Phase unwrapping of MR images using ΦUN

Medical Image Analysis 13 (2009) 257–268
Contents lists available at ScienceDirect
Medical Image Analysis
journal homepage: www.elsevier.com/locate/media
Phase unwrapping of MR images using UUN – A fast and robust region
growing algorithm
Stephan Witoszynskyj a,d,*,1, Alexander Rauscher a,c,1, Jürgen R. Reichenbach a, Markus Barth b
a
Medical Physics Group, Institute for Diagnostic and Interventional Radiology, Friedrich Schiller University, Jena, Germany
Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen, Nijmegen, The Netherlands
c
MRI Research Centre, University of British Columbia, Vancouver, Canada
d
Department of Radiology, Medical University of Vienna, Waehringer Guertel 18-20, 1090 Vienna, Austria
b
a r t i c l e
i n f o
Article history:
Received 25 July 2007
Received in revised form 31 July 2008
Accepted 13 October 2008
Available online 18 October 2008
Keywords:
MRI
SWI
Field maps
Phase imaging
a b s t r a c t
We present a fully automated phase unwrapping algorithm (UUN) which is optimized for high-resolution
magnetic resonance imaging data. The algorithm is a region growing method and uses separate quality
maps for seed finding and unwrapping which are retrieved from the full complex information of the data.
We compared our algorithm with an established method in various phantom and in vivo data and found a
very good agreement between the results of both techniques. UUN, however, was significantly faster at
low signal to noise ratio (SNR) and data with a more complex phase topography, making it particularly
suitable for applications with low SNR and high spatial resolution. UUN is freely available to the scientific
community.
Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
Magnetic resonance data are complex numbers, but most often
only magnitude images are reconstructed. However, phase information can be extremely valuable. For example, phase is very
sensitive to small deviations from the resonance frequency that
can be induced by inhomogeneities of the magnetic field. It is thus
essential for susceptibility weighted imaging (SWI) (Reichenbach
et al., 1997), phase imaging of the human brain (Ogg et al., 1999;
Barth et al., 2003; Rauscher et al., 2005; Duyn et al., 2007;
Koopmans et al., 2008) and the calculation of field maps (Johnson
and Hutchison, 1985; Jezzard and Balaban, 1995). Other
applications that utilize phase information include phase contrast
angiography (Bryant et al., 1984) or MR elastography (Kruse
et al., 2000) where the movement of magnetic moments leads to
phase offsets. Phase images can also facilitate the segmentation
of tissues (Bourgeat et al., 2007).
In all these applications, a physical quantity is mapped into the
phase’s limited domain (p 6 / < p or 0 6 / < 2p depending on
the definition). This mapping can cause aliasing. In the literature,
the resulting ambiguities are usually referred to as phase wraps.
* Corresponding author. Address: Department of Radiology, Medical University of
Vienna, Waehringer Guertel 18-20, 1090 Vienna, Austria. Tel.: +43 1 40 400 5751;
fax: +43 1 40 400 4898.
E-mail addresses: [email protected], stephan.witoszynskyj@
gmx.at (S. Witoszynskyj).
1
Both authors contributed equally.
1361-8415/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.media.2008.10.004
Although the true value of the phase cannot be known unless
additional assumptions are made, the elimination of phase wraps
(i.e. phase unwrapping) often is required before any further postprocessing steps can be applied. The problem of phase unwrapping
arises in various fields, ranging from optics, over radar interferometry, to MRI. Especially in the case of two- or higher dimensional
data it is a non trivial task.
In MRI, there are several situations in which phase unwrapping
can be circumvented. For example field maps can be calculated by
computing the arc-tangent of a complex division of two gradient
echo scans acquired at different echo times. Nevertheless, phase
wraps may remain in areas affected by strong field inhomogeneities (Cusack and Papadakis, 2002; Hutton et al., 2002).
Homodyne detection (Noll et al., 1991) is a technique that removes phase variations with low spatial frequencies and thus reduces the likelihood of phase wraps. It is often used in situations
where phase information of a single high-resolution scan is needed
(e.g. SWI (Reichenbach and Haacke, 2001)). However, it is unsuitable for methods, such as field mapping (Jezzard and Balaban,
1995) or simulations of signal loss based on phase images
(Rauscher et al., 2006), that depend on knowledge of all spatial frequencies of the phase. Furthermore, residual phase wraps may remain. These remaining phase wraps may have singularities which
impede any true phase unwrapping. Completely eliminating phase
wraps without suppressing valuable information is often impossible. In particular, if very long echo times are used to achieve good
susceptibility contrast in phase imaging (Ogg et al., 1999; Barth
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et al., 2003; Rauscher et al., 2005; Duyn et al., 2007), homodyne filtering fails in areas with background field inhomogeneities.
A number of different approaches to true phase unwrapping can
be found in the literature (e.g. Hedley et al., 1992; An et al., 2000,
2002; Jenkinson, 2003; Volkov and Zhu, 2003; Xu and Cumming,
1999). Although much effort has been made to obtain correctly unwrapped phase images, execution times are seldom reported.
However to be useful for applications, an algorithm must not only
be stable and robust but its implementation must result in unwrapped phase images within an acceptable time. This is especially
true for SWI which is based on high-resolution data. While matrix
sizes of 128 64 and a slice thickness of 3 mm can be sufficient for
field mapping to correct for distortions in EPI (Windischberger
et al., 2004), typical SWI data-sets can have matrix sizes of
512 384 96. This increase in data size by about two orders of
magnitude makes fast processing rather critical.
Furthermore, in some applications, such as SWI in general, but
also in EPI-based fMRI (Robinson et al., 2004; Poser et al., 2006)
and especially in contrast-enhanced SWI (CE-SWI) with SPIO, areas
with reduced signal are of interest (Dahnke and Schaeffter, 2005;
Hamans et al., 2006). Phase information is still available in those
regions. Thus, a phase unwrapping algorithm useful for these
applications must perform well in low SNR areas.
Xu and Cumming presented an algorithm for the unwrapping of
synthetic aperture radar (SAR) interferograms, which are known to
have areas of both low phase SNR and high density of phase wraps
(Xu and Cumming, 1999). Their method was shown to venture
deep into noisy areas and the application to SWI data (Rauscher
et al., 2003) leads, in principle, to satisfactory results. Nevertheless,
in areas of steep phase topography or ghosts, the performance of
this method is suboptimal, because only the local coherence of
the phase is considered. Our goal was thus to extend this promising approach such that it exploits additional information available
in MRI to make it better suitable for MRI data. This was done by
optimizing the criteria for placing seed points and for guiding the
unwrapping. We put a strong emphasis on a fast implementation
that would be suitable for large SWI data-sets.
In this paper we present our modified region growing phase
unwrapping method. The emphasis of our approach was to develop
a method that is suitable for high-resolution applications that can
be affected by areas of very low SNR. As mentioned before, this demands high performance in terms of speed and reliable prediction
of the unwrapped phase. For evaluation we compared an implementation of our approach to a well established method namely
PRELUDE (Jenkinson, 2003) which is part of the FSL Analysis Package. The implementation of our method is freely available to the
research community and can be obtained from the authors.
