"A Shape-Based Approach to the Segmentation of Medical Imagery Using Level Sets"

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
137
A Shape-Based Approach to the Segmentation of
Medical Imagery Using Level Sets
Andy Tsai*, Anthony Yezzi, Jr., William Wells, Clare Tempany, Dewey Tucker, Ayres Fan, W. Eric Grimson,
and Alan Willsky
Abstract—We propose a shape-based approach to curve evolution for the segmentation of medical images containing known
object types. In particular, motivated by the work of Leventon,
Grimson, and Faugeras [15], we derive a parametric model for
an implicit representation of the segmenting curve by applying
principal component analysis to a collection of signed distance
representations of the training data. The parameters of this
representation are then manipulated to minimize an objective
function for segmentation. The resulting algorithm is able to
handle multidimensional data, can deal with topological changes
of the curve, is robust to noise and initial contour placements, and
is computationally efficient. At the same time, it avoids the need for
point correspondences during the training phase of the algorithm.
We demonstrate this technique by applying it to two medical
applications; two-dimensional segmentation of cardiac magnetic
resonance imaging (MRI) and three-dimensional segmentation of
prostate MRI.
Index Terms—Active contours, binary image alignment, cardiac
MRI segmentation, curve evolution, deformable model, distance
transforms, eigenshapes, implicit shape representation, medical
image segmentation, parametric shape model, principal component analysis, prostate segmentation, shape prior, statistical shape
model.
I. INTRODUCTION
EDICAL image segmentation algorithms often face difficult challenges such as poor image contrast, noise, and
missing or diffuse boundaries. For example, tissue boundaries
in medical images may be smeared (due to patient movements),
M
Manuscript received September 19, 2001; revised October 11, 2002. This
work was supported in part by the Office of Naval Research (ONR) under Grant
N00014-00-1-0089, in part by the Air Force Office of Scientific Research
(AFOSR) under Grant F49620-98-1-0349, in part by the National Science
Foundation (NSF) under an ERC Grant through Johns Hopkins Agreement
8810274, in part by the National Institutes of Health (NIH) under Grant
1P41RR13218 and NIH R01 Grant AG 19513-01. The Associate Editor
responsible for coordinating the review of this paper and recommending its
publication was J. Duncan. Asterisk indicates corresponding author.
*A. Tsai is with the Massachusetts Institute of Technology, Laboratory for
Information and Decision Systems, Department of Electrical Engineering,
Room 35-427, 127 Massachusetts Ave., Cambridge, MA 02139 USA (e-mail:
atsai@mit.edu).
A. Yezzi, Jr. is with the School of Electrical and Computer Engineering;
Georgia Institute of Technology, Atlanta, GA 30332 USA.
W. Wells is with Brigham and Women’s Hospital/ Harvard Medical School,
Boston, MA 02115 USA, and the Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.
C. Tempany is with Brigham and Women’s Hospital/Harvard Medical
School, Boston, MA 02115 USA.
D. Tucker, A. Fan and A. Willsky are with the Laboratory for Information
and Decision Systems; Massachusetts Institute of Technology, Cambridge, MA
02139 USA.
W. E. Grimson is with the Artificial Intelligence Laboratory; Massachusetts
Institute of Technology, Cambridge, MA 02139 USA.
Digital Object Identifier 10.1109/TMI.2002.808355
missing (due to low SNR of the acquisition apparatus), or
nonexistence (when blended with similar surrounding tissues).
Under such conditions, without a prior model to constrain
the segmentation, most algorithms (including intensity- and
curve-based techniques) fail-mostly due to the under-determined nature of the segmentation process. Similar problems
arise in other imaging applications as well and they also hinder
the segmentation of the image. These image segmentation
problems demand the incorporation of as much prior information as possible to help the segmentation algorithms extract the
tissue of interest. We propose such an algorithm in this paper.
In particular, we derive a model-based, implicit parametric
representation of the segmenting curve and calculate the parameters of this representation via gradient descent to minimize
an energy functional for medical image segmentation.1
A. Relationship to Prior Work
Our work shares common aspects with a number of modelbased image segmentation algorithms in the literature. Chen
et al. [6] employed an “average shape” to serve as the shape
prior term in their geometric active contour model. Cootes et al.
[10] developed a parametric point distribution model for describing the segmenting curve by using linear combinations of
the eigenvectors that reflect variations from the mean shape.
The shape and pose parameters of this point distribution model
are determined to match the points to strong image gradients.
Pentland and Sclaroff [21] later described a variant of this approach. Staib and Duncan [23] introduced a parametric point
model based on an elliptic Fourier decomposition of the landmark points. The parameters of their curve are calculated to
optimize the match between the segmenting curve and the gradient of the image. Chakraborty et al. [4] extended this approach
to a hybrid segmentation model that incorporates both gradient
and region-homogeneity information. More recently, Wang and
Staib [30] developed a statistical point model for the segmenting
curve by applying principal component analysis (PCA) to the
covariance matrices that capture the statistical variations of the
landmark points. They formulated their edge-detection and correspondence-determination problem in a maximum a posteriori
Bayesian framework. Image gradient is used within that framework to calculate the pose and shape parameters that describes
their segmenting curve. Leventon et al. [15] proposed a less restrictive model-based segmenter. They incorporated shape information as a prior model to restrict the flow of the geodesic
active contour [3], [32]. Their prior parametric shape model is
1A
preliminary conference paper based on this work can be found in [26].
0278-0062/03$17.00 © 2003 IEEE
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
derived by performing PCA on a collection of signed distance
maps of the training shape. The segmenting curve then evolves
according to two competing forces: 1) the gradient force of the
image, and 2) the force exerted by the estimated shape where
the parameters of the shape are calculated based on the image
gradients and the current position of the curve.
Our work is also closely related to region-based active contour models [5], [20], [22], [34]. In general, these region-based
models enjoy a number of attractive properties over gradientbased techniques for segmentation, including greater robustness
to noise (by avoiding derivatives of the image intensity) and initial contour placement (by being less local than most edge-based
approaches).
simplifies the alignment task, which we approach from a variational perspective.
