IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 11, NOVEMBER 2006
3431
Unsupervised Variational Image
Segmentation/Classification Using
a Weibull Observation Model
Ismail Ben Ayed, Student Member, IEEE, Nacera Hennane, and Amar Mitiche, Member, IEEE
Abstract—Studies have shown that the Weibull distribution can
model accurately a wide variety of images. Its parameters index
a family of distributions which includes the exponential and approximations of the Gaussian and the Raleigh models widely used
in image segmentation. This study investigates the Weibull distribution in unsupervised image segmentation and classification by a
variational method. The data term of the segmentation functional
measures the conformity of the image intensity in each region to a
Weibull distribution whose parameters are determined jointly with
the segmentation. Minimization of the functional is implemented
by active curves via level sets and consists of iterations of two consecutive steps: curve evolution via Euler–Lagrange descent equations and evaluation of the Weibull distribution parameters. Experiments with synthetic and real images are described which verify
the validity of method and its implementation.
Index Terms—Active curves, classification, image segmentation,
statistical modeling, Weibull distribution.
I. INTRODUCTION
S
EGMENTATION is a fundamental low-level processing
task which occurs in many image interpretation applications. It consists of partitioning an image into segments having
a homogeneous description, generally in terms of a parametric
model of the image. The Weibull distribution has been used in
recent vision studies to model various types of signals, such
as radar [1], sonar [2], and medical images [3], [4], as well as
video shot duration [5], and stochastic textures [6]. It is not
surprising that these studies found the Weibull distribution to
be a good model because the distribution parameters describe
texture contrast, scale, and shape [7], and generate a six-stimulus basis for texture perception which codes the perceptual
properties of regularity, coarseness, contrast, roughness, and
directionality [6], [8], much like the RGB representation is
a tri-stimulus basis for color perception. Also, it has been
shown theoretically and verified experimentally that first order
derivatives of a wide variety of textures also follow the Weibull
distribution [9]–[11]. Variation of the Weibull parameters yields
a spectrum of distributions which includes the exponential and
(the
approximations of the Gaussian and Raleigh. For
Manuscript received January 5, 2006; revised May 2, 2006. This work was
supported by the Natural Sciences and Engineering Research Council of Canada
under Grant OGP0004234. The associate editor coordinating the review of this
manuscript and approving it for publication was Prof. Vicent Caselles.
The authors are with the Institut National de la Recherche Scientifique,
INRS-EMT, Montréal, QC H5A 1K6 Canada (e-mail: benayedi@emt.inrs.ca;
mitiche@emt.inrs.ca).
Digital Object Identifier 10.1109/TIP.2006.881961
Fig. 1. Effect of the shape parameter on the Weibull distribution.
shape parameter) we have the exponential distribution.
gives an approximation of the normal distribution and
an approximation of the Rayleigh distribution (see Fig. 1 for an
illustration). The exponential, Gaussian, and Raleigh distributions have served as image models in numerous studies. All of
this evidence points to the Weibull distribution as a good model
for image segmentation. In this case, the distribution parameters
would serve to distinguish between the segmentation regions.
This paper investigates the Weibull distribution in image
segmentation by a variational formulation with active contours
and level sets. Active contour methods use simple closed plane
curves which evolve to delineate the segmentation regions. The
curve evolution equations are derived from the minimization
of a functional which, generally, contains a term of conformity
of the data to a parametric model and a term of regularization.
The variational formalism with active curves and level sets
is important because several studies have shown that it can
lead to effective segmentation algorithms [12]–[20], as well
as effective algorithms to solve other vision problems such as
classification [21], [20], tracking [22], and texture analysis [23].
The piecewise constant and Gaussian models have been
widely used in active contour and level-set segmentation of
images acquired by conventional cameras [12], [14]–[18], [24].
Although they have been effective in some cases, these models
are not complex enough to be adequately descriptive in general.