2. Theory
2.1. Problem definition
Let’s assume a 2D complex MR image of dimensions Nx Ny.
^i;j reflect the
The real and imaginary signal component of a pixel p
x and y components of the magnetization vector in the rotating
frame of reference (Haacke et al., 1999). The pixel’s complex value
can also be represented by its magnitude qi,j and phase /i,j:
^i;j ¼ qi;j ðcos /i;j þ i sin /i;j Þ ¼ qi;j ei/i;j :
p
ð1Þ
It is apparent that /i,j is ambiguous. If the phase corresponds to a
physical quantity f we can write
/i;j ¼ W½fi;j ;
ð2Þ
where W is the wrapping operator that projects f into the domain of
the phase /. Phase unwrapping tries to regain the information that
is lost by applying the wrapping operator. The ‘‘true” phase can be
written as
/ui;j ¼ /i;j þ 2pmi;j :
ð3Þ
Unwrapping the phase corresponds to finding the correct mi,j, which
can be difficult in the presence of noise.
2.2. Phase unwrapping by region growing
We base our work on a region growing phase unwrapping
algorithm originally developed for SAR interferometry (Xu and
Cumming, 1999). This method calculates a phase prediction /pi;j
for each pixel in the immediate neighborhood of a region of pixels
that have been unwrapped in previous iterations. The prediction
/pi;j is the average of a number of individual predictions /pi;j;i0 ;j0 where
(i0 , j0 ) 2 N(i, j). N(i, j) denotes already unwrapped immediate neighbors of (i, j). The predictions /pi;j;i0 ;j0 are computed by linear extrapolation from the next nearest neighbor if the next nearest neighbor
has already been unwrapped. If the next nearest neighbor has not
been unwrapped /pi;j;i0 ;j0 is set to the unwrapped phase /ui0 ;j0 of (i0 , j0 ).
A pixel (i, j) is unwrapped and added to a region if it passes the
following criteria:
(1) The local coherence (LC), which is a measure of the variation
of the phase in a certain neighborhood, is above a given
threshold at the location (i, j),
(2) the variability of the predictions /pi;j;i0 ;j0 is below a given
threshold, and
(3) the difference between the common prediction /pi;j and the
unwrapped phase /ui;j is within a certain limit.
During unwrapping all thresholds are relaxed step by step. This
ensures that pixels for which reliable predictions can be made easily are unwrapped first.
Unwrapping can be done in several regions at the same time. If
regions overlap and their predictions are in good agreement, they
are merged. If their predictions disagree, the pixels in the overlap
area are removed from both regions in the hope that the two regions will overlap later on with better agreement. As soon as two
regions have been merged, a seed point for a new region is chosen.
2.3. Defining quality measures
Local coherence, which was proposed for seed finding and
guiding unwrapping in the original algorithm, is a measure of
the variation of the phase. It is low in areas with steep phase
topography and in the presence of pure noise. Unfortunately, in
MR imaging, situations exist in which the signal has a very small
magnitude, but the phase is rather coherent. An example of such
a situation are ghosting artifacts. In such a case, the phase can be
very coherent even in areas where the signal’s magnitude is
diminishing (Fig. 1c). Thus, by using local coherence as a quality
criterion, the algorithm tries to unwrap areas with low signal but
coherent phase before it grows into regions with high signal but
steep phase topography. This can lead to artifacts, since the algorithm may accumulate errors in the low signal regions which
then propagate into areas that have a high signal intensity.
To find a possible solution to this problem four more quality criteria were defined and evaluated in addition to the local coherence.
The five quality maps (an example for each map is given in Fig. 1)
are motivated and defined in the following manner:
local coherence (QLC) is a measure for the coherence of the
phase (Fig. 1c). It is highest if the phase is the same in all pixels within a rectangular area around each pixel. It was intro-
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Fig. 1. Overview of the quality maps for an image with ghosting artifacts. The magnitude and phase of this image are displayed in (a) and (b), respectively. All maps were
computed with a rectangular kernel of 5 5 pixels and normalized to [0, 1]. (c) shows the local coherence QLC. The map exhibits the typical properties of QLC: very low values
in areas of steep phase topography even if there is significant signal present. Conversely it can have high values in areas where there is almost no signal as long as the phase is
coherent. (d) depicts the magnitude of the local average of the complex image QACI. QACI suppresses areas with low signal intensities, and has small values in regions with
steep phase topography. The map of the local average of the magnitude QAM is shown in (e). The local variance of the complex image QVCI is given in (f). It separates the object
from the background by a dark rim and has low values in areas of steep phase topography. The variance in noisy areas is very low (corresponding to high pixel intensities)
even compared to homogeneous areas within the object. This does not favor the variance as a quality criterion because it is lower in noise than in the object. (g) is an example
for the local variance of the phase QVPH. It exhibits a behavior similar to QLC.
duced in the original algorithm (Xu and Cumming, 1999) and
is calculated as
Q LC
i;j
iþdx 1
2
1 X
¼
dx dy dx 1
l¼i 2
dy 1
jþ
2
X
dy 1
2
m¼j
^l;m p
;
^l;m j
jp
ð4Þ
where (i, j) are the coordinates of the pixel for which the local
coherence is calculated and dx and dy are the dimensions of the
kernel.
average of the complex image (QACI) reaches its highest values if
the phase is coherent and the signal is high (Fig. 1d). Since it takes
both magnitude and phase into account it should suppress
unwrapping of regions where there is almost no signal present. It
is defined as
Q ACI
i;j
iþdx 1
X
1
2
¼
aACI dx dy dx 1
l¼i 2
X
^
pl;m ;
dy 1
m¼j 2
where aACI is a normalization constant ensuring that Q ACI
i;j lies within
the interval [0, 1].
average of the magnitude (QAM) is a smoothed map of the magnitude image (Fig. 1e). It is normalized such that its values range
from 0 to 1. The justification for this approach is that the variance
of the phase is inversely proportional to the magnitude (Conturo
and Smith, 1990).
variance of the complex image (QVCI) measures the variance of
the complex image within the surroundings of a pixel (Fig. 1f). It
was introduced to allow a better delineation of the object, since
the variance is expected to be highest at the borders of the object.
By using this measure it should be possible to restrict unwrapping
to the object. QVCI is given by
0
Q VCI
i;j ¼
d 1
jþ y2
ð5Þ
1 B
i;j p
i;j @p
aVCI
1
dx dy
iþdx21
X
l¼idx21
jþ
dy 1
2
X
d 1
m¼j y2
1
^l;m p
^l;m C
p
A þ 1;
ð6Þ
i;j is the local average of the complex image computed with
where p
a kernel of the same size (dx dy) and aVCI a normalization constant.
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The definition of QVCI ensures that pixels of highest quality (i.e. lowest variance) have a quality of approximately 1.
variance of the phase (QVPH) is motivated in the same way as QVCI,
but takes only the phase into account (Fig. 1g). It is calculated by
^i;j =jp
^i;j j.
