Data Medical
Resume:
Int J Comput Vis (****) **: *** ***
A Statistical Overlap Prior for Variational Image Segmentation
Ismail Ben Ayed Shuo Li Ian Ross
Received: 30 September 2008 / Accepted: 4 May 2009 / Published online: 13 May 2009
Springer Science+Business Media, LLC 2009
Keywords Image segmentation Distribution metrics
Abstract This study investigates variational image segmen-
Variational methods Active contours Level sets
tation with an original data term, referred to as statistical
Segmentation priors
overlap prior, which measures the conformity of overlap be-
tween the nonparametric distributions of image data within
the segmentation regions to a learned statistical description.
1 Introduction
This leads to image segmentation and distribution track-
ing algorithms that relax the assumption of minimal over-
Image segmentation is a fundamental problem in computer
lap and, as such, are more widely applicable than existing vision and image processing. It serves many important ap-
algorithms. We propose to minimize active curve function- plications such as scene interpretation (Mitiche and Sekkati
als containing the proposed overlap prior, compute the cor- 2006), motion analysis (Vazquez et al. 2006), remote sens-
responding Euler-Lagrange curve evolution equations, and ing (Ben Ayed et al. 2005), content-based image retrieval
give an interpretation of how the overlap prior controls such (Idriss and Panchanathan 1997), medical image analysis
evolution. We model the overlap, measured via the Bhat- (Holtzman-Gazit et al. 2006), and many others.
Following the seminal work of Mumford and Shah
tacharyya coef cient, with a Gaussian prior whose parame-
(1989), variational formulations, which state image segmen-
ters are estimated from a set of relevant training images.
tation as the minimization of a functional, have been the fo-
Quantitative and comparative performance evaluations of
cus of an impressive number of theoretical, methodological,
the proposed algorithms over several experiments demon-
and practical studies (Tai et al. 2007; Ben Ayed and Mitiche
strate the positive effects of the overlap prior in regard to
2008; Aubert and Kornprobst 2006; Samson et al. 2000;
segmentation accuracy and convergence speed.
Martin et al. 2004; Chan and Vese 2001; Vese and Chan
2002; Morel and Solimini 1995; Mumford and Shah 1989;
Cremers et al. 2007; Cremers and Soatto 2005; Gao and
Bui 2005; Holtzman-Gazit et al. 2006; Boykov and Funka-
This study is supported in part by the Natural Sciences and
Lea 2006; Rother et al. 2004; Vicente et al. 2008; Malcolm
Engineering Research Council of Canada (NSERC), under the
Industrial Research Fellowship granted to Ismail Ben Ayed. et al. 2007; Chang et al. 2007; Riklin-Raviv et al. 2008;
S gonne 2008; Rousson and Paragios 2008; Ben Ayed et
I. Ben Ayed S. Li
al. 2006a, 2006b, 2008a, 2008b; Jehan-Besson et al. 2003;
GE Healthcare, 268 Grosvenor, E5-137 London, ON, N6A 4A2,
Canada Mansouri et al. 2006; Zhu and Yuille 1996; Paragios and
e-mail: ******.*******@**.*** Deriche 2000, 2002; Rousson and Cremers 2005; Kadir and
S. Li Brady 2003; Aubert et al. 2003; Freedman and Zhang 2004;
e-mail: ****.**@**.*** Zhang and Freedman 2005; Michailovich et al. 2007;
Georgiou et al. 2007; Kim et al. 2005; Mory et al. 2007;
I. Ross
Awate et al. 2006; Huang and Metaxas 2008). In gen-
University Hospital, London Health Sciences Centre,
eral, these formulations are level set curve evolution vari-
339 Windermere Road, London, ON, N6A 5A5, Canada
116 Int J Comput Vis (2009) 85: 115 132
ants of the Mumford-Shah functional (Tai et al. 2007; data within each region to the piecewise constant model, and
regularization terms related to curve length and region area
Ben Ayed and Mitiche 2008; Aubert and Kornprobst 2006;
Samson et al. 2000; Martin et al. 2004; Chan and Vese
FChan-Vese (, cin, cout )
2001; Vese and Chan 2002; Riklin-Raviv et al. 2008;
S gonne 2008; Rousson and Paragios 2008; Ben Ayed et
= 1 (Ix cin )2 d x + 2 (Ix cout )2 d x
al. 2006a, 2006b, 2008a, 2008b; Jehan-Besson et al. 2003; Rin Rout
Mansouri et al. 2006; Zhu and Yuille 1996; Paragios and
+ ds + d x. (1)
Deriche 2000, 2002; Rousson and Cremers 2005; Kadir and
Rin
Brady 2003; Aubert et al. 2003; Freedman and Zhang 2004;
Zhang and Freedman 2005; Michailovich et al. 2007; The solution is obtained by alternating minimization with
Georgiou et al. 2007; Kim et al. 2005; Mory et al. 2007; respect to and to region parameters, cin and cout, which
Awate et al. 2006; Huang and Metaxas 2008), where the come out to be the means of image data within, respec-
tively, Rin and Rout . 1, 2,, and are positive constants to
solution is given by the evolution equation of an active con-
weigh the contribution of each term. Albeit applicable only
tour. This allows a convenient representation of segmenta-
to segmentation regions where image data is approximately
tion regions the interior (foreground) and exterior (back-
constant, the Chan-Vese model has established the poten-
ground) of the curve at convergence and their boundaries.