Images acquired by sensors other than conventional cameras
often do not follow a Gaussian distribution, as in medical [3],
[4], sonar [2], and radar imagery [1], [25]. Furthermore, the
segmentation regions may require different models. For example, the luminance within shadow regions in sonar imagery
1057-7149/$20.00 © 2006 IEEE
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is well modeled by the Gaussian distribution while the Rayleigh
distribution is more accurate in the reverberation regions [2].
In synthetic aperture radar (SAR) images, the intensity follows
a Gamma distribution in a zone of constant reflectivity and a
K distribution in a zone of textured reflectivity [25]. A few
investigations of level-set segmentation have demonstrated
the benefit of using models other than piecewise constant and
Gaussian. The study in [26], for instance, shows the effect the
noise model can have on segmentation and proposed a two-region formulation adapted to distributions of the exponential
family. In our previous work [20], we investigated the Gamma
distribution for multiregion segmentation of SAR images. SAR
image segmentation using a Gamma model has been investigated in [27] via polygonal snakes, i.e., active sets of connected
line segments. All of these studies assume that the image model
distributions have a known shape, i.e., a shape which does
not depend on the parameters of the distribution. This can be
a significant limitation because there are many applications
where the image model distribution is not determined or varies
with experimental conditions [2].
In the general context of image segmentation/classification,
this study is most related to the investigations of segmentation
based on Euler–Lagrange functional minimization via level
sets in [18], [20], [16], [15], [21]. It is also related to the studies
of segmentation by classical snakes [12] and polygonal snakes
[27]. A polygonal snake is parameter free. Also, it does not
assume the number of regions known, whereas current level-set
methods do. However, they have significant limitations which
level sets remove: they cannot segment regions of arbitrary
topology. In particular, they cannot segment regions composed
of disjoint parts. Snakes do not allow changes of active curve
topology during evolution [30]. These limitations were the
main motivations for the development of level sets.
In this study, we develop a variational level-set segmentation/classification method using a Weibull observation model.
This method is more widely applicable than current ones because the Weibull distribution is a model versatile enough to represent a wide variety of images. The objective functional contains two terms: An original observation term which measures
the conformity of region data to a Weibull distribution representation and a classical length-related term of regularization for
smooth segmentation/classification boundaries. The functional
is efficiently minimized by alternating between the Euler–Lagrange descent equations of curve evolution and a gradient descent update of the shape parameter, with the scale update is
done according to its maximum likelihood relation to the shape
parameter. Experiments are described which verify the method
and its implementation.
The remainder of this paper is organized as follows. The next
section presents the segmentation/classification functional. Section III gives the equations of its minimization, and Section IV
describes experimental results. Section V contains a conclusion.
II. WEIBULL SEGMENTATION/CLASSIFICATION FUNCTIONAL
Let
be an image function. An -region
segmentation/classification of is a partition
of the image domain such that the image is homogeneous with
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 11, NOVEMBER 2006
respect to some characteristics in each region. In general, classification also assigns an identifying label to each region. The
Weibull observation model represents the image in each region
by a Weibull distribution
(1)
is the shape parameter and
the scale paramwhere
eter. Segmentation/classification is stated as the minimization
of a functional containing two characteristic terms: a term of
conformity of the image data within each region to a Weibull
distribution and a regularization term.
Data Term: The data term, , follows a Weibull observation
model. It measures how well the data fits this distribution within
each segmentation region
(2)
where
(3)
and is the function which evaluates the conformity of data to
a Weibull distribution in region
(4)
The evaluation function defined in (4) corresponds to the maximization of the log likelihood.
Regularization Term: We use a classic regularization term for
smooth segmentation boundaries and to avoid small, isolated
segmentation fragments
(5)
The functional to minimize is a weighed sum of the data and
regularization terms
(6)
where
is the boundary of
and is a positive real constant to weigh the relative contribution of the two terms of the
functional. Note that the Weibull distribution parameters in
depend on the partition and, consequently, are to be determined
concurrently with the segmentation/classification.
