Comparaison d`estimateurs de tangente sur les courbes

Transcription

Tangent Estimators + Evaluation + Enhancement
Digital Deformable Model
Onco-SP-IM : Image Restoration
Comparaison d’estimateurs de tangente sur les
courbes discrètes
+
Modèles déformable discrets
+
Onco-SP-IM : Déconvolution
François de Vieilleville
Irit, Université Paul Sabatier
Réunion d’équipe TCI
F. de Vieilleville
Réunion Equipe TCI
Outline
1
Digital Objects - Digital Curves
Classic estimators + Digital Estimators
Evaluation
Enhancement
2
Introduction and Problem Statement
Proposed Solution
Experimental Evaluation
3
F. de Vieilleville
Evaluation
Enhancement
Caractéristiques Géométriques en Géométrie Discrète
Question
Quel estimateur/algorithme choisir pour un descripteur
géométrique donné ? Selon quels critères objectifs ? Comment les
définir ?
F. de Vieilleville
Evaluation
Enhancement
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Aire discrète
Aire
Global
Global
Question
définir ?
F. de Vieilleville
Evaluation
Enhancement
Normales
Local
Normales discrètes
Local
Question
définir ?
F. de Vieilleville
Evaluation
Enhancement
Digital plane ≡ Z2 .
Digital object ≡ Finite set of points of Z2 .
From euclidean shapes to digital shapes : E = X ∩ hZ × hZ.
The digital boundary can be
extracted
using
inter-pixel
contours or cellular decomposition.
Elements of the digital curve are ordered and indexed.
→ Computation of local geometric characteristics use windows
containing finite number of points.
F. de Vieilleville
Evaluation
Enhancement
h = 0.02
Circle
Ellipse
F. de Vieilleville
Flower
Evaluation
Enhancement
Problems When Analyzing Digital Curves
(1) There exists infinitely many continuous Euclidean shapes for a
digitized shape.
(2) How to determine the size of computation window w.r.t. local
geometry ?
(3) Time to spend on computations may be limited.
F. de Vieilleville
Evaluation
Enhancement
Solution 1 : Continuous Estimators and Fixed Sized
Window
(1) Add hypotheses for the reference shape : smoothness,
compactness, convexity, minimal perimeter or maximal area.
→ Brings bias in estimation : special underlying curve.
(2 & 3 ) Fix the size of the computation window.
→ explicit trade-off between time computation and precision, loss
of local geometry consistency.
→ For tangent estimation we chose median filtering, Gaussian
derivative and low order polynomial fitting.
F. de Vieilleville
Evaluation
Enhancement
Solution 2 : Use Maximal Digital Straight Segment Based
Algorithms
(1) No hypotheses on the reference shape : study of the digital
linear part of the digital object.
(2 & 3 ) Size of the computation window computed automatically.
→ At very low resolution, consistency with local geometry ?
→ For tangent estimation we choose the λ-MST tangent estimator
and the Global Minimum Curvature estimator.
F. de Vieilleville
Evaluation
Enhancement
Polynomial Fitting
(Ci = (xi , yi ))1≤i≤2q+1 set of 2q + 1 consecutive samples on
digital curve.
Minimize the functional :
2
P2q+1
Pj=N
E (a0 , . . . , aN ) = i=1 yi − j=0
aj xij .
ELR (a, b) = E (a, b, 0, . . .)
Linear Regression,
EIPF (a, b) = E (0, a, b, 0, . . .)
Implicit Parabola Fitting,
EEPF (a, b, c) = E (a, b, c, 0, . . .)
Explicit Parabola Fitting.
Estimation on whole curve in O ((2q + 1) ∗ number of points)
F. de Vieilleville
Evaluation
Enhancement
Polynomial Fitting
1
Th
EPF q=12
3.25 IPF q=12
RL q=12
3.2
3.15
3.1
3.05
polar
angle
1.5 1.53 1.56 1.59 1.62 1.65 1.68
F. de Vieilleville
AAE of tangent orientation
tangent orientation
3.3
IPF q=8
EPF q=8
LR q=8
IPF q=16
EPF q=16
LR q=16
IPF q=32
EPF q=32
LR q=32
0.1
0.01
0.001
10
100 inv. of grid step
1000
Evaluation
Enhancement
Polynomial Fitting
Done separately on each coordinate (X and Y).