^0i;j ¼ p
applying Eq. (6) on the normalized complex image p
Each map was computed with kernel sizes of 3 3, 5 5 and
7 7 pixels to study the influence of the kernel size on the performance of the algorithm.
for the evaluation (see next section). For this investigation, a seed
point that had a quality measure above a certain threshold but
differed from the seed point with the highest quality measure
(i.e. the seed point that is selected by default) was selected randomly. This procedure was repeated 100 times for each slice. Each
result was compared to the unwrapped phase obtained with the
seed point selected by the algorithm. This was repeated for two
different thresholds (0.95 and 0.99).
3.2. Evaluation of UUN
2.4. Seed finding
Since the placement of seed points is of utmost importance for
the reliability of the prediction of the unwrapped phase, we separated seed finding from the actual unwrapping. It is thus possible
to use different quality measures for seed finding and for unwrapping. We have studied the suitability of all maps defined in the previous section for seed finding. All maps were calculated with kernel
sizes of 5 5 as well as with 15 15. The reason for using larger
kernels is that the resulting map represents a more global quality
measure than the one computed with a smaller kernel. While
unwrapping requires a rather local quality measure to avoid premature termination, seed finding should profit from a large scale
quality measure.
Since searching for the ‘‘best” seed point (i.e. the seed point having the highest quality measure) is rather expensive if it has to be
repeated, we used the following approach: before unwrapping
starts the program creates a list of all possible seed points. This
is done by adding all pixels having a quality measure above a
threshold to a list. For performance reasons the seed points are
not sorted, but divided into small groups having approximately
the same quality. In each group, only the seed points having the
highest quality are sorted (we also investigated the robustness of
the algorithm with respect to the exact location of the seed point.
This is described in a later section). Every time a new seed is
needed (i.e. just before unwrapping starts and after two regions
have been merged) a seed point is taken from the top of the list.
A check whether this seed point or its neighborhood have been unwrapped is then performed. If this is the case it is not selected and
removed from the list of possible seed points and a new seed point
is selected. This procedure avoids the expensive task of merging for
newly created regions.
3. Methods
3.1. Implementation and optimization of the method
The unwrapping algorithm was implemented in C. UUN has a
command-line interface (i.e. stand-alone program) and an interface that allows for an seamless integration into IDL (ITT Visual
Information Solutions, USA) and Matlab (Mathworks, USA).
Two variants of the algorithm were implemented in the same
program: one in which only a single region is used for unwrapping
and one in which multiple regions grow simultaneously. The first
was motivated by a desire for the highest possible processing
speed, the latter by the need to unwrap poorly connected areas.
The influence of the quality maps was studied on brain data-sets
acquired on a 1.5 T system (Magnetom Vision, Siemens Medical
Solutions, Germany) and on a 3 T scanner (Medspec Avance, Bruker
Medical, Germany). All data were acquired with the standard quadrature transmit/receive head coil of each system. From this study a
set of default parameters was defined. This set of default parameters was then used for the evaluation of UUN’s performance.
In addition, we studied the stability of UUN with respect to the
location of the seed point on one of the subject data-sets acquired
To evaluate the performance of UUN we compared its performance in terms of quality of the unwrapped phase images as well
as the required computation time with PRELUDE which is a well
established method for unwrapping MRI data-sets. PRELUDE is
part of the FSL Analysis Package (Oxford Centre for Functional
Magnetic Resonance Imaging of the Brain, United Kingdom).
PRELUDE falls into the class of algorithms commonly known as
‘‘Split and Merge” algorithms. Unlike UUN, PRELUDE allows for
unwrapping to proceed not only in two, but also in three dimensions. Depending on command-line parameters, the unwrapping
proceeds in two dimensions, three dimensions or in a hybrid
mode. In the latter, the regions are labeled (the labeling corresponds to the split phase) in two dimensions while the unwrapping itself is done in three dimensions. According to the
manual, this is the default mode for high-resolution data-sets.
In the case of our data, PRELUDE treated data-sets with dimensions of 256 224 12 (voxel size: 1 1 2 mm3) as high-resolution data-sets.
Both programs were applied to the data using their default
parameters. The only exception were the parameters controlling
the various modes (single region and multiple regions in case of
UUN, and default (hybrid) mode, 2D mode, and 3D mode in case
of PRELUDE).
All data-sets were unwrapped on a multi-processor Linux machine consisting of two dual-core AMD Opteron processors running at 2 GHz. It was ensured that one processor was only used
for unwrapping. At all times the machine had a sufficient amount
of free memory to guarantee that computation times were not
influenced by swapping. Prior to unwrapping the data were copied onto the machine’s local disk to ensure fast and reproducible
data access. The amount of time required to load a full complex
valued data-set and to write back a floating point data-set was
measured and found to be negligible (6.2 103 s for a 256 224 slice).
The results of UUN and PRELUDE in all modes were compared
with respect to the time required for unwrapping, the agreement
of the unwrapped phase images and, in case of disagreements,
the plausibility of the unwrapped phase. Differences between the
unwrapped phase images were identified by subtracting the phase
images from each other. A multiple of 2p was subtracted from the
subtraction image if there was an offset between the images. Pixels
in which the phase difference differed from 0 were then investigated in the magnitude image, the wrapped phase image and the
unwrapped phase images to obtain an understanding for the reasons of the different results.
3.2.1. Phantom data
The phantom consisted of a 5 L glass bowl with a ping-pong ball
immersed in an aqueous solution containing 0.9% NaCl and
0.2 mmol/l Gd-DTPA. A total number of 72 fully flow compensated
3D gradient echo (Reichenbach and Haacke, 2001) data-sets were
acquired on a 1.5 T system (Magnetom Vision, Siemens, Erlangen,
Germany) using different echo times, flip angles and resolutions.
Of those 72 scans, 48 were low resolution scans (FoV:
256 224 96 mm3, Matrix: 256 168 48; i.e. having a voxel
S. Witoszynskyj et al. / Medical Image Analysis 13 (2009) 257–268
dimension of 1 1.33 2 mm3). The echo time TE and flip angle a
of each scan represented a point in the parameter space TE a
with TE 2 {20, 25, 30, 35, 40, 45} ms and a 2 {2, 4, 6, 8, 10, 15,
20, 25}°. Every combination of TE and a was utilized. The FoV
was placed such that the whole ping-pong ball and a substantial
amount of the aqueous solution were covered. Both the magnitude
and the phase of a sagittal and a transverse cut through the phantom can be seen in Fig. 2.
Of the other 24 data-sets 12 were low resolution scans
(FoV: 256 224 48 mm3 Matrix: 256 168 32; i.e. voxel
Fig. 2. A sagittal slice of the phantom (magnitude: (a), wrapped phase: (b)) and a
transversal slice of the phantom (magnitude: (c), wrapped phase: (d)). The arrow
heads in (a) indicate the slices in which the ROIs for calculating the SNR were
placed.
261
dimension: 1 1.33 1.5 mm3). The other 12 were high-resolution scans (FoV: 256 224 48 mm3, Matrix: 512 336 32;
i.e. having a voxel dimension of 0.5 0.67 1.5 mm3). The parameter space TE a was spanned by TE 2 {20, 30, 45} ms and a 2 {2,
4, 8, 16}°. Again, each low resolution and each high-resolution scan
represented one of the possible combinations of TE and a. For these
data-sets the FoV did not cover the full ping-pong ball. The repetition time TR was 60 ms for all scans.
