching algorithm.
Finally, both the images and the manual segmentations were aligned to a common coordinate system using a similarity transformation. The algorithms available in the
Insight Toolkit (http://www.itk.org) were used in these preprocessing steps. Figures 7.1 and 7.2 show sample slices from the database of images after preprocessing.
In our experiments, one of the images was randomly selected as reference and the
remaining datasets were registered to this template using stationary-LDDMM and
diffeomorphic Demons algorithms. Stationary-LDDMM was implemented with GaussNewton optimization where gradient and Hessian were computed from Equations 7.24
and 7.25. Both the inverse consistent version of stationary-LDDMM (IC-LDDMM)
and the symmetric gradient optimization scheme (SG-LDDMM) were considered in
this study. Diffeomorphic Demons was run with the symmetric gradient as proposed
in [251]. Results with the forward versions of these algorithms were not considered
in these experiments as the supremacy of the use inverse consistent and symmetric
gradient in the performance of registration results has been extensively shown in the
literature [42, 251, 16]. The algorithms were stopped when the magnitude of the
update was negligible or after a maximum of 100 iterations. The selection of the
optimal regularization parameters is presented below. Other parameters were fixed to
typical values used in previous publications [11, 106, 251].
7.4.2
Regularization parameters selection
As discussed in Chapter 6, the selection of the regularization parameters is a crucial
step in deformable registration that should be performed previously to the analysis of
any registration result. Therefore, a study for parameter selection was performed in
this work. The carried out experiments were similar to those presented in Chapter 6.
2
to 1.0 in order to handle
For stationary-LDDMM, we fixed the parameter 1/σreg
2
on registraparameters selection more easily and studied the influence of α and 1/σsim
tion results. Table 7.1 shows the metrics for parameter selection for different values of
2
= 1.0e4 as optimal.
these parameters. This led us to select α = 0.0025 and 1/σsim
In diffeomorphic Demons, parameters σϕ and σu control the smoothness of the
diffeomorphism ϕ and the velocity field u, respectively. Therefore, the lower values of
these σ, the higher frequency components are conserved on ϕ and u allowing larger
deformations. In addition, the maximum step length is bounded by uL2 ≤ 0.5 · τ . As
τ increases, the maximum magnitude of the velocity field u remains unbounded which
2
may lead to non-diffeomorphic solutions. In this work we fixed the parameters 1/σreg
and σϕ to 1.0 mm. in order to handle parameter selection more easily. Table 7.2 shows
the metrics for parameter selection for different values of σu and τ . Optimal values
were obtained for σu = 1.0 mm. (close to voxel size) and τ = 0.5 mm.
7.4. Results
198
Table 7.1: Stationary LDDMM registration. Average and standard deviation of the RSSD (%)
2
.
(upper row) and Jmin (lower row) for different values of the regularization parameters α and 1/σsim
Metric values associated to the selected parameters are outlined in boldface. Non-diffeomorphic results
are outlined in red. Note that the algorithms do not converge for values α of order 0.0001.
HH
IC-LDDMM. RSSD =
α
2H
1/σsim
H
1.0e3
1.0e4
1.0e5
1.0
0.01
91.56 ± 3.04
0.60 ± 0.24
90.97 ± 3.11
0.59 ± 0.24
90.97 ± 3.11
0.59 ± 0.24
30.53 ± 3.76
0.44 ± 0.11
24.70 ± 3.10
0.31 ± 0.15
24.70 ± 3.10
0.31 ± 0.15
2
−1
−I0 2 2
1 I0 ◦ϕ−I1 L2 +I1 ◦ϕ
L
2
I0 −I1 2
L2
0.0050
0.0025
.
0.0010
0.0001
21.51 ± 2.37 17.42 ± 4.16 12.18 ± 3.17 100.00 ± 0.00
0.27 ± 0.07
0.17 ± 0.06
-0.17 ± 0.64
1.00 ± 0.00
17.55 ± 2.10 13.88 ± 4.13 9.72 ± 3.72 100.00 ± 0.00
0.19 ± 0.08 0.10 ± 0.05 -3.97 ± 12.28 1.00 ± 0.00
17.55 ± 2.10 13.82 ± 4.00
9.61 ± 3.57 100.00 ± 0.00
0.19 ± 0.08
0.10 ± 0.05
-3.99 ± 12.28 1.00 ± 0.00
SG-LDDMM. RSSD =
I0 ◦ϕ−I1 2 2
L
I0 −I1 2 2
.
L
HH
α
2H
1/σsim
H
1.0e3
1.0e4
1.0e5
1.0
0.01
0.0050
0.0025
91.66 ± 2.88
0.65 ± 0.22
91.11 ± 2.73
0.63 ± 0.23
91.11 ± 2.73
0.63 ± 0.23
30.44 ± 3.44
0.44 ± 0.09
28.61 ± 3.44
0.39 ± 0.12
28.83 ± 3.62
0.39 ± 0.12
22.02 ± 2.33
0.26 ± 0.08
20.99 ± 2.38
0.23 ± 0.10
21.49 ± 2.46
0.25 ± 0.10
15.79 ± 1.68
0.11 ± 0.07
14.81 ± 1.59
0.10 ± 0.07
15.38 ± 2.74
0.11 ± 0.07
0.0010
0.0001
10.88 ± 1.21 100.00 ± 0.00
0.02 ± 0.02
1.00 ± 0.00
10.09 ± 1.35 100.00 ± 0.00
0.01 ± 0.01
1.00 ± 0.00
10.09 ± 1.34 100.00 ± 0.00
0.01 ± 0.01
1.00 ± 0.00
Table 7.2: Diffeomorphic Demons registration. Average and standard deviation of the RSSD
(%) (upper row) and Jmin (lower row) for different values of the regularization parameters σsim and
τ . Metric values associated to the selected parameters are outlined in boldface. Non-diffeomorphic
results are outlined in red.
HH σu
HH
τ
0.5
1.0
2.0
7.4.3
Diffeomorphic Demons. RSSD =
I0 ◦ϕ−I1 2 2
L
I0 −I1 2 2
.
L
1.0
1.5
2.0
2.5
3.0
14.88 ± 1.74
0.07 ± 0.03
9.95 ± 1.18
0.00 ± 0.00
10.91 ± 1.79
-0.01 ± 0.02
20.91 ± 2.50
0.15 ± 0.04
13.99 ± 1.69
0.02 ± 0.02
12.60 ± 1.26
-0.00 ± 0.01
31.58 ± 3.67
0.31 ± 0.05
21.73 ± 2.73
0.11 ± 0.05
18.06 ± 2.15
0.02 ± 0.02
40.71 ± 4.34
0.47 ± 0.06
29.37 ± 3.72
0.23 ± 0.07
24.51 ± 3.01
0.10 ± 0.04
48.15 ± 4.61
0.59 ± 0.06
36.19 ± 4.47
0.35 ± 0.09
30.59 ± 3.75
0.19 ± 0.07
Registration results
Rate of convergence
Figure 7.3 shows the curves of the image similarity metric Ii ◦ ϕi − Iref 2L2 during optimization. We included the curves from IC-LDDMM, SG-LDDMM and diffeomorphic
Demons algorithms. In order to show the improved robustness and efficiency provided
by Gauss-Newton optimization with respect to gradient descent we also included the
curves obtained with the first order optimization scheme for the inverse consistent
199
version of stationary-LDDMM (GD-LDDMM).
While IC-LDDMM showed a monotone decreasing convergence, GD-LDDMM was
frequently prone to get trapped into local minima during a considerable number of
iterations. Besides, IC-LDDMM showed a faster convergence rate than GD-LDDMM
specially at the initial iterations of the algorithm. Therefore, Gauss-Newton resulted
into a more efficient and robust optimization technique for diffeomorphic registration
than gradient descent.
Comparing the curves obtained from second-order optimization methods, it can be
seen that IC-LDDMM provided the highest rate of convergence in all cases. At the
initial stages of optimization, diffeomorphic Demons showed the worst performance in
the great majority of cases, although it usually reached SG-LDDMM performance at
convergence.
In order to compare the image matching results from the different registration algorithms, we used the same metrics proposed in Chapter 6. Figures 7.4, 7.5 and 7.6 show
the residual image differences after registration, Ii ◦ ϕi − Iref . In addition, Figure 7.7
shows the intensity variance associated to these residual images.
Both versions of stationary-LDDMM and diffeomorphic Demons showed a similar
image matching. In general, the performance of all the algorithms diminished at grey
matter area. In some cases, all the algorithms showed a poor image matching at the
lateral ventricles due to the high photometric variations shown between the reference
image and the samples in this subcortical structure in the IBSR database. In particular, diffeomorphic Demons showed a poor matching at ventricle tails in cases with
large ventricle deformation. This is consistent with the intensity variance shown in Figure 7.7, that resulted much higher in these locations for diffeomorphic Demons than
for any of the stationary-LDDMM algorithms.
Figure 7.8 shows the average of the distances dAI computed between the diffeomorphic transformations obtained from IC and SG stationary-LDDMM and diffeomorphic Demons measured on the whole population of patients. In addition, Figure 7.9
shows the average of the distances existing between corresponding grid points, dSSD .
Complementing these results, we show in Figure 7.10 the diffeomorphisms and the
corresponding inverses in a representative example.
Values of dAI showed that the Jacobian matrices at corresponding grid points from
IC and SG-LDDMM were similar except at specific locations. In contrast, Jacobian
matrices between both stationary-LDDMM algorithms and diffeomorphic Demons were
different, specially at the cortex. Interestingly, the distances dSSD between corresponding grid points were, in average, under voxel size. This distance was smaller between
stationary-LDDMM algorithms. This means that although the distance between corresponding points in the transformations is similar, the local structure due to the
7.4. Results
200
differences on transformation characterization and regularization between stationaryLDDMM and diffeomorphic Demons results qualitatively different. This can be appreciated in detail in the representative example shown in Figure 7.10.
7.4.4
Suitability for the computation of statistics
Furthermore, in this experimental section we compare the suitability of stationaryLDDMM and diffeomorphic Demons when dealing with the computation of statistics
on a population of diffeomorphisms. This constitutes a crucial task in the study of the
evolution of growth and disease and the study of anatomical variability existing among
populations. We have focused on two desirable aspects related to the computation of
statistics, namely, the inverse consistency error associated to registration algorithms,
and the smoothness of the elements in the tangent space needed to perform statistics.
The inverse consistency error measures the symmetry of registration results regardless the order of input data. Registration algorithms minimizing inverse consistency
error, have shown to improve the accuracy of correspondences and both forward and
backward image similarity after registration [42, 16]. Therefore, these algorithms are
better suited to provide transformations for Computational Anatomy applications consistent with Grenander deformable template model [96].
In this work, the inverse consistency error was computed from the backward RSSD
−1
between the images before and after registration, RSSD(Iref ◦ ϕ−1
i , Ii ) = Iref ◦ ϕi −
Ii 2L2 /Iref − Ii 2L2 , and the distance between corresponding grid points of ϕi ◦ ϕ−1
i ,
◦
ϕ
,
and
the
identity
transformation
(d
).