→ Parametrisation problem induced : in fact a length estimation
is required.
Iterative refining of the length estimation : tangent → tangent
→ length ....
→ In practice, no real improvement even on constant curvature
shapes (disk)
F. de Vieilleville
Evaluation
Enhancement
Polynomial Fitting
abs. error of tan. orientation
AAE of tan. orientation
0.05
0.045
ICIPF q=8
ITICIPF q=8
0.04
0.035
0.03
0.025
0.02
ICIPF q = 8
ITSIPF q = 8
ICIPF q = 16
ITICIPF q = 16
ICIPF q = 32
ITICIPF q = 32
1
0.1
0.015
0.01
0.005
0.01
polar
angle
inv.of grid step
0
1.2
1.25
1.3
1.35
1.4
1.45
1.5
F. de Vieilleville
10
100
1000
10000
Evaluation
Enhancement
Gaussian Smoothing
Gσ0 (t), first derivative of Gσ (t) =
σq =
√1
σ 2π
Pq
Gσ0 q (−i)Ci
2
,
exp −t
2
2σ
estimated derivative at C0 obtained as :
2q+1
3
Estimation on whole curve in
O ((2q + 1) ∗ number of points).
F. de Vieilleville
i=−q
Evaluation
Enhancement
Median Filtering
Choosing the median orientation among the following 2q vectors
centered on : Ci :
((Ci−q Ci , , ) . . . , (Ci−1 Ci , , ) (Ci Ci+1 , , ) . . . , (Ci Ci+q , ))
F. de Vieilleville
Evaluation
Enhancement
Digital Straight Segment
Digital Straight Segment : piece of the digitization of a ray
D(a, b, µ, ω) = {M ∈ Z2 |µ ≤ aMx − bMy < µ + ω}
(a, b, µ, ω) ∈ Z4 , ba : slope of the line, ω : thickness
ω = |a| + |b| 4-connected line.
→ Exists various classes of DSS, one of interest in the class of the
Maximal Segments.
→ Recognition in linear time, incremental algorithms.
→ Strong link with continued fraction.
F. de Vieilleville
Evaluation
Enhancement
Maximal Segment
→ Recognition of the whole set of MS in linear time on the
whole digital curve.
→ Digital convexity ≡ monotony of slopes of MS[Doerksen
et.al.’04].
→ Average length of maximal segments follows Θ(h−1/3 ),
average number of maximal segments follows Θ(h−2/3 ), on
digitization of shapes with C 3 border.
F. de Vieilleville
Evaluation
Enhancement
Digital Estimator : λ-MST
Weighted average of orientation of MS, weights depend on
point location and λ function.
Exists simple λ function making no false concavities (triangle).
Point to point multigrid convergence proven (th. Θ(h1/3 ), xp.
Θ(h2/3 )).
F. de Vieilleville
Evaluation
Enhancement
Global Minimum Curvature
Choose a shape that minimizes its squared curvature along its
border among the shapes whose digitization is almost the same as
the input digital shape.
Global optimisation scheme, in the space of tangent directions
parametrized by the curvilinear abscissa.
Each MS on the digital curve defines bound in the research
space.
Iterative relaxation scheme.