B. Contributions of Our Work
In our algorithm, we adopt the implicit representation of the
segmenting curve proposed in [15] and calculate the parameters
of this implicit model to minimize the region-based energy functionals proposed in [5] and [34] for image segmentation. The
resulting algorithm is found to be computationally efficient and
robust to noise (since the evolving curve has limited degrees of
freedom), has an extended capture range (because the segmentation functional is region-based instead of edge-based), and does
not require point correspondences (due to an Eulerian representation of the curve). Though in this paper, we only show the development of our technique for two-dimensional (2-D) data, this
algorithm can easily be generalized to handle multidimensional
data. We demonstrate a three–dimensional (3-D) application of
our technique in Section VI. Also, in this paper, we focus on
using the region-based models presented in [5] and [34] . However, it is important to point out that other region-based models
are equally applicable in this framework.
The rest of the paper is organized as follows. Section II describes a gradient-based approach to align all the training shapes
in the database to eliminate variations in pose. Based on this
aligned training set, we show in Section III the development of
an implicit parametric representation of the segmenting curve.
Section IV describes the use of this implicit curve representation in various region-based models for image segmentation.
Section V provides a brief overview to illustrate how the various components mentioned above fit within the scope of our
algorithmic framework. In Section VI, we show the application
of this technique to two medical applications; the segmentation
of the left ventricle from 2-D cardiac MRI and prostate gland
segmentation from 3-D pelvic MRI. We conclude in Section VII
with a summary and some possible future research directions of
this work.
II. SHAPE ALIGNMENT
We begin our shape modeling process with the alignment of
training shapes.2 There have been a number of works dealing
with the alignment of images [6], [8], [11], [17], [28], [29].
For our application, we are interested in aligning binary images
since that is how we encode the training shapes. This greatly
A. Alignment Model
consist of a set of binary images
Let the training set
, each with values of one inside and zero outside the object. The goal is to calculate the set of pose paramused to jointly align the binary imeters
ages, and hence remove any variations in shape due to pose differences. We focus on using similarity transformations to align
these binary images to each other. That is, in two dimensions,
with , , , and corresponding to , -translation, scale, and rotation, respectively. The transformed image
of , based on the pose parameter , is denoted by , and is defined as
where
(1)
The transformation matrix
a translation matrix
in-plane rotation matrix
maps the coordinates
An effective strategy to
is to use gradient descent
functional:
is the product of three matrices:
, a scaling matrix
, and an
. This transformation matrix
into coordinates
.
jointly align the binary images
to minimize the following energy
(2)
where denotes the image domain. Minimizing (2) is equivalent to simultaneously minimizing the difference between any
pair of binary images in the training database. The area normalization term in the denominator of (2) is employed to prevent all
the images from shrinking to improve the cost function.
, taken with respect to
for any , is
The gradient of
given by
2Our method can take advantage of any alignment technique. We need to
employ an alignment technique as a preprocessing step to allow us to capture
shape variations in our database without interference from pose variations.
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(3)
TSAI et al.: A SHAPE-BASED APPROACH TO THE SEGMENTATION OF MEDICAL IMAGERY USING LEVEL SETS
139
Fig. 1. Training data: 12 2-D binary shape models of the fighter jet.
Fig. 2.
Alignment results of the above 12 2-D shape models of the fighter jet.
where
is the gradient of the transformed image taken
with respect to the pose parameter . Using the chain rule, the
th component of
is given by
where
(4a)
(a)
(4b)
(4c)
(4d)
The matrix derivatives in (4) are taken componentwise. Since
the solution of this alignment problem is under-determined, we
regularize the problem by keeping the initial pose of one of
the shapes fixed and calculating the pose parameters for the remaining shapes using the above approach. The initial poses of
the training shapes in are employed as the starting point for
the alignment process and gradient descent is performed until
convergence.
To illustrate this alignment process, a training set, consisting
of 12 binary representations of fighter jets, is shown in Fig. 1. In
this example, the pose parameter of the fighter jet at the far left
.
side of the figure is chosen to be fixed, i.e.,
The aligned version of this data set is shown in Fig. 2. Note
that all the aligned fighter jets share roughly the same center,
are pointing in the same direction, and are approximately equal
in size. One way to judge the effectiveness of this alignment
process is to assess the amount of overlap between the shapes
within the database before and after the alignment process. The
prealignment overlap image, shown in Fig. 3(a), is generated
by stacking together all the binary fighter jets within the database prior to alignment (i.e., the fighter jets shown in Fig. 1),
and adding them together in a pixelwise fashion. The postalignment overlap image, shown in Fig. 3(b), is generated in a similar fashion except that the binary fighter jets used to calculate
the overlap image have already been aligned. Specifically, the
fighter jets used in this case are the ones shown in Fig. 2. By
comparing the two overlap images, there is a dramatic increase
(b)
Fig. 3. Comparison of the amount of shape overlap in the “fighter” database
(a) before alignment and (b) after alignment.
in the amount of overlap between the shapes after the alignment
process suggesting that this method is an effective alignment
technique.
B. Multiresolution Alignment
The nature of the gradient descent approach we just described
allows for only infinitesimal updates of the pose parameters,
thus giving rise to slow convergence properties and increased
sensitivity to local minima. These unattractive features are especially evident when trying to align large and complicated objects. One standard extension to enhance alignment algorithms
is to utilize a multiresolution approach. The basic idea behind
this approach is to employ a coarsened representation of the
training set to obtain a good initial estimate of the pose parameters. We then progressively refine these pose estimates as the
resolution of the objects is increased.
Specifically, given a set of training objects, we repeatedly
subsample all the objects within the training set by a factor of
two in each axis direction to obtain a collection of training sets
with varying resolutions. Initial alignment is performed on the
coarsest resolution set of objects to obtain a good initial estimate of the pose parameters. Operating at such a coarse scale,
we reduce the number of updates required for alignment (since
the domain of the image is reduced) and the sensitivity of the
algorithm to local minima (by allowing the parameter search
to be less local). More importantly though, the computational
burden of alignment at each gradient step is substantially reduced, mostly due to the decreased computational cost associated with calculating (3) on a coarser grid. The pose parameters
estimated on this coarsened set of training objects are appropriately scaled to serve as the starting pose estimates for the next
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
Fig. 4. Training data: 12 2-D binary shape models of the number four with size of 200
Fig. 5.
Lowest resolution representation of the above training data with size of 50
Fig. 6. Alignment results of the above 50
Fig. 7.
2 200 pixels.
2 50 pixels.
2 50 shape models of the number four.