Before reconstruction, the data were zero-filled such that the
in-plane resolution was 1 1 mm2 and 0.5 0.5 mm2, respectively. We ensured that the echo was centered in the z direction.
This minimized the number of wraps in z direction. This was done
to avoid penalizing the three dimensional methods. In the in-plane
phase encoding direction, the echo was shifted from the center to
obtain a more complex phase topography.
After reconstruction, the outermost slices were discarded, because they did not only contain little signal, but also wrap arounds.
For comparing the performance on data-sets with a complex and a
simple phase topography we separated each data-set into two
sub-sets one having a rather homogeneous (13 slices) and one
containing inhomogeneous (20 slices) magnetic field. The homogeneous sub-set did not exhibit significant distortions caused by
the field inhomogeneity. In total, 216 data-sets were created.
Additionally, we studied the dependency of UUN’s and
PRELUDE’s (in 2D mode) performance on the number of wraps
on two neighboring slices from three low resolution sets
(SNR = 5.1, 14.9 and 83.0, respectively). The slices were chosen
such that they were still affected by the field inhomogeneity but
had sufficient SNR to be completely unwrapped by all methods.
Each slice was reconstructed several times. For each reconstruction
the echo was shifted in phase encoding direction by a different
Fig. 3. A typical magnitude (a) and phase (b) image of a data-set that suffered from strong field inhomogeneities in the frontal areas caused by the paranasal sinuses (denoted
by the white rectangle). The image was unwrapped by UUN using QLC, QACI and QAM to guide the unwrapping procedure. The termination criteria were optimized for each
map in such a way that the unwrapping proceeded as far as possible but was limited to the object. The results are shown in the bottom row. By using QLC (c) the algorithm
fails to unwrap the frontal areas. Although, QACI (d) leads to an improved unwrapping of those regions, the best results are obtained with QAM (e).
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amount. The maximum displacement of the echo was 33 voxels.
The number of phase wraps in the resulting images ranged from
6 to 31 depending on how far the echo was shifted off center.
The median distance between wraps ranged from 22.0 to 6.5 voxels, respectively.
Each data-set was unwrapped with the methods described
above. For each data-set the SNR was estimated for a homogeneous
region and for a region that was still subject to the field inhomogeneities caused by the ping-pong ball.
3.2.2. Data of human subjects
Data of five healthy subjects acquired with the same sequence
as above on the same 1.5 T system (Magnetom Vision, Siemens,
Erlangen, Germany) were investigated retrospectively. The parameters were TR = 65 ms, TE = 40 ms, a = 25°, field of view (FoV)
25.6 cm 19.2 cm 6.4 cm, matrix size 512 256 32, acquisition time 10 min, imaging volume parallel to the AC–PC line.
The data were zero-filled before reconstruction to achieve isotropic
in-plane voxel dimensions. All data were unwrapped with UUN in
both single and multiple region mode and PRELUDE in all three
modes. The results were analyzed in the same manner as the phantom data.
4. Results
4.1. Definition of default parameters
Since the extent to which the algorithm unwraps the phase
within the object is controlled by the applied quality map and a
respective threshold, the question of interest was which quality
map allowed for defining a threshold that was applicable to all
data-sets. The performance was evaluated in terms of wrongly unwrapped pixels within the object and how well the algorithm
would unwrap areas of steep phase topography.
Local variance of the complex image QVCI provided a delineation
of the object by a rim of pixels with small intensities. Nevertheless,
this measure turned out to be of limited use for two reasons:
firstly, the variance was higher in the object than in areas containing noise only. Secondly, it was not possible to define general
thresholds because of the large range of values within the maps.
Applying a transformation such as logarithmic scaling instead of
the normalization did not improve the situation.
The local variance of the phase QVPH exhibited the same properties as the local coherence QLC, namely very low values in areas that
had sufficient signal but a steep phase topography while it had
rather high values in areas with no significant signal but coherent
phase due to ghosting artifacts. The expected improvement of the
delineation of the object did not occur. The local average of the
complex image QACI also did not lead to an improvement compared
to the local coherence QLC, because variations of the phase within a
pixel’s surrounding had a larger influence on its values than the
intensity.
Fig. 3 shows the unwrapped phase images obtained with QACI
and QAM which were found to be most suitable as quality measures
compared to QLC. Both, QACI and QAM led to a significantly better unwrapped phase image in areas of steep phase topography (Fig. 3d
and Fig. 3e, respectively). Nevertheless, the algorithm consistently
grew farther into regions with large phase gradients and unwrapped more pixels with QAM than with QACI. In areas where
using all three quality measures resulted in an unwrapped phase,
no difference in the phase images was observed.
In conclusion we found that unwrapping performed best with
local average of magnitude QAM calculated with a kernel size of
5 5 pixels.
However, using QAM for seed finding produced considerable
artifacts in some cases, as it placed seeds into areas that had a high
signal but were poorly connected to the bulk of the object. Thus the
algorithm had to grow along paths with low quality pixels before
unwrapping the rest of the brain. This allowed errors to accumulate and propagate into the bulk of the signal containing areas.
Fig. 4b displays an image that was affected by such an artifact.
The correctly unwrapped phase image is shown in Fig. 4c. For
the latter, QLC was used for seed finding. Using QLC computed with
a comparatively large kernel (15 15) led to the most reliable
placement of seed points.
By using QLC, UUN was very robust with respect to the exact
location of the seed point. Phase images unwrapped using seed
points placed randomly in areas that had a quality measure higher
than 0.99 and 0.95, respectively, did not show significant differences. Depending on the phase topography of the slice, the area
from which a seed point was chosen randomly encompassed between 7% and 42% of the pixels that were unwrapped in case of
QLC > 0.99 and between 45% and 81% for QLC > 0.95. For the higher
threshold, disagreements were observed for three of the 64 slices.
However, not more than 6 pixels at the rim of the brain were affected. In case of the lower threshold, the number of slices in which
disagreements occurred increased to ten. Also in this case, the
number of affected pixels did not exceed six. All affected pixels
were at the boundary of high SNR areas.
In conclusion we found that the use of QLC computed with a
large kernel (15 15) produced the most robust seed points (i.e.
seed points which led to reliable unwrapped phase images).
Fig. 4. Example of an artifact caused by seed points that were placed into a region poorly connected to the rest of the brain. (a) shows the magnitude image, (b) the
unwrapped phase image that is affected by an artifact because unwrapping started in the sagittal sinus, (c) correctly unwrapped phase image. In case of (b) the local average
of magnitude QAM was used for seed finding. (c) was obtained by applying local coherence QLC to seed finding. The arrow heads indicate the areas where seeds were placed.