F. de Vieilleville
Evaluation
Enhancement
3.3
tangent orientation
tangent orientation
Defects of Continuous Methods :Digitized Circle (h=0.01)
Th. Tangent
GD q=3
GD q=7
GD q=11
3.25
3.2
3.15
3.1
3.05
3
3.2
3.15
3.1
3
polar angle
polar angle
2.95
1.4
1.45
1.5
1.55
1.6
1.65
1.7
tangent orientation
tangent orientation
Th. Tangent
ICIPF q=3
ICIPF q=7
ICIPF q=11
3.05
2.95
3.3
3.3
3.25
Th. Tangent
Matas q=3
Matas q=7
Matas q=11
3.25
3.2
3.15
3.1
3.05
3.3
1.4
1.45
1.5
1.55
1.6
1.65
1.7
Theoretical Tangent
GMC
L-MST
3.25
3.2
3.15
3.1
3.05
3
3
polar angle
2.95
polar angle
2.95
1.4
1.45
1.5
1.55
1.6
1.65
1.7
F. de Vieilleville
1.4
1.45
1.5
1.55
1.6
1.65
1.7
Evaluation
Enhancement
Angle deviation
Angle deviation
Isotropy Behaviour :Digitized Circle (h=0.01)
0.1
0.01
0.001
0.001
0.1
0.01
0.001
0
moy MATAS q=3
moy MATAS q=9
max MATAS q=3
max MATAS q=9
0.8 polar angle
0.01
1.6
0
Angle deviation
Angle deviation
0
moy GD q=3
moy GD q=7
max GD q=3
max GD q=7
polar angle
0.1
moy ICIPF q=3
moy ICIPF q=9
max ICIPF q=3
max ICIPF q=9
0.8 polar angle
moy L-MST
max L-MST
moy GMC
max GMC
0.1
0.01
0.001
1.6
F. de Vieilleville
1.6
0
0.8
polar angle
1.6
Evaluation
Enhancement
Perimeter estimation :Digitized Circle(r=50, 100 random
center shifts)
θTAN (Ci ) is the estimated normal at point Ci , the estimated
length, denoted LθTAN (Ci ), equals :
| cos(θTAN (Ci ))| if the associated linel is horizontal,
| sin(θTAN (Ci ))| otherwise.
Win.size
q=1
q=2
q=4
q=8
q=16
q=32
q=64
GD
30.6724
26.9072
6.46475
0.79856
0.12082
0.05776
0.13503
ICIPF
30.6724
2.35272
0.40573
0.11547
0.06791
0.08221
0.24110
MATAS
27.14933
11.32574
3.622088
1.495153
1.336123
1.336123
1.336123
F. de Vieilleville
ITICIPF
29.67998
4.032775
0.897381
0.181935
0.070328
0.145093
0.462371
λ-MST
GMC
0.50378
0.02553
Evaluation
Enhancement
Single Criterion : Multi-Grid Convergence
Euclidean Object (X )
Geometric Descriptor (G)
Evaluation
−→
G (X )
−→
Ĝ (DigGh (X ))
↓
DigGh (·)
↓
,
,· · ·
Digital Object
(DigGh (X ))
Estimation of Geometric
Descriptor (Ĝ)
F. de Vieilleville
Estimation
Evaluation
Enhancement
Single Criterion : Multi-Grid Convergence
Euclidean Object (X )
Geometric Descriptor (G)
−→
↓
DigGh (·)
↓
G (X )
↑
limh→0 ?
↑
−→
,
Evaluation
Ĝ (DigGh (X ))
,· · ·
Digital Object
(DigGh (X ))
Estimation of Geometric
Descriptor (Ĝ)
F. de Vieilleville
Estimation
Evaluation
Enhancement
Multi-Grid Convergence On Digitized Disks : Maximal Absolute Error
1
ME of tangent orientation
1
0.1
0.01
0.001
GD q=1
GD q=4
GD q=16
GD q=64
GD q=128
10
inv. of grid step
100
1000
0.001
ICIPF q=1
ICIPF q=4
ICIPF q=16
ICIPF q=64
ICIPF q=128
10
inv. of grid step
100
1000
100
1000
10000
1
0.01
10000
1
0.1
0.01
0.001
0.1
MATAS q=1
MATAS q=4
MATAS q=16
MATAS q=64