tial of region-based active contours, and has been used in a
Also, the implicit level set representation of curve evolution
considerable number of studies and applications (Cremers
(Sethian 1999) handles automatically arbitrary variations in
et al. 2007; Cremers and Soatto 2005; Gao and Bui 2005;
region topology, and yields a numerically stable representa-
Holtzman-Gazit et al. 2006). More generally, most of seg-
tion of the region membership and boundary, which removes
mentation algorithms are stated as a Bayesian inference
the need of complex data structures.
problem (Cremers et al. 2007; Boykov and Funka-Lea 2006;
The level set active contour framework has resulted in the
Rother et al. 2004; Vicente et al. 2008; Malcolm et al. 2007;
most effective, principled and exible algorithms, mainly
Ben Ayed et al. 2006a, 2006b; Jehan-Besson et al. 2003;
because it is amenable to the introduction of various types
Mansouri et al. 2006; Zhu and Yuille 1996; Paragios and
of constraints in the segmentation functional (Cremers et al.
Deriche 2000, 2002; Rousson and Cremers 2005), where op-
2007). Typical constraints are smoothness of region bound-
timization of the data term amounts at nding the partition
aries, and more importantly, data terms, which measure the
{Rin, Rout } that minimizes minus the image log-likelihood1
conformity of image data within each region to a given sta-
tistical description. Current data terms are based on the fol-
FLikelihood = 1 log P(Ix /Rin )d x
lowing implicit assumption: The overlap between the dis-
Rin
tributions of image data within the object and its back-
2
ground has to be minimal. Such assumption may not be valid log P(Ix /Rout )d x. (2)
Rout
in many important applications. This study investigates an
original data term that embeds statistical information about This corresponds to maximizing the conditional probabil-
the overlap between the distributions within the object and ity of pixel data given the assumed model distributions
the background in active curve segmentation. In the follow- P(I /Rin ) and P(I /Rout ). The way of estimating the model
ing, we discuss current data terms and summarize the con- distributions within the object and its background divides
tributions of this study. segmentation methods into two categories: unsupervised
methods, where the model distributions are estimated from
1.1 The Piecewise Constant Chan-Vese Model and Its the current image along with the segmentation process (Ben
Ayed et al. 2006a, 2006b; Jehan-Besson et al. 2003; Man-
Bayesian Generalizations
souri et al. 2006; Zhu and Yuille 1996; Paragios and De-
riche 2000, 2002; Rousson and Cremers 2005; Kadir and
Let Ix = I (x) : R2 Z be an image function from the
Brady 2003), and methods using data priors, where the
domain to the space Z of a photometric variable such as
model distributions P(I /Rin ) and P(I /Rout ) are learned a
intensity or a color vector. Level set segmentation consists
priori from a set of pre-segmented training images (Samson
of evolving a curve to divide into two regions: Rin,
et al. 2000; Rousson and Cremers 2005), or interactively
corresponding to the interior of (foreground), and Rout,
from user-speci ed regions (Boykov and Funka-Lea 2006;
corresponding to the exterior of (background). The curve
evolution equation is sought following the optimization of a
1 Note
functional. Chan and Vese (2001) pioneered an active con- that the Chan-Vese data term corresponds to a particular case
of the Gaussian image model: P(I /Rin ) = exp (Ix cin ), P(I /Rout ) =
2
tour approach of the Mumford-Shah functional by minimiz-
exp (Ix cout ) .