BEN AYED et al.: UNSUPERVISED VARIATIONAL IMAGE SEGMENTATION/CLASSIFICATION
III. FUNCTIONAL MINIMIZATION
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After some algebraic manipulations, we obtain
For a clearer exposition of the algorithm, we treat the tworegion segmentation/classification problem first (Section III-A).
We generalize to multiple regions in Section III-C. In the case of
multiple regions, we will see that the issue is to guarantee that
the algorithm leads to a partition, i.e., to regions which cover the
image domain without overlap.
(12)
where
is the area of region
(13)
A. Two-Region Segmentation/Classification
In the case of two regions, we consider a closed planar para. Let
be the region in the
metric curve
the region in the exterior. To miniinterior of , and
mize , which depends on and on the distribution parameters
, we adopt an iterative two-step algorithm, with
the functional decreasing at each step: After initializing the parameters, the first step consists of evolving the curve with the parameters fixed and the second step updates the parameters with
the curve fixed.
Step 1: With the parameters fixed, the problem consists of
with respect to . To this end, we derive the
minimizing
Euler–Lagrange equation by embedding the curve
in a
, and
one-parameter family of curves:
solving the partial differential equation
(7)
where
is the functional derivative of with respect
to . The segmentation/classification regions are obtained at
. Using the result in [12] which
convergence, i.e., when
shows that, for a scalar function , the functional derivative with
is
, where is the
respect to curve of
outward unit normal to , we obtain the functional derivative of
the data term
The scale parameters are updated using their maximum likelihood relation to the shape parameters [28]
(14)
B. Level-Set Implementation
Equation (10) can be implemented by an explicit discretization of using a number of points on the curve. A better alternative is to use the level-set representation [30], [32]. In contrast to
this explicit representation, the level-set implementation allows
automatically topological changes of the evolving curve and can
be effected by stable numerical schemes. In the level-set implementation, curve is represented implicitly as the zero level set
, i.e., is the set
. One can
of a function
,
show that [30] if the curve evolves according to
, then the level-set function evolves acwhere
cording to
. Assuming
inside , its outward unit normal can be calculated from by
.
In our case, the level-set function evolution equation corresponding to (10) is given by
(15)
The curvature function
is given by
(8)
(16)
The derivative of the regularization term with respect to is [12]
(9)
where is the mean curvature function of . The final curve
evolution equation of is then given by
(10)
Step 2: We fix the curve and minimize with respect to
the parameters. The descent equations for the shape parameters
are given by
(11)
We should note that, when velocity is defined only for the
evolving curve, the level-set evolution equation applies only for
points on the curve. In such a case, one must define extension
velocities, i.e., proper velocities at points that do not lie on the
evolving curve [30]. For instance, the extension velocity at a
point is the velocity at the point closest to it on the evolving
curve [33]. Proper extension velocities can also be defined so
that the level-set function is at all times the distance function
from the evolving curve [30]. Both of these definitions which
are often implemented via narrow banding, where the evolution
of the level-set function is effected only in a neighborhood of the
zero level set, require that the initial curves intersect the region
they segment. This is important when a region has unconnected
components. We use in this work an alternative robust to initialization, and which extends the expression of velocities on the
evolving curve to the image domain, since this expression can
be evaluated at any point of the image domain [17]–[20]. We
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refer the reader to [30] for a discussion on level sets and related
numerical schemes.
C. Generalization to Multiple Regions
In this section, we generalize the proposed method to an arbitrary but fixed number of regions. In the multiregion case, ambiguities can occur in the segmentation/classification when the
interiors of two or more curves overlap. To guarantee an unambiguous segmentation/classification, i.e., a partition of the
image domain , several solutions have been proposed such
as adding to the functional a term which draws the solution toward a partition [21], using a functional which results in curve
evolution equations where the evolution of a curve involves a
reference to the others [15], or establishing an explicit correspondence between the interior of curves and the regions of segmentation [16], [18]. In this work, we use the representation of
a partition proposed in [18]. For a better understanding of the
advantages of this representation, we refer the reader to [18].