MATAS q=128
10
inv. of grid step
100
1000
0.1
0.01
0.001
10000
F. de Vieilleville
L-MST
GMC
x^(-1.1/2)
x^(-1)
10
inv. of grid step
10000
Evaluation
Enhancement
Multi-Grid Convergence On Digitized Ellipses : Average Absolute Error
0.1
0.01
0.001
GD q=1
GD q=4
GD q=16
GD q=64
GD q=128
ICIPF q=1
ICIPF q=4
ICIPF q=16
ICIPF q=64
ICIPF q=128
inv. of grid step
inv. of grid step
0.001
100
1000
10000
0.1
0.01
0.001
0.01
10
10
0.1
100
1000
100
1000
10000
0.1
0.01
MATAS q=1
MATAS q=4
MATAS q=16
MATAS q=64
MATAS q=128
10
inv. of grid step
100
1000
10000
F. de Vieilleville
0.001
L-MST
GMC
x^(-2/3)
x^(-1.3/3)
10
inv. of grid step
10000
Evaluation
Enhancement
Multi-Grid Convergence On Digitized Flowers : Average Absolute Error
1
1
0.1
0.01
0.001
0.01
GD q=1
GD q=4
GD q=16
GD q=64
GD q=128
10
inv. of grid step
100
1000
0.001
10000
inv. of grid step
100
1000
100
1000
10000
1
ICIPF q=1
ICIPF q=4
ICIPF q=16
ICIPF q=64
ICIPF q=128
10
1
0.1
0.01
0.001
0.1
0.1
0.01
MATAS q=1
MATAS q=4
MATAS q=16
MATAS q=64
MATAS q=128
10
inv. of grid step
100
1000
10000
F. de Vieilleville
0.001
L-MST
GMC
x^(-2.5/3)
x^(-1/2)
10
inv. of grid step
10000
Evaluation
Enhancement
Mixed Criterion : Average Abs. Error * Time
Precision is not the only criterion to be taken into account.
Product of AAE by the time required for the computation.
→ Penalises huge errors on average even if small time.
→ Penalises multi-grid convergence estimators that require too
many computations.
F. de Vieilleville
Evaluation
Enhancement
Asymptotic Behaviour : Average Absolute Error By Time on Circle
AAEBT
AAEBT
100
100
10
10
x^(1/3)
L-MST
GD q=8
GD q=16
GD q=32
GD q=64
GD q=128
1
0.1
10
100
1000
x^(1/3)
L-MST
ICIPF q=8
ICIPF q=16
ICIPF q=32
ICIPF q=64
ICIPF q=128
1
0.1
10
inv. of grid step
AAEBT
AAEBT
100
100
10
10
x^(1/3)
L-MST
MATAS q=8
MATAS q=16
MATAS q=32
MATAS q=64
MATAS q=128
1
0.1
10
100
1000 inv.of grid step
F. de Vieilleville
100
1
0.24 x^(1/3)
17 x^(0.34131)
L-MST
GMC
0.1
10
100
1000 inv. of grid step
Evaluation
Enhancement
Size of Computation Window Required to Reach Best Accuracy for Average
Size of Computation Window
Absolute Error on Circle
0.01
GD q=12
GD q=16
GD q=24
0.001 GD q=32
GD q=48
GD q = 64
GD q=96
GD q=128
x^(2.5/3)
0.0001
10
100
F. de Vieilleville
x^(1/2)
GD
100
inv.of grid step
10
10
100
1000
Evaluation
Enhancement
Computation Window Size : Based on MS ?
Should be in O((1/h)1/2 ).
However, no know digital primitives with such growing speed...
q1−0
q1−1
q1−2
q2
q3
q4
q5
max(||B(Cj )) − Cj ||1 , ||F (Cj )) − Cj ||1 ),
min(||B(Cj )) − Cj ||1 , ||F (Cj )) − Cj ||1 ),
)/2,
(q1−0 + q1−1
1/6
qP
,
1−0 (1/h)
b(
||MS
||
)/nb(MS
i
1
i )c ,
i
j P
k
3/2
=
(( i ||MSi ||1 )/nb(MSi ))
,
= b(||F (Cj )) − B(F (Cj ))||1 − 1)/2c .