Coarse-to-fine multiresolution refinement results of the 200
2 200 shape models of the number four.
higher resolution set of objects.3 By providing a good starting
estimate of the pose parameters at this new scale, only a small
number of updates are required for convergence. This process
of using the pose estimate at one resolution as the starting pose
for the next finer resolution is repeated until the finest resolution set of objects is reached. To illustrate this multiresolution
approach, we show in Fig. 4 a set of 12 binary representations
of the number four. The fours are difficult objects to align due
to the complicated structure of these objects. Fig. 5 shows this
same data set with each shape down sampled by a factor of four
in each direction. Initially, we align the fours in this reduced
image domain. The results of this alignment are shown in Fig. 6.
Next, we appropriately scale the pose parameters to serve as the
starting pose for the next higher resolution. We continue this
process until the finest resolution training set is reached. The
final alignment results are shown in Fig. 7. Fig. 8 shows the
prealignment and postalignment overlap images of the number
four to visually demonstrate the effectiveness of this alignment
process.
III. IMPLICIT PARAMETRIC SHAPE MODEL
As mentioned earlier, a popular and natural approach to represent shapes is via point models where a set of marker points is
used to describe the boundaries of the shape. This approach suffers from problems such as numerical instability, inability to accurately capture high curvature locations, difficulty in handling
topological changes, and the need for point correspondences.
To overcome these problem, we utilize an Eulerian approach to
shape representation based on the level set methods of Osher
and Sethian [19].
3Only the translational components of the pose are scaled up. The scaling and
rotational components of the pose remain fixed.
(a)
(b)
Fig. 8. Comparison of the amount of shape overlap in the “four” database
(a) before alignment and (b) after alignment.
A. Shape Parameters
Following the lead of [15] and [19], we choose the signed
distance function4 as our representation for shape. In particular,
the boundaries of each of the aligned shapes in the database5
are embedded as the zero level set of separate signed distance
with negative distances assigned
functions
to the inside and positive distances assigned to the outside of
the object. Using the technique developed in [15], we compute
, the mean level set function of the shape database, as the av. To
erage of these signed distance functions,
extract the shape variabilities, is subtracted from each of the
signed distance functions to create mean-offset functions
. These mean-offset functions are then used
to capture the variabilities of the training shapes.
9( )
Z
0
4The signed distance
p from an arbitrary point p to a known surface
is the distance between p and the closest point z in , multiplied by 1 or 1,
depending on which side of the surface p lies in [1].
5The shapes in the database are aligned by employing the method presented
in Section II.
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Z
TSAI et al.: A SHAPE-BASED APPROACH TO THE SEGMENTATION OF MEDICAL IMAGERY USING LEVEL SETS
141
. (b) Three-dimensional illustration of +1 8 .
Fig. 9. Three-dimensional visualization of the fighter jet shape variability. (a) The mean level set function 8
(c) Level set of +1 variation of the first principal mode. (d) Three-dimensional illustration of 1 8 . (e) Level set of 1 variation of the first principal mode.
0
Specifically, we form column vectors, , consisting of
samples of each
(using identical sample locations for each
function). The most natural sampling strategy is to utilize the
rectangular grid of the training images to generate
lexicographically ordered samples (where the
columns of the image grid are sequentially stacked on top of one
other to form one large column). Next, define the shape-variability matrix as
An eigenvalue decomposition is employed to factor
as
(5)
is an
matrix whose columns represent the
where
orthogonal modes of variation in the shape and
is an
diagonal matrix whose diagonal elements represent
elements of
the corresponding nonzero eigenvalues. The
the th column of , denoted by , are arranged back into
rectangular image grid (by
the structure of the
undoing the earlier lexicographical concatenation of the grid
columns) to yield , the th principal mode or eigenshape.
Based on this approach, a maximum of different eigenshapes
are generated.
Note that in most cases, the dimension of the matrix
is large
so the calculation of the eigen-
0
vectors and eigenvalues of this matrix is computationally
expensive. In practice, the eigenvectors and eigenvalues of
can be efficiently computed from a much smaller
matrix
given by
It is straightforward to show that if is an eigenvector of
with corresponding eigenvalue , then
is an eigenvector of
with eigenvalue (see [14] for a proof).
, which is selected prior to segmentation, be the
Let
number of modes to consider. Choosing the appropriate in our
model is difficult and beyond the scope of this paper. Suffice it to
say that should be chosen large enough to be able to capture
the prominent shape variations present in the training set, but
not too large that the model begins to capture intricate details
particular to a certain training shape.6 In all of our examples, we
chose empirically. We now introduce a new level set function
(6)
6One way to choose the value of k is by examining the eigenvalues of the corresponding eigenvectors. In some sense, the size of each eigenvalue indicates the
amount of influence or importance its corresponding eigenvector has in determining the shape. Perhapes by looking at a historgram of the eigenvalues, one
can determine the threshold for determining the value of k . However, this approach would be difficult to implement as the threshold value for k varies for
each application. In any case, there is no universal k that can be set.
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
where
are the weights for the eigenshapes with the variances of these weights
given by the eigenvalues calculated earlier. We propose to use
this newly constructed level set function as our implicit representation of shape. Specifically, the zero level set of describes the shape with the shape’s variability directly linked to
the variability of the level set function. Therefore, by varying
, we vary which indirectly varies the shape. Note that the
shape variability we allow in this representation is restricted to
the variability given by the eigenshapes.
Fig. 9 provides some intuition as to how the level set representation of (6) captures shape variability. The set of 12 fighter
jets shown in Fig. 2 is used as the shape training set to obtain
and
. Fig. 9(a) shows
the mean level set function with the red curve outlining the
with
zero level set of . Fig. 9(b) shows the function
the magenta curve outlining the zero crossings of this function.
Notice that most of the spatial variations associated with this
function lie in the area corresponding to the wings of the fighter
jet. Specifically, a large rising “hump” can be seen in those areas.
When this function is added to , a new level set representation of the fighter jet is obtained. This new level set function is
shown in Fig. 9(c) with the blue curve outlining the zero level
to causes the wing size to
set. As expected, adding
shrink, thus yielding a new fighter jet with a much smaller wing
with the maspan. In Fig. 9(d), we show the function
genta curve outlining the zero crossings of this function. This is
simply the negative of Fig. 9(b) and hence adding this function
to causes the wing span of the fighter jet to increase. This
resulting level set function is illustrated in Fig. 9(e) with the
blue curve outlining the zero level set. To further illustrate the
parametric shape encoding scheme of (6), we show in Fig. 10
the mean shape of the fighter jet as well as its shape varia.
tions based on varying its first three principal modes by
As another demonstration, we employ the set of training shapes
shown in Fig. 7 to obtain an implicit parametric representation
of the number four. Fig. 11 shows the mean shape of the number
four as well as its shape variations based on varying its first three
. Notice that by varying the first two
principal modes by
principal modes, the shape of the number four changes topology
going from two curves to one curve. This is an additional advantage of using the Eulerian framework for shape representation
as it can handle topological changes in a seamless fashion. This
ability is of value for biomedical applications. One such application is the tracking of changes in multiple sclerosis lesions over
time (as they shrink, migrate, split, disappear, etc.). Another is
in the segmentation of the pancreas which often presents as one
solid organ. But at times, the pancreas does not fuse in utero
and hence presents as two separate lobes which may require
segmentation algorithms that can deal with topology changes.