In
the
case
of
stationary-LDDMM,
ϕ−1
i
SSD
i
was
computed
using from the group exponential map
the inverse diffeomorphism ϕ−1
i
Exp(−wi ) where wi corresponds to the infinitesimal generator of ϕi . In the case of
diffeomorphic Demons, the inverse was computed from the minimization of the energy
functional
1
E(T ) = φ ◦ T − id22
2
(7.31)
with initial condition id − ϕ and energy gradient given by ∇E(T ) = (Dϕ)t ◦ T · (ϕ ◦ T −
id) [8]. Table 7.3 shows the average and standard deviation of these metrics measured
on the population of patients.
In all cases, the distance dSSD resulted negligible and, therefore, the computation
of the inverse diffeomorphisms could be considered exact (maximum average dSSD
equal to 1.47e−3 ). We found that the inverse consistent version of stationary-LDDMM
(IC-LDDMM) provided the best backward image similarity error as expected (average
RSSD = 14.43%). This error was close to forward RSSD error. Moreover, SGLDDMM provided a better backward image similarity than diffeomorphic Demons
(average RSSD of 15.47 for LDDMM vs 19.00% for Demons). Indeed, average RSSD
differences between methods were statistically different for diffeomorphic Demons.
As will be seen in Chapter 8, the calculus of statistics on a set of diffeomorphisms
are approached through the calculus of statistics on the corresponding logarithms in
201
the tangent space. In the case of stationary-LDDMM, the elements in the tangent
space needed to compute the statistics are directly provided. However, in the case of
diffeomorphic Demons, these elements have to be computed through the group logarithm map of the diffeomorphic transformations. The smoothness of the diffeomorphic
transformations and the corresponding tangent vectors constitutes an important issue
for the computation of proper statistics. In this work, we compared the smoothness of
diffeomorphisms from the Frobenius norm associated to the Jacobian matrices (Equation 7.20). This metric was used for regularization in Diffeomorphic Demons. The
smoothness of the tangent vectors was assessed from the Sobolev metric defined in
Dif f s (Ω) and used in stationary-LDDMM regularization (Equation 7.19). Table 7.4
shows the average and standard deviation of these metrics computed in the whole
population of patients.
The Frobenius norm resulted similar in all cases. This means that both stationaryLDDMM and diffeomorphic Demons transformations show similar smoothness on the
first order derivatives. However, the Sobolev metric · V showed to be much higher
in the case of diffeomorphic Demons. In Figure 7.11 the velocity fields Lw from which
the Sobolev metric is computed are shown in a representative example. Whereas the
continuity of vectors Lw remained bounded for stationary-LDDMM, they showed large
discontinuities in the case of diffeomorphic Demons. This may be due to the absence
of smoothness constraints on the second-order derivatives of Demons transformations.
Table 7.3: Average and standard deviation of the forward and backward RSSD and the distance
−1
dSSD = 12 (ϕi ◦ ϕ−1
i − id2 + ϕi ◦ ϕi − id2 ) used for the assessment of the inverse consistency error.
With IC-LDDMM and SG-LDDMM we indicate the inverse consistent and the symmetric gradient
version of LDDMM, respectively.
RSSD(Ii ◦ ϕi , Iref ) (%) RSSD(Iref ◦ ϕ−1
dSSD
i , Ii ) (%)
−4
IC-LDDMM
13.42 ± 4.23
14.43 ± 4.24
6.66e
± 4.31e−4
−3
SG-LDDMM
14.81 ± 1.59
15.47 ± 2.50
1.47e
± 1.24e−3
Demons
14.88 ± 1.74
19.00 ± 4.72
1.25e−3 ± 1.25e−3
Table 7.4: Average and standard deviation of the metrics used for the assessment of the smoothness
of the diffeomorphisms and the corresponding tangent vectors. With IC-LDDMM and SG-LDDMM
we indicate the inverse consistent and the symmetric gradient version of LDDMM, respectively.
· 2V
· 2
fro
IC-LDDMM 0.17 ± 0.05
140.52 ± 28.84
166.62 ± 12.05
SG-LDDMM 0.20 ± 0.03
Demons
0.13 ± 0.01 2626.70 ± 6069.90
7.4. Results
0
10
20
202
30
40
50
60
70
80
90
100
−50
−40
−30
−20
−10
0
10
20
30
40
50
Figure 7.1: Top row, sagittal, coronal and axial views of the image selected as reference. Left
column, views of the datasets used in this experimental section (patients 1-7). Right column, intensity
differences between the reference and the corresponding dataset on the left.
203
0
10
20
30
40
50
60
70
80
90
100
−50
−40
−30
−20
−10
0
10
20
30
40
50
Figure 7.2: Top row, sagittal, coronal and axial views of the image selected as reference. Left
column, views of the datasets used in this experimental section (patients 8-14). Right column, intensity
differences between the reference and the corresponding dataset on the left.
7.4. Results
204
50
0
20
40
60
80
50
20
50
0
100
0
20
40
60
80
20
40
100
75
60
80
0
40
60
80
0
20
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100
60
80
40
60
80
20
40
100
50
100
50
60
80
100
0
20
40
60
80
100
Iteration number
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
SSD
75
80
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
Iteration number
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
60
25
0
100
40
100
50
Iteration number
100
20
Iteration number
25
25
20
0
SSD
50
0
50
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
SSD
75
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
Iteration number
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
80
25
Iteration number
100
60
75
50
100
40
100
SSD
SSD
20
20
Iteration number
25
0
100
50
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
25
80
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
Iteration number
50
60
25
0
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
40
100
50
Iteration number
100
20
Iteration number
25
25
SSD
80
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
SSD
SSD
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
60
SSD
100
40
Iteration number
Iteration number
SSD
50
25
0
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
25
25
SSD
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
75
SSD
75
SSD
100
GD-LDDMM
IC-LDDMM
SG-LDDMM
Demons
SSD
100
50
25
25
0
20
40
60
Iteration number
80
100
0
20
40
60
80
100
Iteration number
Figure 7.3: Curves of the image similarity, Ii ◦ ϕi − Iref 2L2 , during optimization. GD-LDDMM
denotes the curves for stationary-LDDMM with gradient descent optimization. IC-LDDMM and SGLDDMM denote the curves corresponding to stationary-LDDMM with Gauss-Newton and symmetric
gradient optimization. Demons denotes the curves for diffeomorphic Demons.
205
IC-LDDMM
−50
−40
−30
SG-LDDMM
−20
−10
0
Demons
10
20
30
40
IC-LDDMM
50
−50
−40
−30
SG-LDDMM
−20
−10
0
10
Demons
20
30
40
50
Figure 7.4: Sagittal views of the intensity differences between the reference and the deformed
images obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic Demons (right).
Left column correspond to patients 1-7. Right column corresponds to patients 8-14.
7.4. Results
206
IC-LDDMM
−50
−40
−30
SG-LDDMM
−20
−10
0
10
Demons
20
30
40
IC-LDDMM
50
−50
−40
−30
SG-LDDMM
−20
−10
0
10
Demons
20
30
40
50
Figure 7.5: Coronal views of the intensity differences between the reference and the deformed
images obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic Demons (right).
Left column correspond to patients 1-7. Right column corresponds to patients 8-14.
207
IC-LDDMM
−50
−40
−30
−20
SG-LDDMM
−10
0
10
Demons
20
30
40
IC-LDDMM
50
−50
−40
−30
SG-LDDMM
−20
−10
0
Demons
10
20
30
40
50
Figure 7.6: Axial views of the intensity differences between the reference and the deformed images
obtained with IC-LDDMM (left), SG-LDDMM (center) and Diffeomorphic Demons (right). Left
column correspond to patients 1-7. Right column corresponds to patients 8-14.
7.4. Results
208
500
450
400
350
300
250
200
150
100
50
0
500
450
400
350
300
250
200
150
100
50
0
500
450
400
350
300
250
200
150
100
50
0
Figure 7.7: Illustration of sagittal, coronal and axial views of the intensity variance associated to
the populations of transformed images. First row shows the results for IC-LDDMM, second row shows
the results for SG-LDDMM, and third row shows the results for Diffeomorphic Demons.
209
dAI , IC-LDDMM vs SG-LDDMM results
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
dAI , IC-LDDMM vs diffeomorphic Demons results
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
dAI , SG-LDDMM vs diffeomorphic Demons results
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 7.8: Illustration of sagittal, coronal and axial views of the average of the metric dAI between stationary-LDDMM and diffeomorphic Demons transformations through the database of patients. First row, distance between IC-LDDMM and ESM-LDDMM. Second row, distance between
IC-LDDMM and diffeomorphic Demons. Third row, distance between ESM-LDDMM and diffeomorphic Demons.
7.4. Results
210
dSSD , IC-LDDMM vs SG-LDDMM results
2.5
2
1.5
1
0.5
0
dSSD , IC-LDDMM vs diffeomorphic Demons results
2.5
2
1.5
1
0.5
0
dSSD , SG-LDDMM vs diffeomorphic Demons results
2.5
2
1.5
1
0.5
0
Figure 7.9: Illustration of sagittal, coronal and axial views of the average of the metric dSSD
between stationary-LDDMM and diffeomorphic Demons transformations through the database of patients. First row, distance between IC-LDDMM and ESM-LDDMM. Second row, distance between
IC-LDDMM and diffeomorphic Demons. Third row, distance between ESM-LDDMM and diffeomorphic Demons.
211
IC-LDDMM
SG-LDDMM
Figure 7.10: Illustration of sagittal, coronal and axial views of the 2D projections of the transformations obtained with stationary-LDDMM and diffeomorphic Demons. First group shows the
diffeomorphism and its corresponding inverse obtained with IC-LDDMM. Second group shows the
transformations corresponding to ESM-LDDMM. Third group shows the transformations corresponding to diffeomorphic Demons.
7.4. Results
212
IC-LDDMM
SG-LDDMM
Figure 7.11: Illustration of sagittal, coronal and axial views of the velocity fields Lw obtained with
stationary-LDDMM and diffeomorphic Demons in a representative example (patient #10) First row
shows the results corresponding to IC-LDDMM. Second row corresponds to SG-LDDMM. Third row
corresponds to diffeomorphic Demons.
213
7.4.5
Efficiency
Our experiments were performed on a 2.33 GHZ machine with a C++ implementation
based on the ITK library. We found that the computation time per iteration in a
volume of size 175 × 185 × 144 was approximately 41.54 seconds for diffeomorphic
Demons, 53.23 seconds for SG-LDDMM and 90.64 seconds for IC-LDDMM. However, it
should be noted that if we were also interested in computing the inverse diffeomorphism
or the logarithm from the output of diffeomorphic Demons, the computation time
of the inverse diffeomorphism would take in average 5 706 ± 34 seconds whereas the
computation time for the logarithm would take 17 463 ± 10 681 seconds. In contrast,
the computation time of the exponential takes 28.37 second.