S. Witoszynskyj et al. / Medical Image Analysis 13 (2009) 257–268
Unwrapping time normalized [s/#slices]
0.6
4.2. Evaluation of UUN’s performance on phantom data
PhUN (default parameters)
PhUN (multiple regions, n=100)
Prelude 2D
Prelude 3D
Prelude (default parameters)
0.55
0.5
All data-sets were split into two parts. The first slab contained
the ping-pong ball causing the susceptibility difference and was
thus strongly affected by the field inhomogeneity. These data were
used to investigate the dependency of all methods on computational complexity and SNR. The signal intensity in the second slab
was more homogeneous. The performance with respect to resolution, SNR and phase wrap density was investigated on these data.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
10
20
30
40
50
60
70
80
90
SNR
Fig. 5. Time required for unwrapping the ‘‘homogeneous” slab of the low resolution
data-sets measured with TE = 25 ms normalized to a single slice.
Normalized unwrapping time [s/#slices]
3.5
PhUN (SNR= 5.1)
PhUN (SNR=14.9)
PhUN (SNR=83.0)
PRELUDE 2D (SNR= 5.1)
PRELUDE 2D (SNR=14.9)
PRELUDE 2D (SNR=83.0)
3
2.5
263
4.2.1. Dependency on spatial resolution
UUN and PRELUDE unwrapped both the low and the highresolution data-sets. All differences observed between the performance on the low resolution and high-resolution data-sets could
be attributed to the significantly lower ( a factor 4) SNR of the
high-resolution data-sets and are thus described in the next
section.
The time required by UUN for unwrapping was almost independent of SNR and scaled linearly with the fourfold number of voxels
(0.51 s/slice compared to 0.14 s/slice for single region mode and
0.84 s/slice versus 0.20 s/slice in multi-region mode). PRELUDE in
2 D mode had a similar increase in computation time (0.5 s/slice
vs. 0.14 s/slice) for the high-resolution data-set with the highest
SNR. The increase in computation time for PRELUDE’s 3D and hybrid modes was slightly lower.
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
# wraps
Fig. 6. Dependency of UUN’s and PRELUDE’s (in 2D mode) performance on the
number of wraps in a slice at different SNR levels. The number of wraps was
modulated by shifting the echo off center before reconstruction. While UUN’s speed
is independent of the number of wraps and SNR, the time required by PRELUDE
increases with increasing number of wraps and decreasing SNR.
4.2.2. SNR dependency
The SNR of the low resolution data-sets ranged from 15.4 to
90.6. Even lower SNR (3.7–22.1) was measured in the high-resolution data-sets. Above an SNR of 13 the results of all methods were
in perfect agreement. The unwrapped phases were smooth and all
pixels within the phantom were unwrapped. Compared to
PRELUDE, UUN tended to unwrap an additional one pixel wide
rim at the border of the phantom. This behavior was caused by
slightly different termination conditions.
Below an SNR of 13, UUN still yielded smooth and fully unwrapped phase images. PRELUDE failed to unwrap an increasing number of pixels with decreasing SNR. These pixels were
not connected and appeared like salt and pepper noise. The
Fig. 7. Illustration of the influence of measuring parameters and echo shifts on the phantom data. The top row displays data that were acquired with TE = 20 ms and a flip
angle of a = 25°. The SNR was calculated to be 80.3 in the center of the image. (a) is the magnitude image, (b) the phase image with the echo centered and (c) the phase image
with the echo shifted to achieve the maximum number of phase wraps. The bottom row exhibits data of the same phantom acquired at TE = 45 ms and a = 2°. (d) shows the
respective magnitude image. The SNR in the center of the image is 5.1, (e) and (f) are the phase images with the centered and the maximally off-centered echo, respectively.
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a
b
c
e
d
f
35
Wrapped phase
PhUN
PRELUDE 2D
phase wrap
30
25
20
Phase
15
10
5
0
0
50
100
150
200
250
x
Fig. 8. A slice of the phantom ((a) magnitude image, (b) phase image) unwrapped by UUN (c) and PRELUDE 2D (d). (e) shows the difference between the results obtained with
UUN and PRELUDE. White pixels indicate pixels for which UUN computed an unwrapped phase but not PRELUDE, black pixels those which were unwrapped by PRELUDE but
not by UUN. In (f) the phase along a cut parallel to the x-axis at the center of the slice is shown. At about pixel 150, PRELUDE generates an artificial phase wrap. This wrap is
indicated in (d) and (e) by the arrow heads. It is hardly visible because of its small size and the rather large dynamic range of the image (the lowest value is 51.79, the
highest 42.46).
4.2.3. Sensitivity to phase wrap density
The number of phase wraps was modulated systematically by
shifting the echo from the k-space center in phase encoding
direction. In total, 20 data-sets with 6–31 phase wraps were generated. This corresponded to a median distance between wraps of
22.0 and 6.5 voxels, respectively. Both UUN and PRELUDE (in 2D
Normalized unwrapping time [s/#slices]
number of those pixels increased from a single pixel in single
slices at SNR 13 to 1.5% of the object at the lowest SNR
(SNR 3.7).
Below an SNR of 8, PRELUDE in 3D and hybrid (default) mode
failed to unwrap the data within 24 h and was terminated manually. At these low SNR levels, PRELUDE in 2D mode also tried to unwrap the area outside the phantom. This led to a sharp increase of
time required for unwrapping (a factor 10 at SNR 7 and more
than a factor 1000 at SNR < 4, respectively). At SNR levels below
4, UUN also started unwrapping the area outside of the phantom.
This led to a 10 times slower unwrapping.
UUN’s computation time was independent of SNR, except for
the data-sets with SNR < 4. UUN required 0.14 s for unwrapping
a low resolution slice in single region mode and 0.2 s in multi
region mode. At high SNR, PRELUDE even was slightly faster
0.12 s/slice in 2D and 3D mode. At SNR levels below 50, the
time required by PRELUDE exhibited a strong dependence on
SNR. This is especially obvious in the 3D and the default (hybrid)
mode. Fig. 5 shows the computation times normalized to a single slice for a slab of the low resolution data measured with
TE = 25 ms. For longer TE this behavior was even more pronounced, because the number of phase wraps increased with
longer TE. For TE = 45 ms, PRELUDE required 0.12 s/slice at the
highest SNR and 0.15 s/slice at SNR 15 in 2D mode, between
0.14 s/slice and 0.37 s/slice in 3D mode and between 0.24 s/slice
and 0.81 s/slice in hybrid mode.
100
PhUN (default parameters)
PhUN (multiple regions, n=100)
PRELUDE 2D
PRELUDE 3D
PRELUDE (default parameters)
10
1
0.1
0
10
20
30
40
50
60
70
80
90
SNR
Fig. 9. Time required for unwrapping data-sets containing the source of the field
inhomogeneity normalized to a single slice (on a logarithmic scale) versus the SNR
measured in a ROI affected by the field inhomogeneity. The SNR in this ROI serves as
an estimate for the complexity of the unwrapping problem.
S. Witoszynskyj et al. / Medical Image Analysis 13 (2009) 257–268
mode) unwrapped all images. All unwrapped phase images were
perfectly smooth and in exact agreement (except for the one pixel
wide rim around the phantom described above). The time required to unwrap a slice was comparable up to 17 wraps contained within the object (corresponding to a median distance
between wraps of 12 voxels). Up to this wrap density, both methods needed less than 200 ms for each slice at all SNR levels and
the time was approximately independent of the number of wraps
(see Fig. 6).