=
=
=
=
=
F. de Vieilleville
Evaluation
Enhancement
Adaptivity of the proposed computation windows
Window
Adaptivity
q1−0
Local
q1−1
Local
q1−2
Local
q2
Local
q3
Global
q4
Global
q5
Local
F. de Vieilleville
Average
Expected
Size
1 13
O (h)
1
O ( h1 ) 3
1
O ( h1 ) 3
1
O ( h1 ) 2
1
O ( h1 ) 3
1
O ( h1 ) 2
1
O ( h1 ) 3
Evaluation
Enhancement
Improvements of The Gaussian Derivative Estimator
(a)
3.3
3.2
Th. Tangent
H2 GD
H3 GD
H4 GD
H5 GD
3.25
tangent orientation
3.25
tangent orientation
(b)
3.3
Th. Tangent
H1-0 GD
H1-1 GD
H1-2 GD
3.15
3.1
3.05
3
3.2
3.15
3.1
3.05
3
polar angle
2.95
polar angle
2.95
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.4
moy H1-0 GD
moy H1-1 GD
max H1-0 GD
max H1-1 GD
0.1
0.01
0.001
0
0.8
1.5
1.55
1.6
1.65
1.7
(d)
Angle deviation
Angle deviation
(c)
1.45
polar angle
moy H2 GD
moy H3 GD
max H2 GD
max H3 GD
0.1
0.01
0.001
1.6
F. de Vieilleville
0
0.8
polar angle
1.6
Evaluation
Enhancement
(f)
(e)
Angle deviation
1
0.1
0.01
moy H4 GD
moy H5 GD
max H4 GD
max H5 GD
0.001
0
0.8
polar angle
1.6
0.1
H1-0 GD
H1-1 GD
H1-2 GD
0.01 H2 GD
H3 GD
H4 GD
H5 GD
x^(2.5/3)
x^(1/2)
0.001
10
inv. of grid step
100
1
H1-0 GD
H1-1 GD
H1-2 GD
H2 GD
H3 GD
H4 GD
H5 GD
x^(-2.7/3)
x^(-1/2)
0.1
10000
(h)
(g)
1000
0.01
0.1
H1-0 GD
H1-1 GD
H1-2 GD
H2 GD
H3 GD
H4 GD
H5 GD
x^(-2.7/3)
x^(-1.23/2)
0.001
10
0.01
inv. of grid step
0.001
10
100
1000
10000
F. de Vieilleville
inv. of grid step
100
1000
10000
Evaluation
Enhancement
Experimental results confirm that “classic” estimators can
benefit from digital primitives.
Theoretical results show that q5 is to be multi-grid
convergent.
F. de Vieilleville
Proposed Solution
Active Contours [Kass’88]
Let Ω be a parametrization of the curve embodying the active
contour geometry :
Ω = [0, 1] → R2
x(s)
v (s) =
y (s)
It energy being defined as :
Z 1
0
00
EDM (v ) =
Eint (v (s), v (s), v (s)) + Edata (v (s)) ds
0
v = argminv ∈F EDM (v )
F. de Vieilleville
Proposed Solution
Eint (·) monitored by :
α elasticity,
β rigidity.


2

Eint (v (s), v 0 (s), v 00 (s)) =  α(s) v 0 (s) +
|
{z
}
lenght penalisation
2
β(s) v 00 (s)
|
{z
}

 /2
curvature penalisation
Image term is decreasing with the norm of the image gradient.
Direct computation not tractable, problem rewritten as a
variational problem, solved with finite differences, sometimes with
greedy heuristics on energy.
→ hard to find global minimum, no topological changes.
F. de Vieilleville
Proposed Solution
And so on...
−(αC 0 )0 + (βC 00 )00 + ∇P(C) = 0
Level Sets [Osher, Sethian’89]
→ handle topological
changes
but does not necessarily stops.
2
∂C(s,t)
∂ C
= P(C) ∂s 2 + w~n
∂t
Geodesic Active Contours [Caselles ’95]
→ handle topological changes and stops on strong gradient
2
∂C(s,t)
= P(C) ∂∂sC2 − < ∇P, ~n > ~n
∂t
Minimal Pathes [Cohen, Kimmel ’96]
→ ReachR a global minimum for curve from A to B.
E (C) = Ω (w + P(C(s))) ds
F. de Vieilleville
Proposed Solution
...on Open Digital 4-Connected Path ?
Γ ∈ C(A, B) : Set of simple oriented open 4-connected path from
point A to point B.