7.5
In this Chapter, we have presented a method for second-order optimization in stationaryLDDMM. Methods for non-stationary LDDMM registration have paid few attention
to the optimization strategy where classical gradient descent is often used. A version
of Newton’s method for optimization in stationary-LDDMM has been recently proposed by Ashburner et al. In this work, the computations of the Gâteaux derivatives
are performed in the space of L2 -functions. As alternative, in our method Gâteaux
derivatives are computed in the tangent space of the Riemannian manifold of diffeomorphisms. The most important differences between both approaches are found on
the computation of the action of the linear operator associated to regularization. If
computations are performed in L2 -space, the action has to be formulated using the
matrix representation of convolution leading to a memory demanding algorithm. In
addition, this action results into a high frequency components amplification. This provides convincing arguments for supporting optimization in the tangent space rather
than in L2 .
Simultaneously to these works, an alternative second-order method for efficient
diffeomorphic registration has been proposed in the literature from the extension of
Demons algorithm. Both techniques use stationary vector fields to compute diffeomorphisms and efficient second-order methods during optimization. To our knowledge,
the actual differences between both algorithms have not been studied before. In this
work, we have presented a theoretical and experimental comparison of both stationaryLDDMM and diffeomorphic Demons. We have analyzed the differences in the elements
of both registration scenarios, studied the influence of the regularization parameters on
the quality of the final transformations and evaluated the performance of the registration results. In addition, we have studied the potential of both algorithms for dealing
with the computation of statistics on populations of diffeomorphisms.
For stationary-LDDMM, we considered both the inverse consistent version of the
algorithm and the symmetric gradient optimization scheme (IC-LDDMM and SGLDDMM, respectively). For diffeomorphic Demons we just considered symmetric gradient optimization. Registration results were qualitatively and quantitatively compared
214
in a popular database of anatomical MRI brain images. As in Chapter 6, the metrics
used for comparison aimed at the assessment of image matching quality and local
similarity between transformations.
Previously to the generation of registration results, a parameter selection study
was performed. As discussed in Chapter 6, registration results for stationary-LDDMM
were sensitive to the selection of parameters 1/σ 2 and α. In this work, we found that
parameter τ in diffeomorphic Demons strongly influenced the smoothness of the final
transformation. As happened with α for stationary-LDDMM, there even exist values
of such parameter that provided non diffeomorphic transformations. The minimum average RSSD that was achieved with diffeomorphic transformations was approximately
equal to 13.88 and 14.81% for stationary-LDDMM (IC- and SG-LDDMM, respectively)
and 14.88% for diffeomorphic Demons.
The highest rate of convergence was achieved by the inverse consistent version of
stationary-LDDMM (IC-LDDMM). At the initial stages of optimization, diffeomorphic
Demons showed the worst performance in the great majority of cases although it usually
reached symmetric gradient version of stationary-LDDMM (SG-LDDMM) performance
at convergence. As shown in Figures 7.4, 7.5, and 7.6, both algorithms provided
a similar intensity matching, in general. However, in some cases, both algorithms
locally showed different performance. For example, larger deformations in stationaryLDDMM yielded a higher image matching in locations such as lateral ventricle tails
in some patients. In the comparison of transformation similarity, the values of dSSD
showed that, in average, the distance between corresponding grid points was under
voxel size. However, the values of dAI showed that the first order local structure of the
transformations was different.
The best inverse consistency error was achieved by IC-LDDMM, whereas SGLDDMM provided a slightly better consistent inverse image matching than diffeomorphic Demons (average RSSD of 15.5 for stationary LDDMM vs RSSD of 19.0% for
Demons). The smoothness of the logarithms in the tangent space measured with the
Sobolev metric showed to be much higher in the case of stationary-LDDMM than diffeomorphic Demons. In the last case, logarithms presented considerable discontinuities.
This may be due to the absence of smoothness constraints on the second-order derivatives of the transformations. Smoother elements could be achieved in diffeomorphic
Demons by using a higher standard deviation in the Gaussians involved in regularization. However, this would reduce the performance of the algorithm with respect to
image matching.
Finally, the study of the computational complexity of the registration algorithms
showed that diffeomorphic Demons was 1.28 times faster than SG-LDDMM and 2.18
times faster than IC-LDDMM. However, it should be noted that stationary-LDDMM
provides elements on the tangent space instead of transformations as output. This
allows to compute exponentials and inverses with a low computational cost. On the
contrary, diffeomorphic Demons provides transformations as output and logarithms
and inverses have to be estimated using quite computationally expensive iterative algorithms.
215
Therefore, both methods may be considered close from a theoretical point of view.
Stationary-LDDMM provides a more general framework than diffeomorphic Demons,
as it holds multiple variants in image matching similarity and regularization. Both
methods have shown comparable performance in terms of image matching. Therefore, both techniques could be considered equivalent from a practical point of view
for registration purposes. Diffeomorphic Demons showed just a slight advantage in
terms of computational time. However, transformations resulted qualitatively different. In addition, Diffeomorphic Demons showed a higher inverse consistent error than
stationary-LDDMM. Moreover, this method does not provide as smooth vector fields
as stationary-LDDMM needed to compute proper statistics in the tangent space of the
Riemannian manifold of diffeomorphisms.
In conclusion, it should be advisable to select diffeomorphic Demons for registration
applications where the efficiency of the algorithm is crucial, while stationary LDDMM
should be selected for applications where either the transformation smoothness or the
inverse consistency is important, or if the inverse transformations or logarithm maps
need to be computed. The selection between SG-LDDMM or IC-LDDMM would again
depend on the trade-off between computation time and accuracy for the specific application.
As future directions, it would be interesting to explore whether the versions of diffeomorphic Demons recently introduced in the literature (namely, diffeomorphic registration on DTI and spherical Demons [278, 277]) may be formulated within stationaryLDDMM framework and study if the advantages of stationary-LDDMM with respect
to diffeomorphic Demons still hold in these variants.
216
Appendix
Some of the limitations shown by diffeomorphic Demons have recently motivated the
proposal of an inverse consistent variant of diffeomorphic Demons [252]. As discussed
in Section 7.3.3, the inverse consistent version of the image matching energy in diffeomorphic Demons corresponds to
Esim (I0 , I1 , ϕ) = I0 ◦ ψ ◦ Exp(u) − I1 2L2 + I1 ◦ ζ ◦ Exp(w) − I0 2L2
(7.32)
subject to (ψ◦Exp(u))−1 = ζ ◦Exp(w). The minimization of this energy functional constitutes a constrained complex optimization problem for general expressions of ψ and
ζ. As alternative, Vercauteren et al. proposed to assume that all the diffeomorphisms
involved in Demons update belong to one-parameter subgroups. Thus, diffeomorphic
transformations are represented as the composition of
ϕ = Exp(w) ◦ Exp(uforw )
(7.33)
where w and uforw are vector fields in Ω belonging to a convenient space of vector fields
that guarantee the existence of exponential map and that the composition remains in
Dif f (Ω). Thus, the inverse diffeomorphism is represented as the composition of
ζ = Exp(−w) ◦ Exp(−uback )
(7.34)
With these diffeomorphisms characterization, the inverse consistent version of the image matching energy is simplified to
Esim (I0 , I1 , ϕ) = I0 ◦ Exp(w) ◦ Exp(uforw ) − I1 2L2 +
I1 ◦ Exp(−w) ◦ Exp(−uback ) − I0 2L2 (7.35)
subject to (Exp(w) ◦ Exp(uforw ))−1 = Exp(−w) ◦ Exp(−uback ). The minimization of
this energy functional is approached using constrained optimization where the update
k+1
of uk+1
forw and uback are computed as in the non-symmetric case from Equation 7.30 and
ϕ is approximated in the space of symmetric functions from the average in the tangent
space of the forward and backward updates yielding
k+1
ϕ
= Log
1
k+1
k+1
k
k
(Exp(w ) ◦ Exp( · uforw ) + Exp(−w ) ◦ Exp(− · uback ))
2
(7.36)
217
The authors proposed to use the BCH formula for the computation of the group logarithm in exponential coordinates. It should be noted that with this characterization of
diffeomorphisms, optimization is not performed on the group of diffeomorphisms but
on the tangent space V as happens with stationary-LDDMM. In addition, if first-order
approximation is selected in BCH formula 9 , the update is the one given by stationaryLDDMM and the differences between both algorithms are reduced to the regularization
and the expressions for the search direction. Therefore, we think that it will be of great
interest to include in this study the comparison with this inverse consistent version of
diffeomorphic Demons.
9
In fact, higher orders of BCH formula did not provide any improvement in the final performance
of the algorithm and BCH formula was only considered to theoretically justify the use of the additive
update. Personal communication with T. Vercauteren.
Chapter 8
Generation of 3D anatomical brain
atlases using statistics on groups of
diffeomorphisms
Abstract
As shown in Chapter 5, the main objective of Computational Anatomy is the
study of variability in anatomical structures. Anatomical information is encoded by the diffeomorphic transformations existing between the images and
a template selected as anatomical reference. Statistics on these diffeomorphic
transformations allow the modeling and analysis of the anatomical variability
existing across a population. Although different statistical measurements
have been extensively studied in Euclidean spaces, the need for dealing
with shape representations that lie on non-linear spaces has motivated the
extension of the methods for the study of linear statistics to Riemannian
shape spaces. This has been successfully tackled for some finite dimensional
Riemannian manifolds endowed with an algebraic Lie group structure. In this
Chapter, the fundamental aspects of the computation of non-linear statistics
on infinite dimensional Riemannian manifolds of diffeomorphisms proposed
in the literature are studied. We have discussed the practical limitations
posed by Riemannian calculus on these spaces and studied the alternative of
relying on the algebraic structure of groups of diffeomorphisms. Although this
algebraic structure is close to the one shown in finite dimensional Lie groups,
there exist several theoretical limitations posed by its infinite dimensional
nature. However, the algebraic approach has shown to provide acceptable
results in building a model of the statistical variability of a population of
diffeomorphisms parameterized with one-parameter subgroups. In addition,
this model has been successfully applied to the generation of a 3D anatomical
brain atlases inferred from a population of normal subjects which constitutes
a core step in most Computational Anatomy applications.
8.1. Introduction
220
Contents
8.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Statistics on Riemannian manifolds
. . . . . . . . . . . . . . . . . .
Means in Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . 223
8.2.2
Means in Dif f s (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.2.3
Principal Component Analysis on Riemannian manifolds
8.2.4
Principal Component Analysis in Dif f s (Ω)
Generation of 3D anatomical brain atlases
8.4
Results
8.1
222
8.2.1
8.3
8.5
220
. . . . . . . 229
. . . . . . . . . . . . . . 231
. . . . . . . . . . . . .
232
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
8.4.1
Datasets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.4.2
Statistics on the population of diffeomorphisms and statistical atlas . 234
. . . . . . . . . . . . . . . . . . . . .