While UUN’s unwrapping speed did not show any change at
higher wrap densities, PRELUDE’s unwrapping time exhibited a
strong dependency on both number of wraps in the object and
SNR (the SNR was calculated for the center of the image which
was most strongly affected by the field inhomogeneity). For
high SNR images (SNR = 83.0) unwrapping the images with
the largest number of wraps (31) took PRELUDE approximately
5 times as long as unwrapping the images with up to 17 wraps.
In case of the low SNR images (SNR = 5.1), PRELUDE’s unwrapping was about a factor 20 slower than UUN. Fig. 7 displays
the effects of both measuring parameters and shifting of the
echo.
4.2.4. Computational complexity
Since the phase topography around the ping-pong ball was
complex, the dependence of the performance on the computational
complexity was investigated on the slab containing the ping-pong
ball. The SNR measured in the ROI in proximity to the ping-pong
ball varied with TE and flip angle and provided a measure for the
complexity of the unwrapping problem. The SNR ranged from 5.1
to 84.3 for the low resolution data.
265
The results of all modes of UUN and PRELUDE were in very good
agreement for all SNR levels. The only differences were:
In the areas most strongly affected by the field inhomogeneity (SNR 3), PRELUDE unwrapped further into the region
affected by the signal drop-out, but left small ‘‘holes” of
wrapped pixels. UUN, on the other hand, created a smoother
phase topography,
in some small areas (well below 100 pixels in size) with very
steep phase topography, PRELUDE created phase wraps and
the results of the different modes were not consistent. The
phase images computed by UUN did not show these wraps.
A slice that exhibits these discrepancies is shown in Fig. 8.
The computation time of UUN remained constant with increasing computational complexity (i.e. decreasing SNR and increasing
number of phase wraps). UUN required 0.15 s/slice in single region mode and 0.2 s/slice in multi-region mode. PRELUDE’s
run-time, however, exhibited a strong dependency on computational complexity. In 2D mode, PRELUDE had a performance comparable to UUN (0.26 s/slice) in case of the data with the lowest
complexity (i.e. corresponding to the highest SNR in the region affected by the field inhomogeneity). At the lowest SNR, the time required increased by a factor 2.5. In case of the high SNR data,
PRELUDE in 3D and hybrid mode was more than a factor 14 slower
(2.1 s/slice and 5.15 s/slice). The computation time for those two
modes increased up to 7.38 s/slice and 19.2 s/slice, respectively.
Compared to UUN, this corresponds to an unwrapping time increased by a factor between 50 and 130. Fig. 9 displays the
computation time behavior of UUN and PRELUDE.
Fig. 10. A single slice of one of the subjects ((a) magnitude image, (b) phase image) unwrapped by UUN (c) and PRELUDE 2D (d). (e) shows the difference between the results
obtained with UUN and PRELUDE. White pixels indicate pixels for which UUN computed an unwrapped phase but not PRELUDE, black pixels those for which PRELUDE
calculated an unwrapped phase but not UUN. Except for the fatty area around the brain, of which UUN unwrapped only a small fraction, there are small differences in the
frontal area because of different termination criteria. Within the brain there are four pixels that were not unwrapped by PRELUDE.
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4.3. Performance on subject data
The results of UUN, both in single region as well as in multiregion mode, and PRELUDE in 2D and in hybrid (default) mode
were in very good agreement; within the brain, at maximum single
pixels per slice differed. In those situations, even close inspection
did not reveal which prediction was more likely than the other.
In most cases, the reason for the disagreements was that the pixels
had very low magnitude. In other cases, it was single pixels surrounded by pixels with very low magnitude. Just as in the phantom, PRELUDE tended to grow slightly further into areas where
the signal was very low (Fig. 10).
Although in general, PRELUDE in 3D mode was in good agreement with the other methods, it occasionally created wraps in
places where there were none before unwrapping (Fig. 11).
In all cases, UUN in default mode outpaced all other methods in
terms of speed. UUN needed just below 0.4 s for unwrapping a single 384 512 slice. In multi-region mode UUN required between
0.6 s and 0.8 s. Unwrapping a slice with PRELUDE in 2D mode took
between 2.49 s and 7.59 s per slice. The computation time for each
slice ranged from 29.42 s to 397.56 s for PRELUDE in 3D mode and
for PRELUDE in default (hybrid) mode from 383.16 s to 1509.13 s
(Fig. 12).
5. Discussion
5.1. Algorithm
We proposed a region growing phase unwrapping method
that exploits magnitude information for guiding the unwrapping
and phase information for selecting seed points. Since region
growing algorithms depend on the criteria used for growing
the region and placing seed points, we believe that the choice
of the quality map is crucial as it should reflect the phase
image’s underlying physics and take into account the characteristics of the method for estimating the unwrapped phase. There
is an important difference between using magnitude information
for guiding the growth of regions compared to simply using it as
a termination criterion in the form of a mask. The quality map
can be rather seen as a dynamic mask that is adjusted to the
Fig. 11. Wrapped phase (a) of a subject where PRELUDE in 3D mode generated phase wraps that were not present in the original phase image (b). For comparison the phase
image obtained with UUN is shown in (c). The position of the artifact produced by PRELUDE is indicated by arrow heads in all three images.
Normalized unwrapping time [s/#slices]
1,000
100
10
1
0
Φ UN (default parameters)
Φ UN (multiple regions, n=100)
PRELUDE (default parameters)
PRELUDE 2D
PRELUDE 3D
Fig. 12. Comparison of the time required for unwrapping SWI data-sets of five different subjects. For better comparison the times were normalized to a single slice. While
UUN achieved unwrapping times below 400 ms in single region mode and less than 800 ms in multi-region mode, PRELUDE was about 10 times slower in 2D mode.
Furthermore, PRELUDE used with default parameters and PRELUDE in 3D mode were between 100 and 5000 times slower than the 2D Region Growing Phase Unwrapping
program.
S. Witoszynskyj et al. / Medical Image Analysis 13 (2009) 257–268
unwrapping stage. The argument that magnitude information
should be incorporated in guiding the unwrapping can even be
made without considering that the magnitude is related to
phase SNR. Information of pixels with significant signal strength
is more reliable than of those without. Nevertheless, since a linear prediction is used to estimate the phase, a flat phase topography is preferable at early stages of a region’s growth process.
Since this cannot be ensured by using magnitude information,
we incorporate phase information for seed finding by using local
coherence (QLC).
If the seed points are placed into areas of flat phase topography
the algorithm is very robust with respect to the location of the seed
point. This was shown by randomly selecting seed points. If seed
points were also set in areas with lower QLC the number of phase
images with disagreements increased. This shows that although
the exact location of the seed point is not of great importance
the phase topography around the seed point has to be reasonably
flat.
As any post-processing step, UUN has certain constraints and
requirements on the data it is applied to. The complex images have
to be acquired in a way that the phase information is not distorted.
Although this might seem obvious, it is often not the case. The
most common reasons for distorted phase images are:
(1) Incorrectly combined data obtained with phased array coils
can introduce singularities to the phase image. These singularities are often referred to as open-ended fringe lines
(Chavez et al., 2002). In such situations it is not possible to
describe the phase by a continuous function. The issue of
combining data of the coil elements of a phased array coil
is a very complex one. Its discussion is beyond the scope
of this publication. For a thorough discussion of the theoretical basics we refer the reader to (Roemer et al., 1990).