This representation establishes a correspondence between regions enclosed by closed simple plane curves and regions in the
segmentation/classification which guarantees that at all times
the partition constraint is maintained. The correspondence is as
regions, we consider a family
follows. For a partition into
, of parametric closed plane
of recurves. The correspondence between the family
and the segmentation/classigions enclosed by the curves
fication regions
of the image domain is defined by [18]
(refer to [18, Fig. 1])
Minimization of functional in (18) with respect to curves
is performed by embedding each curve into a oneparameter family
of plane curves, indexed by algorithmic time , and solving
Using the calculus in [18], we obtain the following system of
coupled curve evolution equations:
(19)
is the outward unit normal to and
where
, and
function of , for
the curvature
is given by
(17)
where
is the complementary of
.
The family
thus
obtained is, by construction, a partition of the image domain, for
. With this choice of
any family of closed plane curves
representation of a partition of the image domain into regions,
the energy functional (6) becomes [18]
(18)
Minimization of functional in (18) with respect to the scale
and shape parameters is performed according to (12) and (14)
.
for
For a level-set implementation, we represent each curve
implicitly by the zero level set of a function
,
. The level-set
with the region inside corresponding to
evolution equations corresponding to (19) are then given by the
following system of coupled partial differential equations [18]:
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Fig. 3. Influence of the regularization parameter: (a) final position of curves
with = 0:01; (b) final position of curves with = 2; (c) final position of
curves with = 5; (d) classification with = 0:01; (e) classification with
= 2; (f) classification with = 5.
Fig. 2. Image with different noise models: (a)three regions initialization;
(b) final position of curves; (c) final classification; (d)–(f): regions of segmentation/classification; (g)-(i): results corresponding to the Gaussian distribution.
Image size: 162 158. = 1.
2
where
if
and 0 otherwise, and
is given by
and
is the curvature of the level set of
.
IV. EXPERIMENTATION
To validate the algorithm and its implementation, we ran
several experiments with synthetic and real images. In the
following, we show some representative results.
A. Simulated Data
The purpose of these experiments with simulated data is
to evaluate quantitatively and comparatively the performance
of the method. We have three examples. The image shown in
Fig. 2(a) with initial curves (black and white) is a synthetic
image of three regions with different image models. The image
intensity in the clearer region is generated from the Gaussian
distribution. The gray region is derived from the Rayleigh distribution, and the darker region from the Poisson distribution.
We show the final position of curves in Fig. 2(b), and the final
classification in Fig. 2(c). Fig. 2(d)–(f) shows the corresponding
segmentation/classification regions. The results in Fig. 2 are
. We show also results when a Gaussian
obtained with
distribution is used with the same regularization parameter
[Fig. 2(g)–(i)]. As evident in these figures, the Gaussian model
gives incorrect results.
The role of the boundary length regularization term is to
smooth the segmentation boundaries and prevent the occurrence of small, isolated regions. The higher the weight of the
regularization (parameter ), the smoother the region boundaries. As an illustration, Fig. 3 shows the segmentations and
and
corresponding classifications obtained for
. The partition boundaries are smoother for
, but the
segmentations/classifications are good for both
and
, as are those for the several values between 1 and 5
which we experimented with. In the range between 1 and 5,
, however, small
the differences are quite small. For
islands fragment the segmentation quite noticeably. Table I
lists the percentage of correctly classified pixels for various
values of . Our results in this experiment are consistent with
an minimum description length (MDL) interpretation of the
objective functional, which prescribes a value of parameter
approximately equal to 2 [26]. It is interesting that experimental simulations in [26] show that this value corresponds to
the minimum of the mean number of misclassified pixels.