234
Introduction
Anatomical brain atlases constitute a fundamental tool for studying brain anatomy and
function. Atlases are frequently used in clinical practise for the diagnosis of abnormal
anatomical variations, surgical planning, and the study of the relationships between the
anatomical structures and brain functions (see [237, 89] and references therein). In the
last years, atlases have been crucial for understanding the variability of brain anatomy
in longitudinal and transversal studies and improving the diagnosis of mental disorders
such as schizophrenia or Alzheimer’s disease [283, 32]. Moreover, atlases are involved
in multiple aplications such as template-based segmentation, modeling of anatomical
variability, and brain mapping, among others [236, 102, 274].
Most of the early atlases still used in clinical practise are based on a single subject [230, 213]. However, their applicability is limited to images similar to the atlas and
cannot be generalized to anatomies presenting large variability without introducing an
important bias [236]. To address with this bias, several authors propose to estimate a
statistically representative mean template from populations of images.
The majority of methods for unbiased atlas estimation characterize anatomical variability by the transformations existing between the images [97, 235]. In this context, the
atlas is defined as the anatomical image that minimizes some energy associated to the
amount of deformation necessary to match with all the images simultaneously. From
this definition, two different averaging procedures have been proposed in the literature.
The first one, based on group-wise registration, computes the atlas by simultaneously
warping the images towards the one that minimizes the energy corresponding to the
resulting transformations. The second one, based on pair-wise registration, iterates
221
Chapter 8. Generation of 3D anatomical brain atlases
until convergence between the computation of the mean of the transformations resulting from the registration of the currently estimated atlas and the images, and the
re-estimation of the atlas from this mean.
The computation of the atlas is closely linked to the algorithm used for registration
and the characterization of the transformations. In some works, transformations are
parameterized in the small deformation setting via displacement fields. In the literature, we can find different works where registration methods in the small deformation
setting have been adapted to the group-wise construction of atlases [18, 286]. In the
case of pair-wise registration, the mean transformation is computed from the Euclidean
mean of the displacement fields or the parameters associated to the sample transformations [72, 98, 259, 208]. Although both approaches provide acceptable results with
small enough displacement fields, the invertibility of the computed transformations is
not necessarily preserved, specially in anatomical images with large variability. As will
be shown in this Chapter, inverse transformations are involved in the computation of
proper statistics on transformations.
As alternative, the computation of the atlas is usually approached within the theory
of large diffeomorphic deformations, where the smoothness and invertibility of the
transformations have shown to provide a more appropriate framework when dealing
with statistics on non-linear spaces of transformations. In this setting, transformations
are assumed to belong to a convenient infinite dimensional Riemannian manifold of
diffeomorphisms. The right-invariant Riemannian metric defined in this space allows
measuring distances between the elements on this Riemannian manifold and, therefore,
allows the definition of the atlas associated to a population of images. Some of the works
approach the computation of the atlas by group-wise diffeomorphic registration [123,
154, 86]. These algorithms are robust to initialization, however, they show to require a
huge amount of memory. As alternative, the computation of the mean atlas using pairwise diffeomorphic registration has been approached in [14, 21, 43]. Although these
algorithms require an appropriate initialization [259], they need much less memory
than group-wise approaches.
In some applications, atlases are intended to capture not noly the mean but also
the principal modes of variability from the images. In the small deformation setting,
Principal Component Analysis (PCA) has been applied to the displacement fields or
the parameters associated to the sample transformations [208]. However, these methods do not take the advantage of non-linear computations in the study of the statistical
variability on spaces of transformations. Recently, wavelet-based PCA has been used
in order to simulate non-linear statistical deformations from a population of anatomical
images [274]. In the large diffeomorphic deformation setting, the estimation of statistical atlases has been approached by performing statistics on the Riemannian manifold
of diffeomorphisms [257].
However, as discussed in Chapter 5, the absence of a method for the computation
of the Riemannian logarithm map poses practical limitations to the computation of the
mean which is roughly formulated using first-order approximations. In adition, the absence of a Riemannian bi-invariant metric in Dif f s (Ω) makes the computed mean and
8.2. Statistics on Riemannian manifolds
222
modes of variation lack desirable invariance properties. Therefore, Riemannian framework provides just a partial generalization of linear statistics to infinite dimensional
manifolds of diffeomorphisms.
In this Chapter, we propose a novel method for the generation of the atlas and the
corresponding modes of variation associated to a population of anatomical images. The
computation of the atlas is based on pair-wise diffeomorphic registration and resembles
the methods proposed in [98, 43]. The core differences are found on the characterization
of sample transformations and the adopted framework for computing statistics on these
transformations. In this work, transformations are characterized as diffeomorphisms
belonging to one-parameter subgroups and computed from the efficient stationaryLDDMM method presented in Chapters 6 and 7. Therefore, instead of working with
the Riemannian differential structure defined on the manifold of diffeomorphisms, the
statistical framework comes from the extension of statistics on finite dimensional Lie
groups to groups of diffeomorphisms. Within this framework, we have also computed
the modes of variation associated to the population of anatomical images. Using this
algebraic approach we intend to overcome the practical limitations posed by statistical
calculus on Riemannian manifolds of diffeomorphisms while providing an algorithm
with much less computational requirements.
The method for the generation of 3D atlas was first published in the I3A technical
report [107]. This method was used for the generation of the simulated datasets used
in the work presented at the international workshop MMBIA’07 [106] and provided the
supporting application for the ideas presented by M.N. Bossa in the work published at
the international conference MICCAI’07.
The rest of the Chapter is organized as follows. In Section 8.2 we revisit the theory
of statistics on Riemannian manifolds and discuss the extension of the framework to infinite dimensional manifolds of diffeomorphisms. In Section 8.3 we present the method
for the computation of the atlas and the principal modes of variation. Experiments
are gathered in Section 8.4. Finally, Section 8.5 provides the main conclusions and
perspectives from this work.
8.2
Statistics on Riemannian manifolds
The motivation for the study of statistics on Riemannian manifolds finds its origins
in the need of Computer Vision community of modeling the variability of shapes [128,
221, 64]. With application to any problem where the geometrical comparison of a set
of objects is required, statistical modeling of shape variability has reached a particular
impact in the study of anatomical variability in medical image community, representing
a powerful tool for the analysis and discrimination between healthy and pathological
structures. The most relevant statistical measurements for modeling the variability of
a population are the mean, that represents the central value of the population, and the
modes of variation, that allow to capture the variability of the population with respect
to this central value.
223
The computation of the mean and the modes of variation is closely linked to the
adopted representation of shape. In the literature, different characterizations of shape
have been used, including point clouds, curves, surfaces and even transformations.
The common fact among these representations is that objects belong to shape spaces
that constitute non-linear Riemannian manifolds. Therefore, classical methods for the
computation of linear statistics are not valid anymore in these spaces and the extension
of these methods for statistical calculus in Riemannian manifods becomes necessary.
In this Section, we revisit the fundamental aspects of statistical calculus in Riemannian manifods for the computation of the mean and the modes of variation. This review
is based on the works by Pennec, Fletcher, Woods and collaborators [195, 269, 75, 193,
74] where statistics have been extensively studied for the sphere, projective spaces,
and symmetric groups of matrices, among others. In addition we discuss whether
these calculus techniques are extensible to infinite dimensional Riemannian manifolds
of diffeomorphisms.
8.2.1
Means in Riemannian manifolds
Arithmetic mean
Given a Hilbert vector space H equipped with an inner product ·, ·H , the arithmetic
mean of a set of elements x1 , ..., xN ∈ H corresponds to the expression
N
1 x=
xi
N i=1
(8.1)
It can be easily shown that x is the minimizer of the expression
N
N
1 1 x = arg min
y − xi , y − xi H = arg min
dH (y, xi )2
y∈H N
y∈H N
i=1
i=1
(8.2)
where dH (·, ·) is the distance metric endowed by the inner product defined on H. In
the Euclidean case H = Rd and dH (y, x) = y − x2 . Right-hand of Equation 8.2
at the minimum x corresponds to the expression of sample variance. Therefore, the
arithmetic mean provides a central point that minimizes the dispersion of the sample.
The definition of arithmetic mean can be generalized to continuum random vector
variables in the notion of linear average. In this case, given X a continuum random
variable in H, the linear average is defined as the expectation
x dP (x)
(8.3)
E(X ) =
H
As in the discrete case, this element realizes the global minimum of the variance
2
dH (x, E(X ))2 dP (x)
(8.4)
σX =
H
224
It can be easilly shown that both the arithmetic mean and the linear average present
invariance with respect to addition and multiplication by scalar and, therefore, their
definition is consistent with respect to the algebraic operations induced by the vector
space structure.
Riemannian means
The first difficulty in the definition of a mean in Riemannian manifolds consistent with
the definition of Euclidean mean is that, in general, manifold valued data cannot be
added. As alternative, several definitions based on the properties held by the linear
average have been provided in the literature ([193] and references therein). Among
them, one of the the most popular is Frechet’s average [83] that is defined from the
generalization of expectation in continuum random vector variables (Equation 8.3) to
continuum random manifold valued variables.
Thus, the Riemannian metric defined on the manifold M induces an infinitesimal
volume element dM that locally depends on the metric. The pair (M, dM) constitutes
a measure space. The volume element can be used as a reference to measure random
events from a distribution X on the manifold and define their probability density
function dP (m) = p(m) dM(m), m ∈ M . With the probability measure of a random
element dP (m) the definition of average that extends the definition of expected value
of a random vector distribution is formally given by the set of points in M realizing
the global minimum of the variance
2
2
d(μ, m)2 · p(m) dM(m)
(8.5)
σX (μ) = E(d(μ, X ) ) =
M
Similarly, the empirical Frechet mean of a set of measures m1 , ..., mN from the distribution X is defined as the global minimum of
σX2 (μ)
N
1 =
d(μ, mi )2
N i=1
(8.6)
The mean element minimizes the squared distance to the points in the distribution
measured along geodesics. This definition is general and valid for any Riemannian
manifold and generalizes the definition given for the arithmetic mean in Equation 8.2.
However, as this is a global result of a minimization, existence and uniquenes are not
ensured. Moreover, it has been shown that certain properties of the Frechet mean are
potentially problematic in some Riemannian manifolds [269, 193].
For example (from [269]), considering the sphere and the Frechet mean between
north and south poles, it can be shown that all the points lying at the equator are global
minimizers of the variance. If the point at the north pole is displaced infinitesimally
in one direction, the Frechet mean becomes the infinitesimal displacement in that
direction of the point lying at the equator and the geodesic that joins the displaced
north pole and south pole. However, if the north pole point is displaced in opposite
direction, the Frechet mean becomes the point in the opposite direction. This shows
225
that a minor perturbation on the sample distribution can lead to catastrophic results
on the mean estimation. Furthermore, these situations may be aggravated with the
complexity of the topological and differential structure of the manifold.
In order to overcome with Frechet mean limitations, Karcher et al. proposed to
consider a local minima of the variance on a small enough subdomain of the Riemannian
manifold [125]. In this case, the existence and uniqueness of the Karcher mean can be
guaranteed for sufficiently peaked distributions with small compact supports [125, 128].