Another application might be in segmenting skin lesions. Some
skin pathologies can present both as one confluent lesion or as
an island of lesions.
addition, the implicit description of shape is given by the zero
level set of the following function:
B. Pose Parameters
At this point, our implicit representation of shape cannot accommodate shape variabilities due to differences in pose. To
have the flexibility of handling pose variations, is added as
another parameter to the level set function of (6). With this new
(7)
where
with
defined earlier in (1). The addition of to our parametric shape model enables us to accomodate a larger class of
objects. In particular, the model can now handle object shapes
that may differ from each other in terms of scale, orientation,
or center location. In Section IV, we describe how and are
optimized, via coordinate descent, for image segmentation.
IV. REGION-BASED MODELS FOR SEGMENTATION
In region-based segmentation models [5], [20], [22], [34],
the evolution of the segmenting curve depends upon the pixel
intensities within entire regions. That is, region-based models
regard an image as the composition of a finite number of regions and rely on regional statistics for segmentation. The statistics of entire regions (such as sample mean and variance) are
used to direct the movement of the curve toward the boundaries of the image. This is in sharp contrast to edge-based segmentation models [2], [3], [9], [12], [13], [16], [24], [25], [32]
where the evolution of the curve depends strictly on nearby
pixel intensities (i.e., gradient information). As a result, regionbased models are more global than edge-based models. Furthermore, because of the global nature of region-based models,
these models do not require the use of inflationary terms commonly employed by edge-based techniques to drive the curve toward image boundaries. Region-based models are also more robust to noise since they do not employ gradient operators, which
are inherently sensitive to noise, to explicitly detect the location of edges. In this section, we present three recently developed region-based models for segmentation and describe how
these models fit within the scope of our shape-based curve evolution framework. Specifically, in this section, we present the
Chan-Vese model, the binary mean model, and the binary variance model for image segmentation. However, instead of deriving the evolution equation for the curves used to segment the
image (which is the original design of these models), we derive
gradient descent equations used to optimize the shape and pose
parameters that indirectly describe the segmenting curve.
A. Description of the Models
We begin with a simple synthetic example to present how
region-based segmentation models are incorporated into
our model-based algorithm. Assume that the domain of the
observed image is formed by two regions distinguishable
by some region statistic (e.g., sample mean or variance). We
would like to segment this image via the curve , which in our
framework, is represented by the zero level set of , i.e.,
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Moreover, as a result of this implicit parametric representation
of , the regions inside and outside the curve, denoted, respecand
, are given by
tively, by
menting curve. The gradients of
and , are given by
143
, taken with respect to
(9a)
In our algorithmic framework, we calculate the parameters of
to vary and hence segment the image . These parameters, and , are obtained by minimizing region-based energy
functionals that are constructed using various image statistics.
, are
Some useful image statistics, written in terms of
area in
(9b)
2) The Binary Mean Model: A different strategy was proposed by Yezzi et al. in [34] to segment . They propose to
evolve so as to maximize the distance between and . A
natural cost functional they employed is to minimize the following:
(10)
area in
The authors in [34] called this the binary model (since it is
initially designed to segment images consisting of two distinct
but constant intensity regions). Once again, gradient descent is
and that minimize
employed to calculate the parameters
to implicitly determine the segmenting curve. The gra, taken with respect to and , are given by
dients of
sum intensity in
sum intensity in
sum of squared intensity in
sum of squared intensity in
(11a)
average intensity in
average intensity in
sample variance in
(11b)
sample variance in
3) The Binary Variance Model: So far, we have focused on
using the mean as the image statistic in differentiating the two
regions in . Other image statistics can also be used in a region-based segmentation model. For example, Yezzi et al. in
[34] proposed a segmentation model based on image variances.
Consider the following energy functional for segmentation:
where the Heaviside function
is given by
if
if
.
Chan and Vese in [5], and Yezzi et al. in [34] proposed pure region-based models to segment using these region statistics.
Below, we provide descriptions of their models, describe the
role of and in these models, and detail the optimization of
these models with respect to and (instead of ) for image
segmentation. As detailed in Section III, by calculating the parameters and that optimize the segmentation energy functionals, we have implicitly determined the segmenting curve .
Thus, our segmentation approach can be considered as a parameter optimization technique.
1) The Chan-Vese Model: Chan and Vese in [5] proposed
the following energy functional for segmenting :
(12)
The design of this model is to partition an image into two regions, one of low variance and one of high variance, by maximally separating the sample variances inside and outside the
, taken with respect to and
curve. The gradients of
, are given by
(13a)
(13b)
where
which is equivalent, (up to a term which does not depend upon
the evolving curve), to the energy functional below
(8)
can be viewed as a pieceThe Chan-Vese energy functional
wise constant generalization of the Mumford-Shah functional
[18]. Gradient descent is employed to search for the parameto implicitly determine the segters and that minimize
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B. Gradients of Region Statistics
seven terms: , , and translation; pitch; yaw; roll; and magnification. The shape alignment strategy is to jointly align the
binary volumetric data via gradient descent. Signed distance
function is similarily employed to represent the 3-D shapes. In
particular, the bounding surfaces of each shape is embedded as
the zero level set of a signed distance function with negative
distances assigned to the inside and positive distances assigned
to the outside of the 3-D object. The 3-D shape parameters are
derived in a similar fashion as the 2-D shape parameters. However, these 3-D shape parameters implicitly describe a 3-D segmenting surface rather than a 2-D segmenting curve. The region
statistics used in the region-based models for segmentation are
now calculated over an entire volume rather than over an entire
region.
As shown in (9), (11), and (13), to update the shape and pose
parameters via gradient descent, the gradients of region statis,
, , ,
, and
, taken with respect to and
tics
, are required. Defining the one-dimensional Dirac measure
concentrated at zero by
we can now express the th component of each of the gradient
terms in (9), (11), and (13) as line integrals along
E. Illustration of the Models Using Synthetic Data
where
with
previously defined in (4).