(2) Filters that are applied to the data may cause singularities in
phase images. Especially noteworthy is homodyne detection.
In general, it is not possible to remove phase wraps that
remain after homodyne detection by applying phase
unwrapping, since these phase images usually contain
singularities.
(3) Images which were scaled or rotated by operations that
were applied to the phase and magnitude images separately.
As a result the sharp edges that characterize phase wraps
can be smeared out and thus not appear as a phase wrap.
If image scaling or rotation are to be performed prior to
unwrapping they have to be either done in fourier space
(i.e. by zero filling in case of scaling) or on the real and imaginary part of the complex image.
In images in which the phase information is not distorted by
either a filter or incorrectly combined phase images, open ended
fringe lines appear only in areas with very low SNR. Since the algorithm does not grow into those areas if magnitude information is
used for guiding the unwrapping, UUN does not attempt to unwrap open ended fringe lines.
UUN does not make any assumptions on the orientation in
which the data was acquired. While the orientation of a slice can
have an impact on the phase image itself (especially if the voxels
are anisotropic), the problem of phase unwrapping is not changed.
UUN has successfully unwrapped data acquired in sagittal orientation (Koopmans et al., 2008), for instance. However, since phase
predictions of all prediction lines are regarded in the same manner
for computing a common prediction, isotropic in-plane voxel
dimensions are assumed implicitly. If the voxel dimensions are
very anisotropic it might be necessary to account for this anisotropy by using weights depending on the direction of the prediction
line.
267
MRI data can be volume data in which there is no gap between slices. One could assume that a three dimensional approach would provide a better performance than unwrapping
slice by slice. The performance of PRELUDE in 3D and hybrid
mode show that this is not necessarily the case. The artifacts
produced by PRELUDE in 3D mode are another indication that
three dimensional phase unwrapping does not necessarily lead
to a better unwrapping.
In many cases, fully three dimensional unwrapping is not necessary. As long as each slice has been unwrapped reliably, adding
multiples of 2p removes inconsistencies across slices. Furthermore,
for example in SWI, an important three dimensional application,
only high frequency phase variations are of interest. Thus, because
the phase images are corrected for low frequency phase variations,
any inconsistency across slices is removed.
In principle, extending the algorithm to three dimensions is
straightforward. Nevertheless, we were reluctant to modify our approach mainly because of the following reasons. Firstly, in three
dimensions each voxel has 26 instead of eight neighbors. This increases not only the number of predictions that have to be calculated for each voxel, but also the chances that the common
prediction fails to meet the criteria for being unwrapped. This
would increase the computation time significantly, which is contrary to the aim of our implementation (namely to provide fast
unwrapping). Secondly, in many cases, the slice thickness is much
larger than the in-plane voxel dimensions. This anisotropy would
have to be taken into account. This would add additional complexity and parameters to the problem.
5.2. Evaluation and comparison
We based our evaluation and comparison entirely on measured
and not on simulated data. In measured data the true phase cannot
be known. On the other hand, phase and magnitude images are
mutually entangled, and a simulation would have to take both
properties into account, without losing information on the absolute phase. Furthermore, our results show that a comparison between the algorithms’ performances on phantom data and on
subject data is very difficult. Simulated data would therefore just
give an indication of an algorithm’s performance but would not allow any conclusion about its performance on real data. Because of
these reasons and the fact that the results of all methods were in
very good agreement in almost all situations, we concluded that
knowledge of the ‘‘true” phase was not necessary for our
evaluation.
The biggest difference between the methods did not lie in the
quality of the results but in their speed and its dependency on
the complexity of the phase topography and SNR. A closer investigation of the algorithms’ performance on the phantom data
showed that, in case of high SNR and low complexity of the phase
topography, the times required is approximately the same for all
methods and 0.1–0.2 s per slice. However, even for this rather
homogeneous data, at lower SNR levels, PRELUDE exhibits an
SNR dependency which is largest for the 3D and hybrid algorithm.
On the other hand, the times required by UUN are approximately
independent of SNR. PRELUDE’s SNR dependency was already
described in the original publication (Jenkinson, 2003) where
PRELUDE was compared to a minimum spanning tree (MST) algorithm. PRELUDE’s advantage compared to MST was that, while MST
had an SNR independent run-time, the number of incorrectly
unwrapped pixels increased much faster with decreasing SNR.
UUN had fewer incorrectly unwrapped pixels than PRELUDE at
extremely low SNR values.
Also, while UUN’s performance was approximately independent of the phase topography’s complexity, PRELUDE’s computation time increased with increasing number of phase warps and
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S. Witoszynskyj et al. / Medical Image Analysis 13 (2009) 257–268
complexity of the phase topography. This behavior was even more
pronounced at low SNR and for the 3D and hybrid mode.
The salt and pepper noise like artifacts produced by PRELUDE in
very low SNR images are caused by the way that PRELUDE uses the
magnitude as termination criterion. While UUN bases it’s termination decision on the magnitude averaged over a small area,
PRELUDE only considers the magnitude in a single point. At low
SNR this results in single pixels that are not unwrapped.
This also explains the discrepancies in areas with very low
intensity. In those situations, UUN tends to create smoother
boundaries, while PRELUDE has a tendency to unwrap further into
areas with very low signal. This also causes holes in PRELUDE’s
phase images in the same areas. For subsequent processing UUN’s
smoother boundary might be advantageous, since small holes correspond to high spatial frequencies. Thus those areas would have
to be excluded if a filter is to be applied to the phase image.
UUN has been applied only to phantom and brain data so far.
An application to other tissues and organs should, in principle, be
possible as long as some tissue properties are kept in mind. Brain
tissue, for example, does not contain fatty tissue. In case of organs
containing fatty tissue the phase within the fatty tissue can be
shifted from the phase of the surrounding tissue if the echo time
is chosen appropriately. In this situation the phase topography
may contain discontinuities or, even worse, singularities caused
by partial volume effects. While discontinuities themselves do
not pose a problem if unwrapping is done by multiple regions, singularities hamper any phase unwrapping approach.
6. Conclusions
We have developed and implemented a fast and robust 2D region growing phase unwrapping algorithm optimized for MRI data.
The algorithm’s performance was tested extensively on both
phantom and in vivo data and compared to an established method
(PRELUDE). Both algorithms performed reliable and similar in high
SNR areas, but UUN accomplished unwrapping much faster and
more robust in low SNR areas and in case of complex phase topographies. This makes UUN a suitable application for SWI that contain low SNR areas. UUN is freely available for the scientific
community.
Acknowledgements
This study was supported in parts by the European COST action
B21 (COST-STSM-B21-00690) and the German Research Foundation (DFG RE 1123/7-2).
Stephan Witoszynskyj acknowledges financial support from the
Jena Interdisciplinary Center for Clinical Research (IZKF S10).
Alexander Rauscher acknowledges financial support from Core
Unit MR Methods of the University Jena (BMBF 01ZZ0405).