2
ing a data term, which measures the conformity of image
Int J Comput Vis (2009) 85: 115 132 117
Rother et al. 2004; Vicente et al. 2008; Malcolm et al. 2007). and outside the curve. In Aubert et al. (2003), Freedman
and Zhang (2004), curve evolution distribution tracking is
Embedding likelihood-based data priors in image segmen-
sought by identifying in each frame the region whose sam-
tation has signi cantly improved the performances of unsu-
ple distribution most closely matches a model distribution.
pervised methods (Cremers et al. 2007). It has led to promis-
This corresponds to maximizing a functional which mea-
ing results in natural (Rother et al. 2004), medical (Rous-
sures the similarity between the sample distribution inside
son and Cremers 2005) and texture (Paragios and Deriche
the curve and a model distribution of the object. Similar to
2002) image segmentation, as well as object tracking (Para-
likelihood-based methods using data priors, the model dis-
gios and Deriche 2000). For instance, segmenting a class of
tribution of the object is learned beforehand from a previ-
images with similar photometric patterns occurs in impor-
ously segmented frame. The performance of these distrib-
tant applications such as medical image analysis (Rousson
ution tracking methods based on foreground matching was
and Cremers 2005). In this case, learning model distribu-
improved in Zhang and Freedman (2005) by adding a back-
tions from segmented training images is very useful. Also,
ground mismatching term to the maximized functional. The
interactive foreground/background segmentation, of great
latter aims to maximize the discrepancy between the sam-
practical importance in image editing (Rother et al. 2004),
ple distribution of the background and the model distrib-
and which received a considerable research attention in re-
ution of the object. It has been demonstrated experimen-
cent years (Boykov and Funka-Lea 2006; Rother et al. 2004;
tally (Zhang and Freedman 2005) that curve evolution based
Vicente et al. 2008; Malcolm et al. 2007), uses models dis-
on the Bhattacharyya measure outperforms the likelihood
tributions learned from user-speci ed regions. Data priors
principle when dealing with cluttered backgrounds. Further-
have also enhanced object tracking algorithms, where model
more, it is much less sensitive to inaccuracies in estimating
distributions can be learned from a previously segmented
model distributions (Michailovich et al. 2007).
frame (Paragios and Deriche 2000).
Unfortunately, segmentation based on likelihood-based
1.3 Contributions of This Study
data priors is sensitive to inaccuracies in estimating model
distributions (Michailovich et al. 2007). More importantly, it
Whether using the likelihood principle or distribution met-
can not incorporate information about the overlap between
rics, existing data terms are based on the following implicit
the distributions of image data within the object and the
assumption: The discrepancy between the distributions of
background. Based on the evaluation of a pixelwise corre-
image data within the object and its background has to be
spondence between the image and the models, likelihood-
maximal. Such assumption may not be valid in many impor-
based data priors do not take full advantage of the statisti- tant applications. Although current methods have been ef-
cal information available in the learned distributions and, as fective in some cases, they are not versatile enough to deal
such, the ensuing algorithms assume implicitly that overlap with situations in which a signi cant (cf. the typical exam-
between the distributions of image data within the object and ple in Fig. 1) overlap exists between data distributions within
the background has to be minimal. Unfortunately, this mini- the object and the background. The typical example in Fig. 1
mal overlap assumption may not be valid in many important shows how both likelihood and distribution matching priors
applications (cf. the typical example in Fig. 1). Embedding fail in segmenting the object, the left ventricle cavity in this
overlap information in variational image segmentation is the case, and the background because of the signi cant overlap
main focus of the current study. between their intensity distributions (refer to Fig. 1(b)). As
we will demonstrate in this study, embedding information
1.2 Segmentation with Distribution Metrics about such overlap is important, and would be very useful in
many important applications.