C. Parameter Optimization Via Gradient Descent
The gradients of the various energy functionals taken with
respect to and are given by (9), (11), and (13). For conciseand
as the gradients of any
ness of notation, denote
of the above energy functionals taken with respect to and ,
respectively. With this introduction, the update equations for the
shape and pose parameters in our gradient descent approach are
given by
where
and
are positive step-size paramters, and
and
denote the values of and at the th iteration, respectively. The updated shape and pose parameters are then used
to implicitly determine the updated location of the segmenting
curve.
It is important to note that no special numerics were required
in our proposed technique as it does not involve any partial differential equations. This results in fast and simple implementation of our methodology. In fact, this is one of the main departure
between our model and the earlier one put forth by Leventon
et al. [15]
D. Extension to Three Dimensions
The generalization of this algorithm to three dimensions is
straightforward. The pose parameter is expanded to consist of
,
, and
for
Figs. 12–14 show the use of
segmentation. We show in Fig. 12(a) a fighter jet (that is not part
of the fighter jet database of Fig. 1). Fig. 12(b) shows the same
fighter jet surrounded by horizontal and vertical line clutter. The
presence of these lines creates missing edges in the fighter jet
which can cause problems in conventional segmentation algorithms that do not rely on prior shape information. Fig. 12(c)
shows this line-cluttered fighter jet image contaminated by additive Gaussian noise. The goal is to segment the fighter jet from
this noisy test image. Knowing a priori that the object in the
image is a fighter jet, we employ the database shown in Fig. 2 to
derive an implicit parametric curve model for the fighter jet [in
. The zero level
the form of (7)]. In this example, we use
set of is employed as the starting curve which is illustrated in
Fig. 12(d). The parameters of the segmenting curve, and ,
. Fig. 12(e) shows the final shape
are calculated to minimize
and position of the segmenting curve. Notice that we are able to
successfully find the boundaries of the fighter jet without being
distracted by the line clutter. In Fig. 13, we show a slight variant
of the experiment just described. Specifically, a new fighter jet
(which is also not part of the database of Fig. 1) is employed
is employed as the
as the object in the test image, and
segmentation functional. Using the same
as before, we are able to successfully segment this new object.
Fig. 14 shows a different experiment. The object in this experiment is the number four which is shown in Fig. 14(a). Vertical and horizontal lines are again added to this image to create
missing edges in the object. The resulting line-cluttered image is
shown in Fig. 14(b). This binary mask is used to create the variance image shown in Fig. 14(c) which consists of two regions,
each of identical means but of different variances. The goal is to
segment the object from this noisy test image. Knowing a priori
that the object in the image is a handwritten four, we employ the
database of fours, shown in Fig. 7, to obtain the mean shape and
the eigenshapes for our implicit representation of the object. As
. The zero level set of is employed as the
before, we use
starting curves as illustrated in Fig. 14(d). Notice in this figure
that two curves are used to describe the starting shape. Because
the image statistic that characterizes the two regions in this test
image is variance, the parameters of the segmenting curve,
and , are calculated to minimize
. Fig. 14(e) shows the
successful segmentation of the number four image. Notice that
without any additional effort, the two starting curves merged to
form one single segmenting curve at the end.
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Fig. 10. Shape variability of the fighter jet. (a) The mean shape. (b)
variation of the first principal mode. (c) variation of the first principal mode.
(d)
variation of the second principal mode. (e) variation of the second principal mode. (f) variation of the third principal mode. (g) variation
of the third principal mode. Grossly, the first three principal modes vary the shape and size of the wings as well as the length of the fighter jets.
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Fig. 11. Shape variability of the number four. (a) The mean shape. (b)
variation of the first principal mode. (c) variation of the first principal mode.
(d)
variation of the second principal mode. (e) variation of the second principal mode. (f) variation of the third principal mode. (g) variation
of the third principal mode.
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Fig. 12. Segmentation of a noisy fighter jet with missing edges using E . (a) Original binary image. (b) Original binary image surrounded by line clutter.
(c) Image in (b) with additive Gaussian noise. (d) Blue curve shows the initializing contour. (e) Red curve shows the final contour.
Fig. 13. Segmentation of a noisy fighter jet with missing edges using E
. (a) Original binary image. (b) Original binary image surrounded by line clutter.
(c) Image in (b) with additive Gaussian noise. (d) Blue curve shows the initializing contour. (e) Red curve shows the final contour.
Fig. 14. Segmentation of a noisy number four with missing edges using E
. (a) Original binary image. (b) Original binary image surrounded by line clutter.
(c) Image in (b) with additive Gaussian noise. (d) Blue curve shows the initializing contour. (e) Red curve shows the final contour.
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Fig. 15. A conceptual representation of our algorithmic framework. The top frame summarizes the training phase of our approach (Sections II and III). The
bottom frame illustrates the segmentation phase of our algorithm (Section IV).
These figures demonstrate that our segmentation method is
robust to the presence of clutter pixel-something that can not be
said of many other segmentation algorithms. The reason for this
is that our use of a finitely parameterized shape model makes the
impact of such anomalous pixels much less significant than in
other curve evolution or other segmentation methods.
V. OUTLINE OF THE ALGORITHMIC FRAMEWORK
In this section, we provide a brief overview of our algorithmic
framework. Fig. 15 shows a block diagram to illustrate how
the different components described throughout this paper fit
within the scope of our algorithmic framework. As illustrated
in this diagram, our segmentation algorithm can be divided
into two phases—a training phase and a segmentation phase.
The training phase consists of shape alignment (described
in Section II) and parametric shape modeling (described in
Section III). Given a set of training shapes, gradient descent is
employed to minimize the alignment model of (2) to jointly
align them. Signed distance maps are generated to represent
each of the shapes in the aligned database. By applying PCA
to this collection of distance maps, we extract the mean shape
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Fig. 16. Training data: 2-D binary shape models of the left ventricle based on human interactive segmentations of different spatial and temporal slices of a
patient’s cardiac MRI.
Fig. 17.