The synthetic image shown in Fig. 4(a), with initial curves,
has four regions with data generated from the exponential distribution (the image intensity is modified to enhance the contrast between regions for viewing purposes only). The exponential distribution corresponds to a simulation of a mono-look
SAR image [20]. The synthetic four-region image in Fig. 5(a),
with the initial curves superimposed, is a simulation of an amplitude multilook SAR image with intensities generated from
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TABLE I
PORTION OF CORRECTLY CLASSIFIED PIXELS FOR THE
SYNTHETIC IMAGE WITH DIFFERENT NOISE MODELS
Fig. 5. Simulated multilook SAR image: (a) four regions initialization;
(b) final classification; (c)–(e): regions of segmentation/classification. Image
size: 192 189. = 1.
2
TABLE II
PORTION OF CORRECTLY CLASSIFIED PIXELS FOR THE SIMULATED SAR IMAGES
Fig. 4. Synthetic image with exponential noise: (a) four regions initialization;
(b) final classification; (c)–(e): regions of segmentation/classification. Image
size: 192 189. = 1.
2
the Gamma distribution (
[20]). Both images are generated from the same ideal classification. All the initial curves are
placed arbitrary about the middle of the image. For each region,
the initial shape parameter is set to 1, and the scale parameter
to the mean of the region (this corresponds to the exponential
distribution). Figs. 4(a) and 5(a) show initializations superimposed to the images. Figs. 4(b) and 5(b) show final classifications. Fig. 4(c)–(f) and Fig. 5(c)–(f) show the corresponding regions of segmentation/classification. Table II reports the portion of correctly classified pixels for both the exponential image
and the simulated multilook SAR image. As evident in the results, the algorithm performs well. To validate the joint variational Weibull parameter estimation, we show in Fig. 6 the distribution fits corresponding to some classification regions in the
three synthetic images. Fig. 6(a) displays the fit between the true
Gaussian distribution, i.e., the distribution from which the data
in the Gaussian region [2(f)] is generated, and the estimated
Weibull distribution. Fig. 6(b) shows the fit between the true
exponential distribution of the region in Fig. 4(f) and the estimated Weibull distribution. In Fig. 6(c), we show the fit between
the true Rayleigh distribution of the region in Fig. 2(e) and the
estimated Weibull distribution. Finally, Fig. 6(d) contains the
histogram of amplitudes in a simulated multilook SAR image
region and the corresponding estimated Weibull distribution.
The computed Weibull approximation of the Raleigh data has
a correct shape Fig. 6(c). However, the fit between this approximation and the theoretical distribution is less accurate than for
the other two regions. Such a fit is not surprising for three reasons, the first two being general statistical explanations: 1) the
data is generated from a distribution (Raleigh in this case) and
fit with an approximating distribution (the Weibull distribution
in this case); 2) the approximation of the theoretical distribution
is computed from a single experiment, i.e., the data used is a
single sample from the theoretical distribution; 3) the approximation is computed jointly with the segmentation, the objective
of the algorithm being to obtain a partition of the image domain
into regions which are assumed to differ by the parameters of
their Weibull distribution.
It is important to emphasize that the main purpose, here, is
image segmentation rather than model fitting of data. The desired segmentation is one which minimizes the objective functional (6). This functional embeds the assumption that the desired segmentation regions have regular boundaries and differ
by the parameters of a Weibull approximation of their data. Minimization of this functional does not seek to fit models to the
image data within specific regions. Rather, it seeks simultaneously an image domain partition with regular boundaries and
the Weibull parameters of the partition regions, so as to have a
best overall fit of the image. In this context, Weibull parameters
which produce a correct segmentation are as good as accurate
ones.
The stated function of the algorithm justifies an evaluation
of its performance via an evaluation of the segmentation it
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Fig. 6. Distribution fits between the obtained Weibull distributions and the real distributions.
Fig. 7. Real optical image of two regions: (a) initial curves; (b) intermediate curves; (c) final curve; (d) final classification. Image size: 288
produces. In this experiment with synthetic data, it is done
both by computing the percentage of correctly classified pixels
(Tables I and II) and by inspection of the segmentation boundaries.
The algorithm execution time depends on the size of the
image, the number of regions, and the initialization. On a
1.73-GHZ PC, the algorithm took 67 s to produce the result of
Fig. 2. The algorithm needed 872 s to process the four-region
image of Fig. 5. The size of the image is specified in the caption
of each figure.