Recent studies have shown the advantages and effectiveness This study investigates an original data term, referred to
of using discrepancy measures between distributions in im- as statistical overlap prior, which measures the conformity
age segmentation (Michailovich et al. 2007; Georgiou et al. of overlap between the nonparametric (kernel-based) distri-
2007) and distribution tracking in image sequences (Aubert butions within the object and the background to a learned
et al. 2003; Freedman and Zhang 2004; Zhang and Freed- statistical description. This leads to image segmentation and
man 2005). Possible measures include the Bhattacharyya distribution tracking algorithms that relax the assumption of
coef cient (Zhang and Freedman 2005; Michailovich et al. minimal overlap. The proposed prior can be viewed as a gen-
2007) and the mutual information (Aubert et al. 2003). How- eralization of existing data terms for situations in which an
ever, the Bhattacharyya coef cient has shown superior per- overlap exists between the distributions within different re-
formances over other criteria (Zhang and Freedman 2005; gions. As such, the ensuing algorithms are more widely ap-
Michailovich et al. 2007). In Michailovich et al. (2007), plicable than existing algorithms.
level set segmentation is stated as maximizing the discrep- For image segmentation, we propose to minimize an ac-
tive curve functional containing the proposed overlap prior
ancy between image data distributions sampled from inside
118 Int J Comput Vis (2009) 85: 115 132
Fig. 1 (Color online) A typical example: (a) expected delineation (f) obtained object); fourth column: segmentation result obtained with a
(manually determined) of the left ventricle cavity (red curve); (b) over- distribution matching data term (Zhang and Freedman 2005) ((g) curve
lap between the distributions of the cavity and the nearby background at convergence, (h) obtained object); fth column: result obtained with
(region inside the blue curve in (a)); (c) curve initialization; (d) ground the proposed overlap prior ((i) curve at convergence, (j) obtained ob-
truth: manually segmented cavity; third column: segmentation result ject)
obtained with a likelihood-based data term ((e) curve at convergence,
and classic regularization terms. We compute the corre- corresponding to the interior of (foreground), and
sponding Euler-Lagrange curve evolution equation, and give
Rout = Rc = Rc,
an interpretation of how the overlap prior controls such in
evolution. We model the overlap, measured via the Bhat- corresponding to the exterior of (background). The evo-
tacharyya coef cient, with a Gaussian prior whose parame- lution equation of is sought by optimizing a statistical
ters are estimated from a set of relevant training images. overlap prior. To introduce such prior, we rst consider the
We also describe a quantitative and comparative perfor- following de nitions:
mance evaluation of the proposed method over a represen-
Pout is the nonparametric (kernel-based) estimate of the
tative number of experiments on various medical images.
distribution of image data outside
Evaluation of the proposed prior is supported by compar-
isons with the likelihood prior as well as the Chan-Vese K(z Ix )d x
Rout
z Z Pout (z) =
model. For distribution tracking in image sequences, we (3),
Aout
investigate the minimization of an active curve functional
containing an original overlap prior, a foreground match- where Aout is the area of region Rout
ing term, and a regularization term. In this case, we use a
Aout =
single pre-segmented frame for training. After computing d x. (4)
Rout
the corresponding curve evolution equation, we describe a
representative sample of the results, as well as comparisons Typical choices of K are the Dirac function and the
with related distribution tracking methods. The experiments Gaussian kernel (Georgiou et al. 2007)
demonstrate the positive effects of the overlap prior in regard
2
1 y
to segmentation accuracy and convergence speed. K(y) = exp 2h2 . (5)
2 h2
B (f/g) is the Bhattacharyya coef cient measuring the
2 The Proposed Statistical Overlap Prior
amount of overlap between two statistical samples f
and g
Let Ix = I (x) : R2 Z be an image function from the
domain to the space Z of a photometric variable such as B (f/g) = (6)
f (z)g(z).
intensity or a color vector. Let (s) : [0, 1] be a closed
z Z
planar parametric curve. Our purpose is to evolve in order
Note that the values of B are always in [0, 1], where 0
to divide into two regions:
indicates that there is no overlap, and 1 indicates a perfect
Rin = R, match.
Int J Comput Vis (2009) 85: 115 132 119
We assume that the foreground region, i.e., the target ob- ensuing curve evolution ow and give an interpretation of
ject to be delineated, is characterized by a model distribu- how the overlap prior controls such evolution. Then, we de-
tion, Min, which can be learned over a set of training images scribe a quantitative and comparative performance evalua-
and segmentation examples. Consider the following mea- tion over a representative number of experiments on vari-
sure of overlap between the sample distribution outside the ous medical images. In the following, evaluation of the seg-
curve (background) and the model distribution of the object mentation method we propose, referred to as Overlap Prior
(foreground) Method (OPM), is supported by comparisons with a Like-
lihood Prior Method (LPM). Note that the likelihood prior
O(, Min ) = B (Pout /Min ) = Pout (z)Min (z). (7) has been commonly used in image segmentation as a data
z Z term.