Alignment results of the 50 2-D binary shape models of the left ventricle.
and the eigenshapes particular to this shape database. The
mean shape and the eigenshapes are used to form the implicit
parametric shape representation described in (7). The next
part of our algorithm, the segmentation phase (described in
and , the parameters of
Section IV), involves calculating
our implicit shape representation, to minimize a segmentation
functional. This minimization is performed as an iterative
process using gradient descent. At each gradient step, and
are updated to generate a new level set
. The segmenting
curve is implicitly determined by this new level set. Based
on the new position and shape of , we recalculate the image
statistic inside and outside the curve. These newly computed
statistics are used in the segmentation functional to determine
the update rules for and . We continue this iterative scheme
until convergence is reached for segmentation.
VI. APPLICATIONS TO MEDICAL IMAGERY
We now apply the model-based curve evolution technique derived in this paper to two medical applications. Section VI-A
illustrates a 2-D example (cardiac MRI segmentation), while
Section VI-B illustrates a 3-D example (prostate gland segmentation from pelvic MRI).
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B. A 3-D Example: Prostate Segmentation of Pelvic MRI
Taken With Endorectal Coil
(a)
(b)
Fig. 18. Comparison of the amount of shape overlap in the cardiac database
(a) before alignment and (b) after alignment.
A. A 2-D Example: Left Ventricle Segmentation of Cardiac
MRI
Cardiac MRI is an important clinical tool used to provide
four–dimensional (4-D) (temporal as well as spatial) information about the heart. Typically, one study generates 80–120 2-D
images of a patient’s heart. In a variety of clinical scenarios
(such as assessing cardiac function and diagnosing cardiac diseases), it is important to extract the boundaries of the left ventricle from this data set. For example, the segmentation of the
left ventricle is a prerequisite in calculating important physiological parameters such as ejection fraction and stroke volume.
Manual tracing of the left ventricle from such a large data set is
both tedious and time-consuming. A robust automated segmentation algorithm of the left ventricle would be preferred.
Conventional automated segmentation techniques usually
encounter difficulties in segmenting the left ventricle because
1) the intensity contrast between the ventricle and the myocardium is low (due to the smearing of the blood pool in the
ventricle into the myocardium), and 2) the boundaries of the
left ventricle are missing at certain locations due to the presence
of protruding papillary muscles which have the same intensity
profile as the myocardium.
In the experiment to illustrate our technique, we equally divided the 100 2-D images from a single patient’s cardiac MRI
into two sets: a training set and a test set. Fifty 4-D interactive
segmentations of the left ventricle from the training set form the
2-D shape database shown in Fig. 16. This particular database
is employed to allow our model to capture both the spatial and
the temporal variations of the left ventricle. Fig. 17 shows the
aligned version of this database. Fig. 18 compares the overlap
images of the left ventricle database before and after alignment.
Using the aligned database, we derived the mean level set and
the eigenshapes to form the implicit shape model of the left ven. Fig. 19 shows the mean shape of the left
tricle using
ventricle as well as its shape variations by varying the first three
. The parameters of this implicit parametric
eigenshapes by
using statistics
representation are calculated to minimize
calculated in the entire region both inside and outside the curve.
Fig. 20 shows the segmentation result of the testing set by our algorithm (red curves). These results are comparable with the ones
given by a 4-D interactive cardiac MRI segmenter [33] (green
curves) which utilizes a 4-D conformal surface shrinking technqiue based upon the models outlined in [32].
Pelvic MRI, when taken in conjunction with an endorectal
coil (ERC) (a receive-only surface coil placed within the
rectum) using T1 and T2 weighting, provides high-resolution
images of the prostate with smaller field of view and thinner
slice thickness than previously attainable. Because of the
high-quality anatomical images obtainable by this technique,
it may become the imaging modality of choice in the future
for detection and staging of prostate cancer [7], [31]. For
assignment of appropriate radiation therapy after cancer
detection, the segmentation of the prostate gland from these
pelvic MRI images is required. Manual outlining of sequential
cross-sectional slices of the prostate images is currently used to
identify the prostate gland and its substructures, but this process
is difficult, time-consuming, and tedious. The idea of being
able to automatically segment the prostate is very attractive.
Automatic segmentation of the prostate is difficult because
the prostate is a small glandular structure buried deep within
the pelvic region and surrounded by a variety of different tissues which show up as varying intensity levels on the MRI.
This segmentation problem is further complicated by an artifact called the near-field effect which is caused by the use of
the ERC. The near-field effect causes an intensity artifact to appear in the tissues surrounding the ERC. This can be seen as a
white circular halo surrounding the rectum in each image slice
of Figs. 27 and 30. The intensity artifact can bleach out the borders of the prostate near the rectum, making the prostate segmentation problem even more difficult.
We employ a 3-D version of our shape-based curve evolution
technique to segment the prostate gland. By utilizing a surface
(instead of a curve), the segmentation algorithm is able to utilize
the full 3-D spatial information to extract the boundaries of the
prostate gland. Fig. 21 shows the prostate training data we use
which consists of eight 3-D binary shape models of the prostate
gland-obtained by stacking together 2-D expert hand segmentations of eight patients’ pelvic MRIs taken with an ERC. The
alignment results of these 3-D models are shown in Fig. 21. To
evaluate the alignment process, Fig. 23 shows 12 consecutive
axial slice overlap images of the eight 3-D prostate gland models
prior to alignment. And Fig. 24 shows the same 12 overlap images after alignment for comparison. Prior to shape training,
these 3-D shape models are smoothed to remove the “step-like”
artifact along the axial direction of the prostate. Based on these
3-D models, we derived the mean level set and the eigenshapes
to form the implicit shape model of the prostate gland using
. Fig. 25 shows the mean shape of the prostate gland as
well as its shape variations based on varying the first three eigen.
shapes by
In this particular application, it is important to realize that
despite the fact that the prostate gland is mostly deformed by
its neighboring structures, the prostate shape parameters are
still very important in describing its shape. In our method, by
capturing how its surrounding structures deform the prostate
gland, we obtain shape parameters that can effectively describe
the deformations of the prostate gland. Specifically, we looked
at a population of patients and learned the total net resultant
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Fig. 19. Shape variability of the left ventricle. (a) The mean shape. (b)
variation of the first principal mode. (c) variation of the first principal mode.
(d)
variation of the second principal mode. (e) variation of the second principal mode. (f) variation of the third principal mode. (g) variation
of the third principal mode.
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Fig. 20. Left ventricle segmentation of cardiac MRI. The segmentation by our algorithm (red curves) is compared to the segmentation by an interactive 4-D
cardiac MRI segmenter (green curves).
Fig. 21.