Given boundary conditions, finite number of possibilities !
Classical mathematics resolution techniques (DPE stuff) not
available...
→ Greedy heuristics based on energy.
→ Problem is how to define energies with good properties.
F. de Vieilleville
Proposed Solution
Requirements for Internal Energy
D
Eint
(Γ) = αL̂(Γ)
Internal energy relying only on length estimation and requires :
convex functional,
low variability of global minimum,
has to be a good length estimator.
F. de Vieilleville
Proposed Solution
Internal Energy : Length Estimation
P
Using DSS-Based Methods : L̂(Γ) = s∈surfel(Γ) < s, τ̂ (s) >
with θ(τ̂ (s)) equals to atan(slope) of rightmost max dss of surfel s.
n + 2 points
Γ
2 +2
L̂(Γ) = √(n−1)
(n−1)2 +1
Γ’
2 +1
L̂(Γ0 ) = √(n−2)
(n−2)2 +1
For n = 10, L̂(Γ) ≈ 9, 15 and L̂(Γ0 ) ≈ 9, 47.
→ Does not bring a convex functional.
F. de Vieilleville
+
√
2
Proposed Solution
Naive length estimation using elementary surfel length :
B
A
→ Classic Euclidean distance but with horizontal and vertical steps,
→ brings a global minimum,
→ but too many of them ! !
F. de Vieilleville
Proposed Solution
D
Eint
(Γ) = αL2 (CMLP(Γ))
→ Known as a good estimator, multi-grid convergent estimator
[Klette’99],
→ convex functional,
→ global minimum is very close to the digitization of the euclidean
segment joining A et B.
F. de Vieilleville
Proposed Solution
Data Energy
Regarding the image energy term we define it as :
X
D
EData
(Γ) =
max(||∇I ||) − ||∇I (c)||,
c∈Γ
in order for the energy to be positive everywhere and decreasing
with the norm of the image gradient.
Our digital combinatorial minimization problem becomes :
Γ∗ = arg
min
Γ∈C(A,B)
D
D
D
D
EDM
(Γ), where EDM
(Γ) = EInt
(Γ) + EData
(Γ).
F. de Vieilleville
Proposed Solution
Elementary Deformations
Outside corners,
−
+
−
−
−
0
−
0
−
0
+
+
0
+
0
Flip +
0
−
+
into
0
−
+
+
.
0
+
0
+
Inside corners,
−
+
−
−
−
0
−
0
+
+
0
−
0
0
+
Flip −
−
−
+
into
0
+
0
+
.
Inside
or outside bumps,
− −
Flat to ().
Flat outside parts,
+
+
−
+
+ +
0
0
0
Bump +
outside to
Flat inside parts,
−
− −
−
−
0
0
0
+
+
+
−
−
+
− −
0
0
0
Bump −
to
F. de Vieilleville
0
0
0
+
+
+
.
−
+
−
+
.
Proposed Solution
Examples
−
0
+
−
0
+
0
0
−0
−
−
+
F. de Vieilleville
+
Proposed Solution
Constrain Topology
Maintain a simple 4-connected path from A to B :
check if deformation is valid, does not collide with the
contour,
filtered when listing the possible deformations of Γ.
F. de Vieilleville
Proposed Solution
CMLP and Path
Convex vertices correspond to outside corners or outside
bumps, convex deformation are Flip + or Flat.
Concave vertices correspond to the inside corners or inside
bumps, concave deformation are Flip − or Flat.
−
−
+
F. de Vieilleville
+
+
Proposed Solution
Internal Energy Descent
Theorem
Let Γ ∈ C(A, B) being finite, choosing at each step any one of the
valid convex or concave deformation leads to CMLP(Γ) = [AB]
after a finite number of steps.
[Requires]
Proposition
Let Γ ∈ C(A, B) being finite , if CMLP(Γ) 6= [AB] there exists at
least one valid convex deformation on a convex vertex or one valid
concave deformation on a concave vertex of CMLP(Γ).
F. de Vieilleville
Proposed Solution
Proposition
Let Γ ∈ C(A, B) being finite , if CMLP(Γ) 6= [AB] there exists at
least one valid convex deformation on a convex vertex or one valid
concave deformation on a concave vertex of CMLP(Γ).