In Sect. 5, we devise the proposed overlap prior for distri-
In order to incorporate prior information about the photo-
bution tracking in image sequences. For such purpose, is
metric similarities between the object and the background,
estimated from a given segmentation of a single frame, the
we assume that O(, Min ) is a random variable following
rst frame in the sequence for instance, and is set equal
a Gaussian distribution
to 1 . We propose to minimize an active curve functional
2
(O (,Min ) )2
1 containing an original overlap prior, a foreground match-
G (O(, Min ),, ) = 2 2
exp (8)
.
ing term which measures the similarity between the sam-
2 2
ple distribution inside the curve (foreground) and the model
Parameters and are learned beforehand over a set of
distribution of the target object, and a regularization term
relevant training images and segmentation examples. Para-
for smooth segmentation boundaries. After computing the
metric distributions other than the Gaussian distribution can
corresponding curve evolution equation, we describe a rep-
be employed to model the overlap prior. This would change
resentative sample of the results along with comparisons
the nal curve evolution equation, but would not change
with related distribution tracking methods, which demon-
the method conceptually. We will show in the experiments
strate clearly the positive effect of the overlap prior. We
that the Gaussian assumption is suf cient in many practical
also demonstrate experimentally that the overlap prior has
cases such as in medical image analysis. For instance, the
a computational advantage: it speeds up signi cantly curve
overlap between the intensity distributions within the heart
evolution.
myocardium and the background in cardiac Magnetic Res-
onance Images (MRI) can be modeled accurately with the
Gaussian distribution (refer to Fig. 5(b)).
3 Image Segmentation with Statistical Overlap Priors
We propose to minimize with respect to a statistical
overlap prior which measures the conformity of O(, Min )
3.1 Segmentation Functional
to a Gaussian model described by and
F (O(, Min ),, ) = log G (O(, Min ),, ). In this section, we propose an active curve segmentation
(9)
functional containing an original statistical overlap prior and
The statistical overlap prior can be viewed as a generaliza- classic regularization terms (Chan and Vese 2001), namely,
tion of existing data terms used in image segmentation and the length of curve and/or the area of the region within
distribution tracking. The particular case corresponding to
= 0 is an explicit form of assuming that the overlap be- ES = F (O(, Min ),, ) + ds + d x. (10)
tween the intensity distributions of the object and the back- Rin
ground is minimal. Such assumption is implicit in existing
3.2 Minimization Equation via Curve Evolution
methods. The overlap prior is more versatile than existing
data terms. It addresses cases in which an overlap exists be-
The curve evolution equation is obtained by minimizing ES
tween the distributions within the object and the background
with respect to . To this end, we derive the Euler-Lagrange
and, as such, the ensuing algorithms are more widely ap-
gradient descent equation by embedding curve in a one-
plicable than existing algorithms.
parameter family of curves: (s, t) : [0, 1] R+, and
In the next section, we devise the proposed statistical
solving the partial differential equation:
overlap prior for image segmentation. In this case, parame-
ters and are estimated using a set of relevant training
ES F (O(, Min ),, ) ds
= =
images independent from the test images (images of inter-
t
est). Used in conjunction with classic regularization terms,
dx
our statistical overlap prior is minimized by curve evolu- Rin
(11),
tion via the Euler-Lagrange equation. We rst compute the
120 Int J Comput Vis (2009) 85: 115 132
O(, Min ) 1
where t is an arti cial time parameterizing the descent di- = O(, Min )
rection, and ES denotes the functional derivative of ES with 2Aout
(s, t)
respect to .
Min (z)
K(z I
To derive the nal curve evolution equation, we need to (s,t) ) dz
Pout (z)
z Z
compute the functional derivative of the statistical overlap
n(s, t).
prior with respect to . We have (15)
F (O(, Min ),, ) (O(, Min ) ) Using (15) and the classic derivative of the regularization
= 2 F (O (, M ),, )
2
terms (Chan and Vese 2001) in (12) gives the nal curve
in
O(, Min ) evolution equation:
(12)
.
(O(, Min ) )
(s, t)
=
2 2 F (O(, Min ),, ) 2Aout
O (,Min ) t
Now we need to compute in (12). We have
Global overlap test
O(, Min ) 1 Min (z) Pout (z)
Min (z)
= (13)
.