Training data: eight 3-D shape models of the prostate gland obtained based on axially stacking together 2-D expert hand segmentations of the prostate.
effect of the surrounding structures in deforming the prostate
gland, and incorporated this information within the prostate
shape parameters. Thus, instead of looking at how the prostate
gland deforms in a vacuum by itself, we have taken into ac-
count how the prostate deforms in vivo by the surrounding
structures.
To accentuate the boundaries of the prostate gland as well as
to minimize the intensity artifact caused by the ERC, the pelvic
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
Fig. 22.
Alignment results of the eight 3-D shape models of the prostate gland.
Fig. 23.
Overlap images of consecutive axial slices of the eight 3-D prostate models prior to alignment.
MRI data set
is transformed to a bimodal data set
applying the following map:
by
where
here denotes a 3-D gradient operator. This mapping
was employed because: 1) the interior of the prostate is homogeneous in intensity, so with this mapping, the interior regions
of the prostate are mapped to low values while the boundaries
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TSAI et al.: A SHAPE-BASED APPROACH TO THE SEGMENTATION OF MEDICAL IMAGERY USING LEVEL SETS
Fig. 24.
151
Overlap images of consecutive axial slices of the eight 3-D prostate models after alignment.
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Fig. 25. Shape variability of the prostate. (a) The mean shape. (b)
variation of the first principal mode. (c) variation of the first principal mode. (d) variation of the second principal mode. (e)
variation of the second principal mode. (f) variation of the third principal mode. (g) variation of the
third principal mode.
01
of the prostate are mapped to high values; and 2) this mapping
is robust to the smooth spatially varying intensity artifact cause
by the ERC. We segment the prostate gland by minimizing
using the transformed data set . The statistics used in
are
calculated in the entire volumetric data both inside and outside
was emthe segmenting surface. The energy functional
ployed in this application because we found it to be more robust
empirically. We start by initializing the segmenting surface to
be within the interior of the prostate gland so that the evolving
surface does not get distracted by various other high gradient
features surrounding the prostate (such as interfaces between
various hard and soft tissue types). With each iteration, the segmenting surface moves outward to capture more and more of the
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low-valued region in the transformed data (which corresponds
to the prostate gland). Eventually, the segmenting surface converges to a local minimum near the boundaries of the prostate
(corresponding to high values in the transformed data).
Twelve contiguous axial slices of patient A’s and B’s MRI
data set containing the prostate gland are displayed in Figs. 26
and 29, respectively. These two data sets are not part of the
training database of Fig. 21. We show in Figs. 27 and 30 the
prostate segmentation results of patient A’s and B’s MRI data
set, respectively. In each of these figures, the MRI data set containing the prostate gland are displayed along with the segmentation by our algorithm (outlined in red), and the segmentation
by a radiologist from Brigham and Women’s Hospital (outlined
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Fig. 26. Prostate images of patient A. These images represent consecutive axial slices of the prostate. Segmenting curves were not superimposed on the images
for better visualization of the prostate organ.
Fig. 27.
Prostate segmentation of patient A. The segmentation by the radiologist (green curves) is compared to the segmentation by our algorithm (red curves).
in green). Another radiologist, also from Brigham and Women’s
Hospital, rated the first radiologist’s segmentation of data set A
to be slightly better than our algorithm’s, and rated our algorithm’s segmentation of data set B to be slightly better than the
radiologist’s. For visual comparison, Figs. 28 and 31 show the
3-D models of the prostate gland generated by our algorithm
and by stacking together 2-D expert hand segmentations. Notice that by employing a surface to capture the prostate gland,
our 3-D model does not display any of the “step-like” artifacts
that mar the radiologist’s 3-D rendition of the prostate gland. In
addition, working in 3-D space allows our algorithm to utilize
the full 3-D structural information of the prostate for segmentation (instead of just the information from neighboring slices
which are typically used by the radiologists).
VII. CONCLUSION AND FUTURE RESEARCH DIRECTIONS
We have outlined a statistically robust and computationally
efficient model-based segmentation algorithm using an implicit
representation of the segmenting curve. Because this implicit
representation is set in an Eulerian framework, it does not require point correspondences during the training phase of the algorithm and can be used to handle topological changes of the
(a)
(b)
Fig. 28. Three-dimensional models of patient A’s prostate gland. (a) Based on
our segmentation algorithm. (b) Based on the radiologist’s segmentation.
segmenting curve in a seamless fashion. This algorithmic framework is capable of segmenting images contaminated by heavy
noise and delineate structures complicated by missing or diffuse
edges. In addition, this framework is flexible, both in terms of
its ability to model and segment complicated shapes (as long
as the shape variations are consistent with the training data), as
well as its ability to accommodate the segmentation of multidimensional data sets. Furthermore, by employing a region-based
segmentation functional, our algorithm is more global, exhibits
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Fig. 29. Prostate images of patient B. These images represent consecutive axial slices of the prostate. Segmenting curves were not superimposed on the images
for better visualization of the prostate organ.
Fig. 30.
Prostate segmentation of patient B. The segmentation by the radiologist (green curves) is compared to the segmentation by our algorithm (red curves).
(a)
(b)
Fig. 31. Three–dimensional models of patient B’s prostate gland. (a) Based
on our segmentation algorithm. (b) Based on the radiologist’s segmentation.
our method by constructing different segmentation functionals
based on first (and maybe higher) order statistics such as skewness, kurtosis, and entropy.
In this paper, we discussed the use of signed distance functions as a way to represent shapes. However, because distance
functions are not closed under linear operations, the level set
representation of our segmenting curve, based on the PCA approach described in Section III, is not a distance function. This
gives rise to an inconsistent framework for shape modeling. This
intellectual issue remains an important and challenging problem
(indeed one on which we are now working ourselves), but the
method developed in this paper stands on its performance in
practice.
ACKNOWLEDGMENT
increased robustness to noise, displays extensive capture range,
and is less sensitive to initial contour placements compared with
other model-based segmentation algorithms.
The performance of our model-based curve evolution technique depends largely upon how well the chosen set of statistics
is able to distinguish the various regions within a given image.
In this paper, we detailed the use of means and variances as
the discriminating statistics. However, this approach may be applied to any computed statistics. We are interested in extending
The authors would like to thank the anonymous reviewers for
their valuable comments and thoughtful suggestions.
REFERENCES
[1] G. Borgefors, “Distance transformations in digital images,” CVGIP:
Image Understanding, vol. 34, pp. 344–371, 1986.