Moreover, Karcher mean overcomes with the limitations derived from the globality of
the Frechet mean.
To practically compute the Karcher mean, a Gauss-Newton iterative algorithm was
proposed for the minimization of Equation 8.6 on Riemannian manifolds that also have
a Lie group structure. This algorithm was introduced for computing the mean in Lie
groups of rotations and rigid-body motions in [195] and further extended to general
Lie groups in [193]. It was also used for the computation of Riemannian means in the
manifold of medial atom representations [75] and diffusion tensors [74]. Thus, given
g1 , ..., gN a sample of points from the distribution X defined on a Lie group G, the
(left/right) Karcher mean is computed from the minimization scheme
N
1
log(π((μn )−1 , gi ))
(8.7)
μn+1 = π μn , exp
N i=1
where π denotes the (left/right) group multiplication and exp and log denote the
(left/right) Riemannian exponential and logarithm maps at the identity. This algorithm essentially alternates the computation of the barycenter of the sample distribution in exponential coordinates and a re-centering step to the current mean in each
iteration.
The Karcher mean inherits the invariance properties from the associated Riemannian metric. Karcher mean is bi-invariant in Lie groups where a bi-invariant Riemannian metric exists, and therefore, the mean is invariant with respect to both left and
right multiplication and with respect to inversion. In these cases, Karcher mean is
consistent with respect to the algebraic operations induced by the Lie group structure
as happened with the arithmetic mean in linear spaces. However, although either left
or right Riemannian invariant metrics do exist in general Lie groups, the existence
of bi-invariant Riemannian metrics is only guaranteed for compact finite dimensional
Lie groups [62]. There exist Lie groups where such metrics do not exist or are still
unknown. For example, Arsigny et al. [7] showed that no bi-invariant metric exists for
rigid transformations, which form one of the most simple Lie groups used in Computer
Vision applications. In these cases, the mean is compatible with just either right or
left multiplication and, therefore, it lacks desirable invariance properties.
Log-Euclidean mean
Log-Euclidean metrics were introduced as a simple and efficient framework for processing diffusion tensor data that can be considered elements in the Lie group of symmetric
226
positive definite matrices Sym+ (d) with the left multiplication of matrices [9].
Given g1 , g2 defined in a Lie group G, the Log-Euclidean distance is defined as
d(g1 , g2 ) = Log(g1 ) − Log(g2 )V
(8.8)
where Log represents the group logarithm at the identity element, and · V is the
norm defined in the Lie algebra V . The Log-Euclidean mean of g1 , ..., gN is defined as
N
1 μ = Exp
Log(gi )
(8.9)
N i=1
It can be easily shown that this mean inherits all the invariance properties of the
Log-Euclidean metric.
In the Lie group of symmetric positive definite matrices, Sym+ (d), the group logarithm is identified with Riemannian logarithm that is the matrix logarithm. The
Log-Euclidean metric is also a Riemannian one, i.e. it is associated to a scalar product defined in the tangent space. In addition, this metric is invariant with respect to
inversion and matrix multiplication which implies consistency with respect to group
operations. However, although the Log-Euclidean framework constitutes a powerful
technique for processing in Sym+ (d) and can be extended to any Lie group in a straightforward manner, the desirable invariance properties of the metric and the mean often
do not hold.
Bi-invariant mean
In those Lie groups where bi-invariant metrics are not defined, to define a mean compatible with group operations it was proposed to use alternative characterizations that
do not require any bi-invariant Riemannian metric. Arsigny et al. [7] defined the biinvariant mean for a general Lie group G as the solution of the barycentric equation
N
Log(π(μ−1 , gi )) = 0
(8.10)
i=1
where Log is the group logarithm map at the identity element. A similar notion of
bi-invariant mean was first defined by Woods et al. [269] in the case of matrices groups
endowed with a semi-Riemannian structure. The definition proposed by Arsigny et
al. does not rely on any Riemannian structure but on the algebraic properties of the
group. Moreover, the resulting mean is invariant with respect to group operations and
inversion. It should be noted that, in the particular case of groups endowed with a biinvariant Riemannian metric, the Riemannian and group logarithm maps are identical
and the Karcher mean provides the unique solution of the barycentric equation [193].
Therefore, this mean constitutes a generalization of Karcher mean in Lie groups where
a bi-invariant Riemannian metric is not available endowed with desirable invariance
properties.
227
To practically compute the solution of the barycentric equation, the Gauss-Newton
iterative algorithm proposed for the minimization of Equation 8.10 can be adapted
yielding to
N
1
Log(π((μn )−1 , gi ))
(8.11)
μn+1 = π μn , Exp
N i=1
where Exp and Log correspond to the group exponential and logarithm maps at the
identity. Relying on the analiticity of the BCH formula in the group G, Arsigny et al.
prooved the existence and uniqueness of the solution of the barycentric equation and
the convergence of the iterative algorithm towards the bi-invariant mean [7].
8.2.2
Means in Dif f s (Ω)
Riemannian mean
As discussed in Chapter 5, the Riemannian manifold of diffeomorphisms Dif f s (Ω) used
in Computational Anatomy applications is endowed with the right invariant metric
v, wTϕ (Dif f s (Ω)) = (dRϕ−1 )ϕ v, (dRϕ−1 )ϕ wV = L((dRϕ−1 )ϕ v), L((dRϕ−1 )ϕ w)L2
(8.12)
where L is a linear differentiable operator related to the physical deformation model
imposed on Ω. Given ϕ1 , ..., ϕN a sample of points on Dif f s (Ω) in a small enough
neighborhood, the Karcher mean is defined as the element that minimizes the squared
distances to the sample measured along right-geodesics. It should be noted that this
mean is right invariant but not inverse invariant as right invariance and inverse invariance would imply the left invariance of the Riemannian metric leading to a contradiction [7]. Karcher mean results from the convergence of the iterative scheme
N
1
log(ϕi ◦ (μn )−1 ) ◦ exp(μn )
(8.13)
μn+1 = exp
N i=1
where exp and log are the Riemannian exponential and logarithm maps associated to
this metric at the identity.
The exponential map has been characterized from the momentum conservation
equation along right geodesics in Dif f s (Ω). Thus, given v0 ∈ V , the exponential map
can be computed as the geodesic parameterized from the flow of vector fields v(t) with
v(0) = v0 verifying
L† Lv(t) = Dφ−1 (t)T · (L† Lv(0)) ◦ φ−1 (t) · det(Dφ−1 (t))
(8.14)
However, no algorithm for the computation of v(0) for a given diffeomorphism has been
provided yet. This hinders approaching the computation of the Riemannian mean in
the manifold of diffeomorphisms with the usual iterative approach.
228
As alternative, in the LDDMM literature, the Riemannian mean of a sample of
points ϕ1 , ..., ϕN on Dif f s (Ω) is defined from
N
1 log(ϕi )
(8.15)
μ = exp
N i=1
where exp and log constitute the Riemannian exponential and logarithm maps at the
identity [14, 21]. This definition is similar to the Log-Euclidean mean. The only
differences can be found on the use of Riemannian instead of group exponential and
logarithm maps. In this case, the computation of the Riemannian logarithm map is
not problematic, as LDDMM registration algorithms used for the computation of ϕi in
Computational Anatomy directly provide the elements vi (0) (identified with log(ϕi ))
in the tangent space. This mean is inverse invariant as
N
N
1
1 log(ϕ−1
= exp
−log(ϕi ) = μ−1
(8.16)
exp
i )
N i=1
N i=1
However, it should be noted that it does not present either right or left invariance
properties.
Log-Euclidean mean
The framework for the computation of the Log-Euclidean mean on the grop of diffeomorphisms was proposed by Arsigny et al. in [8]. In this work, the elements of
the Log-Euclidean framework for diffeomorphisms were identified and algorithms for
the computation of the group exponential and logarithm maps were proposed. These
algorithms have been described in Section 5.2, Chapter 5.
The Log-Euclidean mean for diffeomorphisms is defined as
N
1 Log(ϕi )
(8.17)
μ = Exp
N i=1
where Log is the group logarithm map at the identity element. As in the Riemannian
case, this mean is inverse but not either right or left invariant.
It should be noted that our stationary-LDDMM method for efficient diffeomorphic
registration presented in Chapters 6 and 7 provide directly the elements wi identified
with Log(ϕi ) for the computation of the Log-Euclidean mean.
Bi-invariant mean
The bi-invariant mean for diffeomorphisms is defined from the solution of the barycentric equation
N
i=1
Log(ϕi ◦ μ−1 ) = 0
(8.18)
229
where Log is the group logarithm map at the identity element. The bi-invariant mean
can be computed as the steady-state of the iterative algorithm
N
1
Log(ϕi ◦ (μn )−1 ) ◦ μn
(8.19)
μn+1 = Exp
N i=1
This mean is invariant with respect to inversion and left and right composition.
8.2.3
Principal Component Analysis on Riemannian
manifolds
Linear PCA
Given a population of samples x1 , ..., xN defined in a Hilbert vector space H, the
statistical variability of the data is encoded in the residuals x"1 = x1 −x, ..., x#
N = xN −x
where x is the sample mean. The principal modes of variation are the directions that
concentrate the maximum variability of the population measured in terms of the sample
covariance. In Euclidean spaces, Principal Component Analysis (PCA) is used for the
computation of these modes of variation.
This method assumes the statistical relevance of the mean and the covariance matrix
on the sample variability that is usually assumed to belong to a Gaussian distribution.
PCA aims at the generation of an orthonormal basis spanning the vector space. This
basis is computed from the eigenvectors of the covariance matrix ordered with respect
to the corresponding eigenvalues energy. The eigenvectors with the largest eigenvalues
correspond to the dimensions that have the strongest variance.
Thus, given the residuals x"1 , ..., x#
N arranged in the columns of the data matrix X,
the modes of variation U are the eigenvectors associated to the non-null eigenvalues of
the data covariance matrix
1
(X · X t )
(8.20)
N −1
However, if the data dimension is large, this method becomes an intractable numerical
problem. This can be overcome by the eigenanalysis of the covariance matrix associated
to X t instead
1
t
U = X · eig
(X · X)
(8.21)
N −1
The subspaces generated by the principal modes of variation and their corresponding eigenvalues verify the following relationships with the population of samples. The
first mode of variation u1 generates the one-dimensional subspace that minimizes the
squared distance from all sample elements to this subspace
S=
u1 = arg min
u=1
N
i=1
x"i − x"i , u · u22
(8.22)
230
The eigenvalue associated to the first mode is given by the sum of squared distances
from the sample to the subspace Hu1
λ1 =
N
1 x"i , u1 22
N i=1
(8.23)
Recursivelly, the k-th mode of variation generates the one-dimensional subspace Huk
that minimizes the distance from all sample elements to the subspace spanned by the
sub-basis uk , ..., uN uk = arg min
N
u=1
i=1
(x"i −
k−1
x"i , uj · uj ) − x"i , u · u22
(8.24)
j=1
The eigenvalue associated to the k-th mode is given by the sum of squared distances
from the sample to the subspace Huk . These relationships allow the extension of the
definition of the modes of variation to samples living in a Riemannian manifold.