B. Real Data
We applied the algorithm to three real images. The first image
is an optical image of a plane, shown in Fig. 7. The second image
is a mono-look SAR image corrupted with a high multiplicative
speckle noise. It is well known that this kind of images is accurately modeled by an exponential distribution. The third image
represented in Fig. 9(a), and which contains the initial curves
is an an extract of 256 256 pixels from a real multilook SAR
image. For the image of the plane, we show the initialization
superimposed on the image, in Fig. 7(a), an intermediate step
of curve evolution in 7(b), and the final curve in Fig. 7(c). The
two classified regions represented by their mean gray value are
2 193. = 1.
displayed in Fig. 7(d). Results related to the mono-look SAR
image are shown in Fig. 8. Fig. 8(a) shows the initial position
of the curve. Fig. 8(b) shows an intermediate step of the curve
evolution. Fig. 8(c) shows the final curve. The two classified
regions represented by their mean gray value are displayed in
Fig. 8(d). The choice of the number of regions is based on visual inspections. The multilook SAR image is classified into
four regions. Fig. 9(a) shows initializations superimposed to the
image. Fig. 9(b) contains final classification and Fig. 9(c)–(f)
contains the various computed regions individually. The algorithm took 16s to produce the result of Fig. 7. The simulation
in Fig. 8 needed 103 s, and the simulation in Fig. 9 took 994
s. The results demonstrate the efficiency of the method and its
adaptivity to different kinds of real images.
V. CONCLUSION
We presented a level-set segmentation algorithm adapted to
a variety of imaging noise by the use of the Weibull distribution. The parameters of the Weibull distribution are updated
iteratively along with the segmentation/classification using the
Euler–Lagrange descent equations corresponding to a functional containing a term of conformity of the data to the Weibull
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Fig. 8. Mono-look SAR image: (a) two regions initialization; (b), (c) two steps in the curve evolution; (d) final classification. Image size: 192
Fig. 9. Extract of a multilook SAR image: (a) four regions initialization;
(b) final classification; (c)–(e): regions of segmentation/classification. Image
size: 204 206. = 1.
2
model and a term of regularization. The algorithm is implemented via the level-set method. Results were shown on both
synthetic and real images which demonstrate the efficiency of
the method and its capacity to adapt to different image models.
ACKNOWLEDGMENT
The authors would like to thank C. Vázquez for letting them
use his level-set implementation programs.
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3439
Ismail Ben Ayed (S’04) received the B.S. degree
and the M.S. degree in telecommunications from
the Ecole Superieure des Communications de Tunis,
Tunisia, in 2002 and 2003, respectively. He is
currently pursuing the Ph.D. degree in the Telecommunications Department, Institut National de la
Recherche Scientifique, Montréal, QC, Canada.
He has authored over ten research papers in leading
journals and conferences. His research interests include image and motion segmentation with a focus on
variational techniques, statistical modeling and shape
representation for image analysis, and remote sensing.
Nacera Hennane received the engineering degree in electronic from the Science and Technology University of Blida, Blida, Algeria, in 1995, and the B.Sc.
degree in microelectronic from the Universite du Quebec a Montreal, Montréal,
QC, Canada, in 2002. She is currently pursuing the M.Sc. degree at the Institut
National de la Recherche Scientifique, INRS-EMT, Montréal.
Her current research interests focus on image segmentation.
Amar Mitiche (M’03) received the Licence És Sciences in mathematics from the University of Algiers,
Algiers, Algeria, and the Ph.D. degree in computer
science from the University of Texas, Austin.
He is currently a Professor in the Department of
telecommunications, Institut National de Recherche
Scientifique, Montréal, QC, Canada. His research
interests include computer vision, motion analysis
in monocular and stereoscopic image sequences
(detection, estimation, segmentation, and tracking)
with a focus on methods based on level-set PDEs,
and written text recognition with a focus on neural networks methods.