[2] V. Caselles, F. Catte, T. Coll, and F. Dibos, “A geometric model for
active contours in image processing,” Numerische Mathematik, vol. 66,
pp. 1–31, 1993.
Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on October 6, 2008 at 19:7 from IEEE Xplore. Restrictions apply.
154
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 2, FEBRUARY 2003
[3] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic snakes,” Int. J.
Comput. Vis., 1998.
[4] A. Chakraborty, L. Staib, and J. Duncan, “An integrated approach to
boundary finding in medical images,” in Proc. IEEE Workshop Biomedical Image Analysis, 1994, pp. 13–22.
[5] T. Chan and L. Vese, “Active contours without edges,” IEEE Trans.
Image Processing, vol. 10, pp. 266–277, Feb. 2001.
[6] Y. Chen, S. Thiruenkadam, H. Tagare, F. Huang, D. Wilson, and E.
Geiser, “On the incorporation of shape priors into geometric active contours,” in Proc. IEEE Workshop Variational and Level Set Methods,
2001, pp. 145–152.
[7] D. Cheng and C. Tempany, “MR imaging of the prostate and bladder,”
Seminars Ultrasound, CT, MRI, vol. 19, no. 1, pp. 67–89, 1998.
[8] G. Christensen et al., “Consistent linear-elastic transformation for image
matching,” in Lecture Notes in Computer Science, A. Kuba et al., Eds.,
1999, vol. 1613, Information Processing in Medical Imaging (Proc. 16th
Int. Conf.), pp. 224–237.
[9] L. Cohen, “On active contour models and ballooms,” CVGIP: Image
Understanding, vol. 53, pp. 211–218, 1991.
[10] T. Cootes, C. Taylor, D. Cooper, and J. Graham, “Active shape modelstheir training and application,” Comput. Vis. Image Understanding, vol.
61, pp. 38–59, 1995.
[11] B. Frey and N. Jojic, “Estimating mixture models of images and
inferring spatial transformations using the EM algorithm,” in Proc.
IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, 1999, pp.
416–422.
[12] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour
models,” Int. J. Comput. Vis., vol. 1, pp. 321–331, 1987.
[13] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi,
“Conformal curvature flows: From phase transitions to active vision,”
Arch. Rational Mech. Anal., vol. 134, pp. 275–301, 1996.
[14] M. Leveton, “Statistical models in medical image analysis,” Ph.D. dissertation, Massachusetts Inst. Technol, Dept. Elect. Eng., 2000.
[15] M. Leventon, E. Grimson, and O. Faugeras, “Statistical shape influence
in geodesic active contours,” in Proc. IEEE Conf. Computer Vision and
Pattern Recognition, vol. 1, 2000, pp. 316–323.
[16] R. Malladi, J. Sethian, and B. Vemuri, “Shape modeling with front propagation: A level set approach,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 158–175, Feb. 1995.
[17] E. Miller, N. Matsakis, and P. Viola, “Learning from one example
through shared densities on transforms,” in Proc. IEEE Conf. Computer
Vision and Pattern Recognition, vol. 1, 2000, pp. 464–471.
[18] D. Mumford and J. Shah, “Optimal approximations by piecewise
smooth functions and associated variational problems,” Comm. Pure
Appl. Math., vol. 42, pp. 577–685, 1989.
[19] S. Osher and J. Sethian, “Fronts propagation with curvature dependent
speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comput.
Phys., vol. 79, pp. 12–49, 1988.
[20] N. Paragios and R. Deriche, “Geodesic Active Regions for Texture Segmentation,” INRIA, Sophia Antipolis, France, Res. Rep. 3440, 1998.
[21] A. Pentland and S. Sclaroff, “Closed-form solutions for physically based
shape modeling and recognition,” IEEE Trans. Pattern Anal. Machine
Intell., vol. 13, pp. 715–729, July 1991.
[22] R. Ronfard, “Region-based strategies for active contour models,” Int. J.
Comput. Vis., vol. 13, pp. 229–251, 1994.
[23] L. Staib and J. Duncan, “Boundary finding with parametrically deformable contour models,” IEEE Trans. Pattern Anal. Machine Intell.,
vol. 14, pp. 1061–1075, Nov. 1992.
[24] H. Tek and B. Kimia, “Image segmentation by reaction diffusion bubbles,” in Proc. Int. Conf. Computer Vision, 1995, pp. 156–162.
[25] D. Terzopoulos and A. Witkin, “Constraints on deformable models: Recovering shape and nonrigid motion,” Artif. Intell., vol. 36, pp. 91–123,
1988.
[26] A. Tsai, A. Yezzi, W. Wells, C. Tempany, D. Tucker, A. Fan, E. Grimson,
and A. Willsky, “Model-based curve evolution technique for image segmentation,” in IEEE Conf. Computer Vision and Pattern Recognition,
vol. 1, 2001, pp. 463–468.
[27] A. Tsai, A. Yezzi, and A. Willsky, “A curve evolution approach to
smoothing and segmentation using the mumford-shah functional,” in
IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, 2000, pp.
1119–1124.
[28] T. Vetter, M. Jones, and T. Poggio, “A bootstrapping algorithm for
learning linear models of object classes,” in IEEE Conf. Computer
Vision and Pattern Recognition, vol. 1, 1997, pp. 40–46.
[29] P. Viola and W. Wells, “Mutual information: An approach for the registration of object models and images,” Int. J. Comput. Vis., 1997.
[30] Y. Wang and L. Staib, “Boundary finding with correspondence using
statistical shape models,” in IEEE Conf. Computer Vision and Pattern
Recognition, 1998, pp. 338–345.
[31] T. Wong, G. Silverman, J. Fielding, C. Tempany, K. Hynynen, and F.
Jolesz, “Open-configuration MR imaging, intervention, and surgery of
the urinary tract,” Urologic Clin. No. Amer., vol. 25, pp. 113–122, 1998.
[32] A. Yezzi, S. Kichenassamy, A. Kumar, P. Olver, and A. Tannenbaum,
“A geometric snake model for segmentation of medical imagery,” IEEE
Trans. Med. Imag., vol. 16, pp. 199–209, Apr. 1997.
[33] A. Yezzi and A. Tannenbaum, “4D active surfaces for cardiac segmentation,” Med. Image Computing Comput. Assist. Intervention, pp.
667–673, 2002, submitted for publication.
[34] A. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for
bimodal and trimodal imagery,” in Proc. Int. Conf. Computer Vision, vol.
2, 1999, pp. 898–903.
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