Riemannian PCA
Similarly to linear average, PCA cannot be directly applied to manifold value data.
Fletcher et al. proposed the extension of the characterization of the subspaces generated
by the principal modes of variation in linear spaces to the computation of principal
components in Riemannian manifolds with bi-invariant metric endowed with a Lie
group algebraic structure, G [76, 75].
Thus, the role of one-dimensional subspaces in the linear case is represented by the
one-dimensional submanifolds
Gu = {exp(t · u)|t ∈ R}
(8.25)
where exp is the Riemannian exponential map and u constitutes a vector in the tangent
space. In this case, Gu is a geodesic curve where u provides its tangent direction at
the identity. For this reason the method was named as Principal Geodesic Analysis
(PGA). The distance from any element g ∈ G to this geodesic is given by d(g, Gu ) =
mint d(g, exp(t · u)) where d is the Riemannian distance defined on the manifold.
−1
Therefore, given the residuals g"1 = π(g1 , g −1 ), ..., g#
N = π(gN , g ), where g represents the empirical Riemannian mean, the first principal mode of variation is characterized as the geodesic Gu1 that minimizes the distance from all sample elements to
this submanifold, where u1 is defined from
u1 = arg min
u=1
N
d("
gi , Gu )2
(8.26)
i=1
The eigenvalue associated to the first mode is given by the sum of squared distances
from the sample to this submanifold. The k-th mode of variation is defined recursiv-
231
elly as in the linear case. In this framework, the subspaces Guk are called Principal
Geodesics.
In practise, the numerical implementation of this approach becomes a non-linear
optimization problem. Depending on the complexity of the Lie group, optimization
can be impractical, specially in high dimensions. For a more efficient computation,
Fletcher et al. proposed to approximate the computation of the Principal Geodesics
gN ).
by performing PCA on the logarithm of the residuals log(g"1 ), ..., log(#
It should be noted that in the case of Lie groups where no bi-invariant metric is
available, this method provides left/right principal geodesics, that are different from
right/left principal geodesics and the principal one-parameter subgroups.
PCA on one-parameter subgroups
The computation of principal modes of variation in the Log-Euclidean framework was
proposed by Arsigny et al. for statistics in diffusion tensor data [9]. As in the Riemannian case, the modes of variation are computed by performing PCA on Log(g"1 ), ...,
Log(#
gN ). In this case, the residuals are computed with respect to the Log-Euclidean
mean and the group logarithm is used in order to map group elements into the tangent
space. It should be noted that the one-dimensional submanifolds Gu generated by the
modes of variation constitute one-parameter subgroups.
In addition, the residuals can be computed with respect to the bi-invariant mean,
gN ).
and the modes of variation can be computed by performing PCA on Log(g"1 ), ..., Log(#
As happened with the bi-invariant mean, this constitutes a generalization of Riemannian PCA to Lie groups where no bi-invariant metric is available, endowed with desirable invariance properties.
8.2.4
Principal Component Analysis in Dif f s (Ω)
Riemannian PCA
The computation of the modes of variation associated to a population of diffeomorphisms was introduced by Vaillant et al. for diffeomorphic transformations existing between point distributions [243]. The algorithm has been recently adapted to deal with
diffeomorphisms parameterized by time-varying vector fields in [257]. Given ϕ1 , ..., ϕN
a set of diffeomorphisms with respective parameterizations v1 (t), ..., vN (t), the computation of the Principal Geodesics is aproached by performing PCA on the residuals
of the initial vector fields vi (0), solving the momentum conservation equation for the
modes of variation ui (0) and computing the transport equation associated to ui (t).
Residuals are computed by the substraction of the mean of the initial vector fields. As
PCA is performed on the tangent space V where the scalar product is associated to the
Riemannian metric via operator L, the first stage of the algorithm is equivalent to perform linear PCA on the residuals of the respective initial momenta Lv1 (0), ..., LvN (0).
It should be noted that as no bi-invariant metric is available in Dif f s (Ω), this method
provides right principal geodesics different from principal one-parameter subgroups.
8.3. Generation of 3D anatomical brain atlases
232
PCA on one-parameter subgroups
As in the finite dimensional case, the principal modes of variation can be computed
restricted to one-parameter subgroups by performing PCA on the residuals of Log(ϕ1 ),
..., Log(ϕN ). These residuals can be computed either with respect to the Log-Euclidean
or the bi-invariant means. It should be noted that the scalar product defined on the
Lie algebra V depends on operator L. Therefore, PCA should be performed with
respect to this scalar product. This is equivalent to performing PCA on the residuals
of L(Log(ϕ1 )), ..., L(Log(ϕN )).
8.3
Generation of 3D anatomical brain atlases
In this section we describe the proposed method for the generation of anatomical brain
atlases. The computation of the mean and the modes of variation of a population of
images is performed within Grenander deformable template model formulation [97].
Images are considered as diffeomorphic deformations of a reference image plus additive
noise
I = Iref ◦ ϕ + n
(8.27)
The noise term accounts for anatomical details that cannot be explained by the transformation ϕ as well as for photometric variations. In this work, diffeomorphisms are
restricted to belong to one-parameter subgroups, i.e. ϕ = Exp(w) where w ∈ V is the
infinitesimal generator of the subgroup.
The notion of similarity between two images I and J is defined from the minimal
norm of all the stationary diffeomorphisms connecting them,
min w2V
w∈V
such that I = J ◦ Exp(w) + n
(8.28)
This allows to translate the study of the anatomical variability from the sample of images to the corresponding minimal norm stationary diffeomorphisms. Hence, the atlas
can be defined as the anatomical image that minimizes the norm of the stationary diffeomorphisms needed to match with all the sample images simultaneously. The modes
of variation can be defined as the images that concentrate the maximum variance of
the population measured in terms of the variance associated to these diffeomorphisms.
More preciselly, given I1 , ..., IN a set of sample images, the estimation of the atlas
¯
I is translated to the computation of the mean diffeomorphism ϕ̄ associated to the
diffeomorphisms ϕi = Exp(wi ) such that
Ii = Iref ◦ ϕi + ni , i = 1, ..., N
(8.29)
where Iref represents the initial estimate of the atlas. In our method, the mean diffeomorphism should be selected as the one satisfying most of the desirable invariance
233
I1
I2
I3
IN
...
_
I i ϕ−1i ϕ
ϕ3
ϕN
ϕ2
ϕ1
_
ϕ
I ref
Figure 8.1: Diagram of the algorithm for the computation of the average atlas.
properties. In Dif f (Ω) the ideal candidate would be the bi-invariant mean. However,
we found that the computation of the bi-invariant mean from the solution of the iterative algorithm given in Equation 8.19 was quite sensistive to the initialization and
prone to get trapped into a local minimum at the first iteration. For this reason, we
propose to use preliminarily the Log-Euclidean mean instead, and leave a deeper investigation of the algorithm for the computation of the bi-invariant mean for future
work.
The algorithm is initialized with Iref equal to the element from the sample images
that minimizes
i=j
wi 2V +
1
2
2
(Ii ◦ ϕ−1
i − Ij L2 + Ij ◦ ϕi − Ij L2 )
2
σ
(8.30)
In order to further reduce the bias introduced by the selection of Iref , we include
the computation of I¯ and ϕ̄ in an iterative algorithm that re-estimates Iref and ϕi
subsequently as follows. The images Ii ◦ ϕ−1
◦ ϕ̄, that can be expressed as Iref ◦
i
◦
ϕ̄,
constitute
a
cloud
of
points
in
a neighborhood of Iref ◦ ϕ̄ 10 . In this
ϕ̄ + ni ◦ ϕ−1
i
−1
neighborhood we can approximate the new Iref by the linear mean N1 N
i=1 (Ii ◦ϕi ◦ ϕ̄),
(k)
(k−1)
thus reducing the noise variance. The convergence is reached when Iref − Iref 2L2 is
under a tolerance value. The final estimated atlas I¯ can be found in Iref . A diagram
synthetizing the computation of the average atlas is shown in Figure 8.1.
At this point, the principal modes of variation associated to the sample images
are estimated from PCA on one-parameter subgroups. Given (u(k) , λ(k) )k=1,...,K the
dominant eigenvectors and corresponding eigenvalues, new instances in the statistical
atlas can be generated from I = I¯ ◦ ϕ where
10
It should be noted that both forward and backward transformed images are computing during
the computation of the average atlas. Therefore, the inverse consistency of the registration algorithm
is a highly desirable property.
8.4. Results
234
ϕ = Exp
and α(i) are drawn from N (0, 3 λ(i) ).
8.4
K
α(i) u(i)
(8.31)
i=1
Results
In this section, we present the results obtained from the computation of statistics on
stationary diffeomorphisms and its application to the construction of the corresponding
statistical atlas associated to the population of images.
8.4.1
Datasets
The computation of the statistical atlas was performed in the 14 T1-MRI images used in
Chapter 6. The anatomical variability of this population of images can be appreciated
in Figures 6.2 and 6.3 at that chapter.
8.4.2
Statistics on the population of diffeomorphisms and
statistical atlas
Figure 8.2 shows the mean diffeomorphism associated to the diffeomorphic transformations and the first three principal modes of variation. In addition, Figures 8.3, 8.4
and 8.5 show the corresponding statistical atlas. Figures associated to the first mode of
variation show that the main variability of the population was located at the parietal
lobe, the cerebellum, the corpus callosum truncus and lateral ventricle tails. The main
variability associated to the second mode of variation was located at frontal and occipital lobes, the cerebellum, the splenium (corpus callosum tail) and the whole lateral
ventricle. Since the third mode of variation, the variability was mainly focused at grey
matter and subcortical structures. The resulting mean and modes of variation seemed
to be consistent with the anatomical variability found on the database of subjects. In
addition, non-plausible images were absent from the modes of variation.
8.5
In this chapter, we have presented a method for the generation of statistical atlases associated to a population of images. Our method for the computation of the mean atlas
iteratively combines the computation of the mean associated to the diffeomorphisms
existing between the sample images and a reference, and the update of this reference
image in order to reduce the bias introduced by the selection of the initial reference.
The modes of variation are subsequently computed from PCA on the corresponding
diffeomorphic transformations.