Markus Barth acknowledges financial support from the Austrian
Science Foundation (FWF Fonds No. J2439-B02).
References
An, L., Xiang, Q.-S., Chavez, S., 2000. A fast implementation of the minimum
spanning tree method for phaseunwrapping. IEEE Trans. Med. Imaging 19 (8),
805–808.
Barth, M., Nöbauer-Huhmann, I.M., Reichenbach, J.R., Mlynarik, V., Schöggl, A.,
Matula, C., Trattnig, S., 2003. High-resolution three-dimensional contrastenhanced blood oxygenation level-dependent magnetic resonance venography
of brain tumors at 3 Tesla: first clinical experience and comparison with 1.5
Tesla. Invest. Radiol. 38 (7), 409–414. Jul.
Bourgeat, P., Fripp, J., Stanwell, P., Ramadan, S., Ourselin, S., 2007. MR image
segmentation of the knee bone using phase information. Med. Image. Anal. 11
(4), 325–335. August.
Bryant, D.J., Payne, J.A., Firmin, D.N., Longmore, D.B., 1984. Measurement of flow
with NMR imaging using a gradient pulse and phase difference technique. J.
Comput. Assist. Tomogr. 8 (4), 588–593. August.
Chavez, S., Xiang, Q.-S., An, L., 2002. Understanding phase maps in MRI: a new
cutline phase unwrapping method. IEEE Trans. Med. Imaging 21 (8), 966–977.
August.
Conturo, T.E., Smith, G.D., 1990. Signal-to-noise in phase angle reconstruction:
dynamic range extension using phase reference offsets. Magn. Reson. Med. 15
(3), 420–437. September.
Cusack, R., Papadakis, N., 2002. New robust 3-D phase unwrapping algorithms:
application to magnetic field mapping and undistorting echoplanar images.
Neuroimage 16, 754–764.
Dahnke, H., Schaeffter, T., 2005. Limits of detection of SPIO at 3.0 T using T2
relaxometry. Magn. Reson. Med. 53 (5), 1202–1206. May.
Duyn, J.H., van Gelderen, P., Li, T.-Q., de Zwart, J.A., Koretsky, A.P., Fukunaga, M.,
2007. High-field mri of brain cortical substructure based on signal phase. Proc.
Natl. Acad. Sci. USA 104 (28), 11796–11801. June.
Haacke, E.M., Brown, R.W., Thompson, M.R., Venkatesan, R., 1999. Magnetic
Resonance Imaging – Physical Principles and Sequence Design. Wiley-Liss.
Hamans, B.C., Barth, M., Leenders, W.P., Heerschap, A., 2006. Contrast enhanced
susceptibility weighted imaging (CE-SWI) of the mouse brain using ultrasmall
superparamagnetic ironoxide particles (USPIO). Z. Med. Phys. 16 (4), 269–274.
Hedley, M., Rosenfeld, D., 1992. A new two-dimensional phase unwrapping
algorithm for MRI images. Magn. Reson. Med. 24 (1), 177–181. May.
Hutton, C., Bork, A., Josephs, O., Deichmann, R., Ashburner, J., Turner, R., 2002. Image
distortion correction in fMRI: a quantitative evaluation. Neuroimage 16 (1),
217–240.
Jenkinson, M., 2003. Fast, automated, N-dimensional phase-unwrapping algorithm.
Magn. Reson. Med. 49, 193–197. January.
Jezzard, P., Balaban, R.S., 1995. Correction for geometric distortion in echo planar
images from B0 field variations. Magn. Reson. Med. 34, 65–73.
Johnson, G., Hutchison, J.M.S., 1985. The limitations of NMR recalled-echo imaging
techniques. J. Magn. Reson. 63 (1), 14–30.
Koopmans, P.J., Manniesing, R., Niessen, W.J., Viergever, M.A., Barth, M., 2008. MR
venography of the human brain using susceptibility weighted imaging at very
high field strength. Magn. Reson. Mater. Phys. 21 (1–2), 149–158. March.
Kruse, S.A., Smith, J.A., Lawrence, A.J., Dresner, M.A., Manduca, A., Greenleaf, J.F.,
Ehman, R.L., 2000. Tissue characterization using magnetic resonance
elastography: preliminary results. Phys. Med. Biol. 45 (6), 1579–1590. June.
Noll, D.C., Nishimura, D.G., Makovski, A., 1991. Homodyne detection in magnetic
resonance imaging. IEEE Trans. Med. Imaging 10, 154–163.
Ogg, R.J., Langston, J.W., Haacke, E.M., Steen, R.G., Taylor, J.S., 1999. The correlation
between phase shifts in gradient-echo MR images and regional brain iron
concentration. Magn. Reson. Imaging 17 (8), 1141–1148. October.
Poser, B.A., Versluis, M.J., Hoogduin, J.M., Norris, D.G., 2006. BOLD contrast
sensitivity enhancement and artifact reduction with multiecho EPI: parallelacquired inhomogeneity-desensitized fMRI. Magn. Reson. Med. 55 (6), 1227–
1235. June.
Rauscher, A., Barth, M., Reichenbach, J.R., Stollberger, R., Moser, E., 2003. Automated
unwrapping of MR phase images applied to BOLD MR-venography at 3 Tesla. J.
Magn. Reson. Imaging 18 (2), 175–180. August.
Rauscher, A., Sedlacik, J., Barth, M., Mentzel, H.J., Reichenbach, J.R., 2005. Magnetic
susceptibility-weighted MR phase imaging of the human brain. Am. J.
Neuroradiol. 26 (4), 736–742. April.
Rauscher, A., Sedlacik, J., Barth, M., Witoszynskyj, S., Reichenbach, J.R., 2006. Phase
map based investigation of intravoxel signal dephasing in gradient echo MRI.
In: Proc. Intl. Soc. Mag. Reson. Med., vol. 14, p. 2792.
Reichenbach, J.R., Haacke, E.M., 2001. High-resolution BOLD venographic imaging: a
window into brain function. NMR Biomed. 14, 453–467.
Reichenbach, J.R., Venkatesan, R., Schillinger, D., Haacke, E.M., 1997. Small vessels in
the human brain: MR-venography with deoxyhemoglobin as an intrinsic
contrast agent. Radiology 204, 272–277.
Robinson, S., Windischberger, C., Rauscher, A., Moser, E., 2004. Optimized 3 T EPI of
the amygdalae. Neuroimage 22 (1), 203–210. May.
Roemer, P.B., Edelstein, W.A., Hayes, C.E., Souza, S.P., Mueller, O.M., 1990. The NMR
phased array. Magn. Reson. Med. 16 (2), 192–225.
Volkov, V.V., Zhu, Y., 2003. Deterministic phase unwrapping in the presence of
noise. Opt. Lett. 28 (22), 2156–2158. November.
Windischberger, C., Robinson, S., Rauscher, A., Barth, M., Moser, E., 2004. Robust
field map generation using a triple-echo acquisition. J. Magn. Reson. Imaging 20
(4), 730–734. October.
Xu, W., Cumming, I., 1999. A region-growing algorithm for InSAR phase
unwrapping. IEEE Trans. Geosci. Remote Sens. 37 (1), 124–134. January.