K(z I dz O(, Min )
(s,t) )
2 Pout (z)
z Z
Pout (z)
z Z
O (,Min ) Pixelwise hypothesis tests
To derive the nal expression of, we need to com-
pute P (z) . To this end, we consider the following propo-
out
+ (s, t) n(s, t) (16)
sition, which will be used also in the rest of the computation
in this paper.
where (s, t) is the mean curvature function of . Note
that O(, Min ), Pout, and Aout depend on the curve and,
Proposition 1 For a scalar function g and a curve, the
consequently, need to be updated along with the evolution
functional derivative with respect to of the integral of g
process.
over the region enclosed by, i.e., R = Rin, is given by
3.3 Link to Statistical Hypothesis Testing
g(x)d x
Rin
= g ( (s, t)) n(s, t)
(s, t)
Let us examine how the statistical overlap prior controls
curve evolution, as well as the link between (16) and the
where n(s, t) is the outward unit normal to at (s, t). Ap-
plying this result to the complement of Rin, Rout = Rc, classical theory of hypothesis testing (Lehmann 1986). The
curve ow obtained in (16) is the product of two terms, one
yields
is coordinate dependent and can be viewed as a pixelwise
g(x)d x
hypothesis testing which amounts to a classical likelihood
Rout
= g ( (s, t)) n(s, t).
ratio test for each pixel, whereas the other is independent of
(s, t)
the spatial coordinate and can be viewed as a global over-
The rightmost minus sign in the equation above is due to the lap test. As such, the proposed model can be viewed as a
fact that n being the external unit normal to Rin, the external hypothesis testing controlled by an overlap test. Given the
unit normal to its complement Rout is n. learned overlap mean and the initial curve, we have two
cases:
A proof of this result, based on the Green s theorem
and the Euler-Lagrange equations, can be found in Zhu and Case 1 The curve is initialized so as to de ne a positive
overlap test, i.e., O(, Min ) > (the overlap measure is
Yuille (1996).
Applying this proposition to Aout and Rout K(z Ix )d x superior to its expected value ).
Pout (z)
in yields, after some algebraic manipulations
In this case, the obtained curve evolution amounts to a
classi cation by separating distributions, whereas the over-
Pout (z) 1
= (s,t) ) K(z I (s,t) )
Pout (I n(s, t). lap test term forces the curve evolution to stop when the
Aout
(s, t)
overlap measure reaches its expected value . Ignoring the
(14)
regularization terms, the curve evolution is guided by the
sign of the hypothesis testing term in (16). At each iteration
Embedding (14) into (13), and after some algebraic manip-
t and for each pixel s on the curve, the evolution equation
ulations, we obtain:
Int J Comput Vis (2009) 85: 115 132 121
performs the following likelihood ratio test,2 which consists 3.4 In uence of the Learned Variance
in separating the null hypothesis, whose likelihood is speci-
ed by the background distribution Pout, and the alternative The learned variance affects the weight of the overlap
hypothesis, whose likelihood is speci ed by the foreground prior. The smaller, the higher such weight. This is intuitive
model Min : because a small variance indicates that is a reliable esti-
M (I mation of the overlap and, as such, it gives more importance
)
If Pout (I (s,t) ) > O2, the velocity is a positive function
in (s,t)
to the overlap prior ow. On the contrary, a high variance
multiplied by the outward normal. Therefore, the curve
gives more importance to the other functional terms.
expands to include pixel s within the foreground, thereby
rejecting the null hypothesis (background) at pixel s .
3.5 Level-Set Implementation
Min (I (s,t) )
If Pout (I (s,t) ) 0. This representation has well-known
tion decreases during curve evolution until it becomes equal advantages over an explicit discretization of using a num-
to its expected value . When the overlap reaches, i.e., ber of points on the curve. It handles automatically topolog-
O(, Min ) =, the overlap test term equals zero, thereby ical changes of the evolving curve ( may split and merge
forcing curve evolution to stop. while u remains a function), and can be effected by stable
It is worth noting that in the particular case = 0, hy- numerical schemes (Sethian 1999). When curve evolves
pothesis testing continues until the curve reaches a max- following (Sethian 1999)
imum discrepancy between the distributions, i.e., a maxi-
(s, t)
mum separation between the null hypothesis and the alterna-
= V n(s, t), (17)
tive hypothesis. In this particular case, the proposed model t
reduces to a classi cation by maximally separating the dis-
where V : R R, the corresponding level set function u
tributions.
evolves according to
Case 2 The curve is initialized so as to de ne a negative u
(x, t) = V u(x, t) . (18)
overlap test, i.e., O(, Min )
Contact this candidate