235
Instead of working with the usual Riemannian structure defined on the manifold of
diffeomorphisms for the computation of statistics on transformations, we relied on algebraic oriented calculus. Thus, transformations are assumed to belong to one-parameter
subgroups of diffeomorphisms computed from the efficient stationary-LDDMM algorithm presented in Chapter 7. In addition, the mean and modes of variation are
computed from the extension of statistical calculus in finite dimensional Lie groups
to groups of diffeomorphisms. This turn of perspective was introduced by Arsigny et
al. [7] and successfuly applied in situations where either the Riemannian framework
resulted quite computationaly expensive or the resulting statistics lack of desirable
invariance properties. Through this Thesis, we have pointed out that this is preciselly the case of Riemannian calculus in diffeomorphisms. Although working with
the group of diffeomorphisms still poses some theoretical limitations due to the infinite
dimensional nature of this space and the lack of surjectivity of the group exponential
map, the estimated mean and modes of variation seemed to be consistent with the
anatomical variability found on the population of patients. In addition, non-plausible
instances were absent from generated instances. Therefore, these experiments have
shown promising results for statistical atlas generation that leave open the possibility
of dealing with this algebraic structure in other Computational Anatomy applications.
However, there are still some issues that still remain open at the end of this Thesis
for future work. In first place, further investigation is needed in order to provide
stable numerical methods for the computation of the bi-invariant mean. In addition,
it would be of great interest to quantitatively assess whether the model for anatomical
variability is consistent with the anatomy of the underlying population and to compare
with the model obtained from the Riemannian framework. It would be also of great
interest to study if the anatomical findings associated to the model may improve other
alternative methods commonly used for the analysis of anatomical variability.
236
mean diffeomorphism
first mode
second mode
third mode
Figure 8.2: Statistics on a population of diffeomorphisms. Upper row, illustration of sagittal,
coronal and axial views of the mean transformation. Next three rows, illustration of sagittal, coronal
and axial views of the first three Principal Components. Both grids and glyphs are colored with
respect to displacement and vector magnitude, respectively.
237
√
−2.5 λ1
√
−1.5 λ1
atlas
√
1.5 λ1
√
2.5 λ1
Figure 8.3: Statistics on a population of images. Mean atlas and first mode of variation.
238
√
−2.5 λ2
√
−1.5 λ2
atlas
√
1.5 λ2
√
2.5 λ2
Figure 8.4: Statistics on a population of images. Mean atlas and second mode of variation.
239
√
−2.5 λ3
√
−1.5 λ3
atlas
√
1.5 λ3
√
2.5 λ3
Figure 8.5: Statistics on a population of images. Mean atlas and third mode of variation.
Chapter 9
Contributions, conclusions and
perspectives
Contents
9.1
9.2
Contributions and conclusions
. . . . . . . . . . . . . . . . . . . . .
241
9.1.1
9.1.2
Registration of medical images . . . . . . . . . . . . . . . . . . . . . . 245
Perspectives
. . . . . . . . . . . . . . . . . . . . . 241
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
9.2.1
. . . . . . . . . . . . . . . . . . . . . 249
9.2.2
Registration of medical images . . . . . . . . . . . . . . . . . . . . . . 249
The contribution of this Ph.D. Thesis focuses on the the development of novel
methods for segmentation and registration of medical images in the framework of challenging applications. The methods have been set within a variational formulation in
their respective solution spaces. The algorithm for segmentation has been specifically
devised for the automatic segmentation of vascular structures. It has been successfully
applied to the generation of 3D models of cerebral aneurysms in different angiographic
modalities. The method for registration has been devised as an efficient method for
diffeomorphic registration intended to spread the use of diffeomorphisms in clinical
research studies and Computational Anatomy applications. It has been successfully
applied to the generation of 3D brain atlases computed from statistical calculus on
the group of diffeomorphisms. In this final Chapter, we summarize the main contributions and conclusions derived from the work developed in this Thesis together
with the description of the scenarios for future methodological extensions and possible
applications.
9.1
9.1.1
Contributions and conclusions
The first Part of this Thesis has been devoted to the segmentation of vascular structures
with application to cerebral aneurysms. This application is particularly important in
9.1. Contributions and conclusions
242
clinical practice, as the analysis of the 3D geometry of the aneurysm and its surrounding vascular tree is increasingly playing an important role in treatment selection and
interventional planning. In addition, the availability of 3D models of brain aneurysms
is making possible the investigation of novel geometric descriptors or hemodynamic
parameters associated to the aneurysm development and the risk of rupture.
Non-parametric vessel enhancement filter
In Chapter 3 we have presented a novel method for probabilistic vessel enhancement
intended to improve the visualization of vascular structures in different angiographic
images. Probabilistic vessel enhancement filters usually assume a parametric model on
the distribution of image intensities. In contrast, we have proposed a non-parametric
model for probability estimation. This way, statistical estimation is performed by
learning the distribution of probabilities associated to the main tissue types presented in
a significative sample of images. The feature space is composed of high-order differential
image descriptors in a multi-scale framework. In this Thesis we have investigated
different feature spaces based on three different state of the art representations of the
multi-scale second-order local structure of the image, namely, the ellipsoid, prototype
and differential invariant representations. Parzen windows is the machine learning
algorithm selected for non-parametric estimation that, due to the sparseness of the
training data in the feature space, is approximated by k-Nearest Neighbors.
Our vessel enhancement filter was applied to the visualization of different angiographic modalities (MRA, CTA and 3D-RA) using patient specific training sets. In
addition, visualization results were compared to reference multi-scale and parametric
vessel enhancement filters. In all cases, our non-parametric method outperformed the
other considered techniques providing acceptable results in the most challenging examples. More precisely, results in MRA and 3D-RA showed a high performance, including
3D-RA images acquired under low contrast dose or after coil implant. Performance in
CTA at vessel tissues was also high, although the method showed high values of probability for vessel at bone tissue surrounding the carotid grooves and at the Turkish
saddle. The three feature spaces provided similar results in MRA and 3D-RA. In the
case of CTA, the prototype space representation showed a lower rate of erroneously
estimated probabilities.
In Chapter 4 we have presented a method for non-parametric Geodesic Active Regions
with application to automatic cerebral aneurysm segmentation in 3D-RA and CTA.
Region descriptors were defined from the non-parametric vessel enhancement filter
presented in Chapter 3. In this work, instead of using patient-specific training sets, we
have relied on the generation of a single training set from a representative sample of
images following a protocol provided for datasets selection and training point sampling.
Once the supervised part of the algorithm was carried out, our method allowed us to
243
Chapter 9. Contributions, conclusions and perspectives
automatically obtain 3D models of cerebral aneurysms from images with characteristics
similar to those in the datasets used for training.
In addition, the sensitivity of the algorithm to the selection of the training datasets
was studied. A feature space selection depending on the image modality was carried
out in order to achieve an optimal probability estimation for each of the tissue classes
presented in the images. In 3D-RA, neither dataset nor feature space selection did
significantly influence the final outcome of the probability estimation. Due to the lower
computation time spent on the kd-tree search, the feature space based on the differential
invariants was selected for this image modality. In CTA, probability estimation in
partial volume voxels of bone tissue area resulted strongly dependent on the selection of
the datasets. However, probability estimation at vessel tissue was in general consistent
despite of dataset selection. Due to the lower rate of errors shown in bone tissue area,
the feature space based on the prototype representation was selected for this image
modality.
Finally, the method was evaluated using manual segmentations as ground truth
and compared to reference region-based implicit deformable models. In 3D-RA, our
method showed the highest overlap with the ground truth segmentations outperforming
the rest of region-based algorithms. Similar conclusions could be inferred from CTA
examples that did not include bone tissue attached to the aneurysm. In the case of
CTA examples including bone tissue attached to the aneurysm, the performance in
all segmentation algorithms was highly degraded. However, our method showed to
minimize the size of bone attached to vessel tissue, resulting easily removable with
postprocessing algorithms or surface editing tools.
As a result of a collaboration with Dr. J.R. Cebral, our method for automatic
segmentation was included as initial stage into an efficient pipeline for Computational
Fluid Dynamics (CFD) simulation intended to the generation of patient-specific models of blood flow inside the aneurysms [37, 35]. In addition, the models generated
from this pipeline were used in the geometric characterization of cerebral aneurysms
using momentum invariants [169, 168]. Both constitute current active areas of research
towards the understanding of aneurysm development and the assessment of rupture
risk.
Publications
• M. Hernandez, R. Barrena, G. Hernandez, G. Sapiro, and A.F. Frangi.
Pre-clinical evaluation of implicit deformable models for 3D-segmentation of brain
aneurysms from CTA images.
SPIE’03, 5032:1264 – 1274, 2003.
• M. Hernandez, G. Sapiro, and A.F. Frangi.
Three-dimensional segmentation of brain aneurysms in CTA using non-parametric
region-based information and implicit deformable models: Method and evaluation.
244
MICCAI’03, Lecture Notes in Computer Science (LNCS), Springer-Verlag, Berlin,
Germany, 2879:594 – 602, 2003.
• J. R. Cebral, M. Hernandez, and A.F. Frangi.
Computational analysis of blood flow dynamics in cerebral aneurysms from CTA and
3D rotational angiography image data.
ICCB’03, 2003.
• J. R. Cebral, M. Hernandez, A. F. Frangi, C. Putman, R. Pergolizzi, and J. Burgess.
SPIE’04, 5369:319 – 327, 2004.
• M. Hernandez and A. F. Frangi.
Geodesic active regions using non-parametric statistical regional description and their
application to aneurysm segmentation from CTA.
MIAR’04, Lecture Notes in Computer Science (LNCS), Springer-Verlag, Berlin, Germany, 3150:94 – 102, 2004.
• M. Hernandez, A.F. Frangi, and G. Sapiro.
Quantification of cerebral aneurysm 3D morphology from CTA based on non-parametric,
region-based level-set techniques.
Handbook of Biomedical Image Analysis, Kluwer Academic Press, New York, 2005.
Brain aneurysm segmentation in CTA and 3DRA using geodesic active regions based
on second order prototype features and non parametric density estimation.
SPIE’05, 5747:514 – 525, 2005.
• R. D. Millan, M. Hernandez, D. Gallardo, J. R. Cebral, and A. F. Frangi.
SPIE’05, 5747:743 – 754, 2005.
Non-parametric geodesic active regions: Method and evaluation for cerebral aneurysms
segmentation in 3DRA and CTA.
Med. Image. Anal., 11(3):224 – 241, 2007.
• X. Mellado, M. Hernandez, I. Larrabide, and A.F. Frangi.
Flux driven medial curve extraction.
The Insight Journal, 2007.
245
Software
The computation of the nearest-neighbors search involved in non-parametric estimation was implemented using the ANN C++ library (http://www.cs.umd.edu/~mount/
ANN). This library provides an efficient implementation of the kNN algorithm using
kd-trees. For the management of the training and test sets, and cross validation implementation, Torch library was used in our algorithms (http://www.torch.ch [46]).
The rest of codes were implemented into a C++ library within the Insight Toolkit
framework (http://www.itk.org). This library included several modules devoted to
general image processing, computation of level set flows, non-parametric estimation of
probabilities, segmentation performance evaluation and skeleton computation. Codes
for skeleton computation were published at the Insight Journal. The remaining codes
have been recently integrated into a platform for segmentation and geometrical analysis
of cerebral aneurysms from angiographic images in the framework of AneurIST European project [82]. This ongoing research is being currently carried out at the Pompeu
Fabra University.