→ Flat ok, remains Flip
if Γ made of 2 Freeman code, Flip ok
else contour would be infinite since :
→ Flip not valid → contour has 2 consecutive outside corners
associated to 2 convex vertices of the CMLP (+0m +) and m ≥ 1
→ Flip not valid → contour has 2 consecutive outside corners
associated to 2 convex vertices of the CMLP (+0m +) and m ≥ 1
...
F. de Vieilleville
Proposed Solution
Theorem
Let Γ ∈ C(A, B) being finite, choosing at each step any one of the
valid convex or concave deformation leads to CMLP(Γ) = [AB]
after a finite number of steps.
−
−
+
F. de Vieilleville
+
+
Proposed Solution
Greedy1 Algorithm : Lowest Energy Among all
Deformations
Input: Γ : a Digital Deformable Model
Output: Γ0 : when returning true, elementary deformation of Γ with
D
D
EDM
(Γ0 ) < EDM
(Γ), otherwise Γ0 = Γ and it is a local minimum.
Data: Q : Queue of (Deformation, double) ;
D
E0 ← EDM
(Γ);
foreach valid Deformation d on Γ do
Γ.applyDeformation(d);
D
Q.push back(d, EDM
(Γ) );
Γ.revertLastDeformation();
end
(d, E1 ) = SelectDeformationWithLowestEnergy (Q);
Γ0 ← Γ;
if E1 < E0 then Γ0 .applyDeformation(d);
return E1 < E0 ;
F. de Vieilleville
Proposed Solution
Greedy2 Algorithm : First Valid Decreasing the Energy
Input: Γ : a Digital Deformable Model
Output: Γ0 : when returning true, elementary deformation of Γ with
D
D
(Γ), otherwise Γ0 = Γ and it is a local minimum.
EDM
(Γ0 ) < EDM
D
E0 ← EDM (Γ);
foreach valid Deformation d on Γ do
Γ.applyDeformation(d);
D
if EDM
(Γ) < E0 then Γ0 ← Γ;
return true;
Γ.revertLastDeformation();
end
Γ0 ← Γ;
return false;
F. de Vieilleville
Proposed Solution
On Synthetic Images
Initial
α=0
α = 200
α = 300
F. de Vieilleville
Proposed Solution
On Synthetic Images
Inital
α=0
α
=
800, 4000
α
=
1600, 8000
F. de Vieilleville
Proposed Solution
On Real Images : Top-Down Approach
Binary Mask Image
Initialisation
Minimisation
32x
32
Initialisation From Scaling
Minimisation
Initialisation From Scaling
Minimisation
64x64
128x128
F. de Vieilleville
Proposed Solution
Initial
Minimisation
α = 400, internal energy term is based on the CMLP. The high
value of α penalise the length of the contours, over-smoothing the
contours, only top-left contour seems to be correctly delineated.
F. de Vieilleville
Proposed Solution
Initial
Minimisation
α = 150. The internal energy term is based on the CMLP. Top-left
and bottom-left contours are correctly delineated.
F. de Vieilleville
Proposed Solution
Initial
Minimisation
α = 100. The internal energy term is based on the CMLP. Global
delineation of contours is correct.
F. de Vieilleville
Proposed Solution
Initial
Minimisation
α = 100. The internal energy term is based on the length of the
freeman code of the path, other positive values for the balance
term bring very similar contours. Although the global delineation
seems correct, the obtained contours are not smoothed at all.
F. de Vieilleville
Proposed Solution
Initial
Minimisation
Results of the top-down segmentation process using the
deformable model described in [William’92]. Parameters are such
that (α, β) equal (0.1, 0.45), the neighborhood is chosen as a 3 × 3
square, and minimisation is done until no smaller energy can be
found. Top-right contour and middle contours are well delineated,
top left and bottom left are partially correctly delineated.
F. de Vieilleville
Proposed Solution
SpiraleFIRST.avi
→ Keep the topology of Γ.