9.1.2
The second Part of this manuscript has been devoted to the devise of efficient methods
for diffeomorphic registration with application to the generation of statistical models
of the anatomical variability of a population. This may be considered one of the core
applications at Computational Anatomy, essential for the understanding of anatomical
variability among populations, the study of brain development and the diagnosis of
pathologies from images, among others.
Study of Riemannian manifolds and groups of diffeomorphisms
In Chapter 5 we have studied the fundamental aspects of the differential structure of
infinite dimensional Riemannian manifolds of diffeomorphisms in relation to Computational Anatomy. Although our study has been based in the work of different authors,
the compilation of very disperse results from different sources could be considered a
contribution by itself.
Riemannian manifolds of diffeomorphisms are infinite dimensional entities, therefore
they show a much more complex differential structure than the finite dimensional case.
Among them, Computational Anatomy deals with Sobolev diffeomorphisms where
thanks to the Hilbert differentiable structure of the tangent space many of the results
valid for finite dimensions can be extended to infinite dimensions. The Riemannian
exponential map associated to the metric has been characterized in Continuum Mechanics from the solution of the Euler-Poincare equation for diffeomorphisms. However,
the absence of a method for the computation of the Riemannian logarithm together
with the lack of bi-invariant Riemannian metric pose some limitations to the calculus
of statistics on this Riemannian manifold. As alternative, it has been proposed the
possibility of relying on the algebraic structure of the diffeomorphism group where
246
the group exponential and logarithm can be computed and, therefore, Log-Euclidean
statistics can be performed [8, 7].
However, the infinite dimensional nature of the group of diffeomorphisms poses
several theoretical limitations to working with the algebraic structure. Perhaps, the
most serious limitation is that the group exponential map is not surjective. Therefore,
the set of diffeomorphisms obtained from this map does not comprise all elements in
the manifold of diffeomorphisms. This means that there may exist some aspects of the
anatomical variability in a population that could be better explained by non-stationary
than by stationary diffeomorphisms. On the other hand, there may also exist the
possibility that the space of diffeomorphisms would be too complex for dealing with
anatomical variability and smaller groups of diffeomorphisms would provide a valid
framework for encoding the whole anatomical variability of a population in a rigorous
algebraic way. As will be discussed below, the work developed in this Thesis has
provided evidences consistent with this last hypothesis.
Inclusion of the stationary parameterization of diffeomorphisms in
LDDMM
In Chapter 6 we have presented a novel method for diffeomorphic registration intended
to reduce the computational complexity of classical paradigms for Large Deformation
Diffeomorphic Metric Mapping (LDDMM) registration. In LDDMM, diffeomorphisms
are usually parameterized from time-varying or non-stationary flows of vector fields.
However, the large computational complexity inherent to the use of the non-stationary
parameterization makes this methodology not much attractive for applications requiring fast registration algorithms.
As alternative, we have included the stationary parameterization of diffeomorphisms
in the LDDMM framework for registration. To this end, we have formulated the
variational problem related to the registration scenario and derived the associated
Euler-Lagrange equations. The use of the stationary parameterization has shown to
drastically alleviate the computational complexity of non-stationary LDDMM whereas
providing similar registration results.
It should be noted that due to the non-surjectivity of the group exponential map,
non-stationary LDDMM should provide much better registration results than our algorithm. However, the results reported in this Chapter have shown that, at least for
MRI brain anatomical images, one can find transformations from both parameterizations that provide similar acceptable results. These results may provide support to
the possibility that the space of transformations involved in Computational Anatomy
applications may constitute a considerably smaller subset of diffeomorphisms.
Second-order optimization in diffeomorphic registration
In Chapter 7 we have introduced efficiency and robustness into stationary-LDDMM
proposing a Gauss-Newton method for optimization. The energy gradient and Hessian computations are directly performed on the tangent space. This provides a stable
247
numerical scheme that can be executed in standard machines. A similar method for
second-order optimization was previously proposed in the literature. However, computations were performed in the space of square integrable functions leading to a more
unstable and memory demanding algorithm.
In addition, resulting stationary-LDDMM has been compared to diffeomorphic Demons. This method was proposed for efficient diffeomorphic registration from the
extension of Demons algorithm. Although both methods arose from different backgrounds, they may be considered close from a theoretical point of view. In addition,
they showed comparable results in terms of image matching with slight differences in
the computation time. Therefore, they could be considered equivalent from a practical point of view for registration purposes. However, diffeomorphic Demons showed a
lower inverse consistency performance than stationary-LDDMM. In addition, transformations resulted qualitatively different. In particular, diffeomorphic Demons did not
provide as smooth transformations as stationary-LDDMM needed to compute proper
statistics in the tangent space of the manifold of diffeomorphisms. Some of these limitations have recently motivated the proposal of a symmetric variant of diffeomorphic
Demons that approaches to our efficient stationary-LDDMM [252].
Statistics on diffeomorphisms and generation of anatomical atlases
Finally, in Chapter 8 we have focused on the generation of statistical models of anatomical variability from diffeomorphisms. Instead of working with the usual Riemannian
manifold of diffeomorphisms we have relied on statistical calculus on the group structure. To this end, we have extended the computation of statistics from finite dimensional Lie groups to diffeomorphisms. In particular, we have work with the extension
of the Log-Euclidean framework. The model has been applied to the generation of
statistical atlases associated to a population of anatomical images. For this application, the resulting mean and the modes of variation showed to be consistent with the
anatomical variability found on the database of patients. In addition, non-plausible
images were absent from the modes of variation.
Publications
IEEE 11th International Conference on Computer Vision, 2007 (ICCV’07). Workshop
on Mathematical Methods in Biomedical Image Analysis (MMBIA’07), 2007. Best
MMBIA 2007 Paper Award.
9.2. Perspectives
248
• M. N. Bossa, M. Hernandez, and S. Olmos.
Contributions to 3D diffeomorphic atlas estimation: Application to brain images.
Proc. of the 10th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI’07), Lecture Notes in Computer Science (LNCS),
Springer-Verlag, Berlin, Germany, 4791:667 – 674, 2007. MICCAI 2007 Young Scientist Awards runner up.
• M. Hernandez and S. Olmos.
5th IEEE International Symposium on Biomedical Imaging (ISBI’08), 2008.
• M. Hernandez, S. Olmos and X. Pennec.
Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons.
11th International Conference on Medical Image Computing and Computer Assisted
Intervention (MICCAI’08). 2nd Workshop on Mathematical Foundations on Computational Anatomy (MFCA’08), 2008.
• M. Hernandez, M.N. Bossa, and S. Olmos.
Registration of anatomical images using paths of diffeomorphisms parameterized with
stationary vector fields.
Int. J. Comput. Vis., 2009.
In press.
Software
Algorithms were implemented into a C++ library coded within the Insight Toolkit
framework (http://www.itk.org). This library included several modules devoted to
diffeomorphic image registration, statistics on transformations and input/output communication between ITK and VTK formats for visualization purposes. We are currently
working on adapting our codes for their publication at the Insight Journal. Codes for
diffeomorphic Demons registration available at [250] were adapted into our library in
order to make both algorithms fully comparable.
9.2
Perspectives
The methods developed in this Thesis have opened a wide range of possibilities that
will be explored in future work. The most interesting ones are presented below.
249
9.2.1
• Perform improvements in the non-parametric vessel enhancement filter. This would include the definition of different metrics in the space of features
that better discriminate intensity and gradient from second-order features in the
different tissues present in angiographic images, and the exploration of alternative
machine learning algorithms for improving estimation.
• Combine non-parametric Geodesic Active Regions with vessel specific
flows. Simultaneously to our work, several flows for the segmentation of tubular
structures have been proposed in the literature. It would be quite interesting
to properly combine them with our non-parametric region-based flow for the
improvement of segmentation results inside the thinnest vessels.
• Model the geometry of cerebral aneurysms. Segmentations could be useful
in the generation of 3D models useful for CFD simulations or the computation
of geometrical descriptors.
• Segmentation in CTA. Although our method significantly improved results
from existing techniques, segmentation of the whole Circle of Willis from CTA
may still be considered an open problem due to the low performance achieved in
locations close to the carotid grooves and the Turkish saddle. It may be quite
interesting to restrict the topology of front evolution or to introduce geometrical
models learned from 3D-RA in level set evolution at these specific locations.
• Introduction in clinical practice. Our algorithms for vessel enhancement and
segmentation may be included into a platform used in clinical practice for treatment selection and planning of cerebral aneurysms in endovascular procedures.
• Extension to other applications. Our algorithm could be applied to the
segmentation from different angiographic modalities of coronaries, aorta and renal
arteries, among others.
9.2.2
• Introduce variations on stationary-LDDMM scenario. This variations
would include the use of different image similarity energies, like mutual information or cross correlation, among others, and different regularization energies.
• Improve the algorithms for calculus of statistics on diffeomorphism
groups. As discussed in Chapter 8, the bi-invariant mean should be used for
the computation of a mean consistent with the algebraic operations of the group.
However, the extension of the iterative algorithm for the computation of the biinvariant mean proposed for finite dimensional Lie groups to diffeomorphisms
resulted quite sensitive to the initialization and prone to get trapped into a local
9.2. Perspectives
250
minimum at the first iteration. Therefore, further investigation in the devise of a
robust algorithm for the computation of the bi-invariant mean is needed. As alternative, the Log-Euclidean mean was considered in this Thesis as an acceptable
initial approximation.
• Compare the models of anatomical variability generated from statistical calculus on Riemannian manifolds and groups of diffeomorphisms.
Due to the lack of a bi-invariant Riemannian metric defined on the manifold of
diffeomorphisms, right-geodesics and one-parameter subgroups constitute different subspaces. Therefore, statistics computed from Riemannian and algebraical
calculus constitute two different approaches for modeling the anatomical variability. It becomes necessary to compare whether both models of variability are
consistent with the actual anatomical variability found on a population.
• Perform morphometric studies. As alternative to voxel based morphometry,
deformation and tensor based morphometric studies are increasingly used in clinical research studies for the assessment of anatomical differences between groups.
It would be interesting to analyze the results from morphometric studies associated to the diffeomorphic transformations obtained with stationary-LDDMM
registration. In addition, it would be of great interest to investigate the performance of novel metrics defined on the elements of the tangent space.
• Re-introduce learned models of anatomical variability into registration. Models of anatomical variability may be used as feedback in registration in
order to introduce robustness in the algorithm. In addition, these models would
be useful to incorporate anatomical constraints that would improve registration
performance in cortical and subcortical structures and serve as a guidance for
diffusion tensor registration.
• Atlas applications. Our method for anatomical atlas generation may be applied
to customary applications involving atlases in clinical practice as, for example,
atlas based segmentation. In addition, it could be included as a template estimation method in algorithms for the generation of temporal regression models.
• How to overcome with the limitations posed by the infinite dimensional
nature of diffeomorphisms? Last but no least, it would be of great interest
to keep exploring how to overcome with the theoretical and practical limitations
found in both non-stationary and stationary LDDMM paradigms.
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