F. de Vieilleville
Proposed Solution
Conclusion
Proposed method is as good as classic snake formulation but
inherits its drawback : tuning of parameter α.
Probabilistic deformation scheme ? Dynamic CMLP algorithm ?
More comparison with “classic” active contours ?
F. de Vieilleville
Introduction-Motivation
Images obtenues par un microscope sur des objets marqués par des
fluorophores :
Laser gaussien + lentille cylindrique+ objectif à l’emission,
chambre d’immersion (4 degrés de liberté)+ agar + spécimen,
objectif d’imagerie (grossissement) + filtre + caméra CCD.
Illumination sélective de l’objet par feuilles de lumière :
Moins phototoxique, possibilité d’imager en temps réel,
d’imager plus en profondeur dans le spécimen.
Pas de photo-blanchiment des fluorophores.
Coupes 2D d’un objet 3D.
F. de Vieilleville
Introduction-Motivation
Restaurer les images (3D) ayant subi des dégradations lors du
processus de formation : déconvolution.
Connaissance a priori sur la fonction de dégradation de l’objet
et le processus de formation de l’image.
→ type de bruit de la caméra : Poissonien + thermique
→ optique linéaire, psf théorique connue
F. de Vieilleville
Déconvolution : Richardson-Lucy
Bruit Poissonien : f (λ, k) =
λk exp−λ
k!
Modèle th. de formation : i(x) = P(o ⊗ h)(X )
Approche Bayésienne : p(o|i) =
p(i|o)p(o)
p(i)
MaximiserQla vraisemblance
selon la loi de Poisson :
i(x)
−(h⊗o)(x)
p(i|o) = x [(h⊗o)(x)] i(x)!exp
Minimiser la log-vraisemblance
P
−log (p(i|o) = x [(h ⊗ o)(x) − i(x) log(h ⊗ o)(x)
F. de Vieilleville
MinimiserR la fonctionnelle :
J1 (o) = x [(h ⊗ o)(x) − i(x) log(h ⊗ o)(x)]dx
CN : ∇J1 (o) = 0 ⇔ h(−x) ⊗
i(x)
(h⊗p)(x)
=1
Algorithme itératif
h multiplicatif : i
i(x)
ok+1 (x) = ok (x) h(−x) ⊗ (h⊗o
k )(x)
F. de Vieilleville
Algorithme multiplicatif non-stable :
Applification du bruit après quelques itérations
Problèmes numériques
Algorithme additif (descente en gradient) :
i
h
i(x)
ok+1 (x) = ok (x) + δt 1 − h(−x) ⊗ (h⊗o
k )(x)
Nécessite une régularisation !
F. de Vieilleville
Déconvolution : Richardson-Lucy avec Total Variation
Minimiser la fonctionnelle
:
R
J2 (o) = J1 (o) + λTV x |∇(o)(x)|dx
CN :
∇J2 (o) = 0 ⇔ 1 − h(−x) ⊗
i(x)
(h⊗p)(x)
− λTV div
ok+1 (x) = h
∇o
ok (x) + δt −1 + λTV div |∇o|
+ h(−x) ⊗
F. de Vieilleville
∇o
|∇o|
i(x)
(h⊗ok )(x)
i
=0
Descente en gradient avec deux paramètres
Balance de régularisation
Paramètre d’intégration
Performance de la méthode liée au schéma numérique employé
Calcul des différences finies Schéma ROF [Rudin et al.’92]
on retombe dans des problèmes d’estimation ! !
F. de Vieilleville
F. de Vieilleville
Déconvolution d’images SPIM : Problèmes a posteriori
Les images présente des raies sombres et claires :
Absorption dues à des éléments non imagés ! !
Focalisation
Dirigées selon la direction de la feuille de lumière.
F. de Vieilleville
Déconvolution d’images SPIM
Création d’images synthétiques pour obtenir une “vérité
terrain” afin de mieux comparer les méthodes.
Tests d’autres schémas numériques pour comparaison et
évaluation des performances.
Séparer les problèmes : débruitage, suppression des raies,
déconvolution.
F. de Vieilleville

Comparaison d`estimateurs de tangente sur les courbes

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