Numer. Math. Theor. Meth. Appl. Vol. 2, No. 4, pp.

445-468
doi: 10.4208/nmtma.2009.m9007s November 2009
Fast Texture Segmentation Based on Semi-Local
Region Descriptor and Active Contour
Nawal Houhou
1,
, Jean-Philippe Thiran
1
and Xavier Bresson
2
1
Signal Processing Laboratory 5, Ecole Polytechnique Fdrale de Lausanne (EPFL),
Lausanne, Switzerland.
2
Department of Mathematics, University of California, Los Angeles,
CA 90095-1555, USA.
Received 26 March 2009; Accepted (in revised version) 22 July 2009
Abstract. In this paper, we present an efcient approach for unsupervised segmenta-
tion of natural and textural images based on the extraction of image features and a
fast active contour segmentation model. We address the problem of textures where
neither the gray-level information nor the boundary information is adequate for object
extraction. This is often the case of natural images composed of both homogeneous and
textured regions. Because these images cannot be in general directly processed by the
gray-level information, we propose a new texture descriptor which intrinsically denes
the geometry of textures using semi-local image information and tools from differen-
tial geometry. Then, we use the popular Kullback-Leibler distance to design an active
contour model which distinguishes the background and textures of interest. The exis-
tence of a minimizing solution to the proposed segmentation model is proven. Finally,
a texture segmentation algorithm based on the Split-Bregman method is introduced to
extract meaningful objects in a fast way. Promising synthetic and real-world results for
gray-scale and color images are presented.
AMS subject classications: 65M10, 78A48
Key words: Semi-local image information, Beltrami framework, metric tensor, active contour,
Kullback-Leibler distance, split-Bregman method.
1. Introduction
Texture segmentation is among the most challenging problems in image segmentation.
The problem already begins with the denition of textures. The human eye can easily
recognize different textures, but it is quite difcult to dene them in mathematical terms.
Then, there is a deliberate vagueness in the denition of textures, which explains the
difculty to conceptualize a model able to describe it. Besides, textures raise the problem
of non-existence of signicant edges and the non-homogeneity of intensity distributions

Corresponding author. Email addresses: (N. Houhou),

(J.-P. Thiran), (X. Bresson)
http://www.global-sci.org/nmtma 445 c 2009 Global-Science Press
446 N. Houhou, J.-P. Thiran and X. Bresson
lying in images. However, it is consensually admitted (e.g. [3]) that textures are ne
scale-details, usually with some periodicity and oscillatory nature.
Different approaches deal with the extraction of homogeneous features from textures.
Among these texture descriptors, some are statistical-based or transformed-based as the
structure tensor [8, 54] or the Gabor lters [38]. Some recent methods propose to use the
Beltrami framework [47], [27]. In this paper we propose a new method for the extraction
of homogeneous regions or textures based on the Beltrami framework and on semi-local
image information. We will see that our texture feature is more robust with respect to
noise than the one we have previously presented in [27].
Once the feature extraction step is completed, a fast and robust algorithm to perform
texture segmentation/extraction is needed. To reach this objective, we develop an algo-
rithm based on the active contour model. In this image segmentation method, an initial
contour is subjected to a speed term which drives the contour toward the boundary of
the object to be segmented. The speed term is determined by the minimization of an
energy functional composed of a contour smoothing term (internal energy) and an attrac-
tion term that pulls the contour towards the object boundaries (external energy). Active
contour models have a long history. This method was originally introduced by Kass et al.
in [31]. Then, Caselles et al. in [10] and Kichenassamy et al. in [32] proposed an intrinsic
geometrical model, called geodesic/geometric active contours, where the curve evolution
is handled by the level set method introduced by Osher and Sethian in [40]. Thus, the
rst generation of active contour segmentation methods [10, 31, 32] was based on edge
detection. However, boundary-based segmentation is often too limited in many applica-
tions such as in medical image segmentation where fuzzy contours can be encountered or
in natural images with textures. To deal with these problems, segmentation models based
on region descriptors such as mean [14] or variance [29,55] have been developed. Among
the region descriptors, the probability density function (pdf) (or histogram) is proven to
be an efcient and general region descriptor.
Zhu and Yuille in [56] and Paragios, Rousson and Deriche in [41, 45] approximated the
pdf of the image regions by mixture of Gaussians, each one representing a homogeneous
intensity region to be segmented. Jehan-Besson, Aubert, Barlaud, Faugeras and Herbulot
in [1, 26, 29] used the pdf of evolving regions of interest as a general region descriptor.
The regions of interest are given by the minimization of region-based energies using the
shape derivative tool dened by Delfour and Zolesio in [16]. Aubert, Barlaud, Faugeras
and Jehan-Besson proved in [1] that minimizing a region-based functional with the shape
derivative tool is equivalent to minimizing boundary-based functional with the calculus of
variations. These authors applied in [1,29] the shape derivation tool to the image segmen-
tation problem. Along the same lines, Herbulot et al. in [26] used the shape derivative
tool and information theoretical concepts (entropy and mutual information) to also per-
form image segmentation.
All these methods enable us to nd local minimizers of the segmentation problem. Local
minimizers mean that the quality of the segmentation results depends on the choice of the
initial condition. In other words, the initial contour is critical to get satisfactory segmenta-
tion results, and to avoid bad local minimizers. The authors of [5, 12] proposed to refor-
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 447
mulate the active contour problem in a convex optimization framework, providing a global
minimizer to the active contour model. Besides, a fast minimization scheme

based on a
dual approach of the TV norm and a splitting technique was introduced in [5], dening a
fast segmentation algorithm. In [27], an unsupervised segmentation method based on the
Kullback-Leibler distance and on non-parametric estimation of the pdf was introduced and
solved with the fast minimization scheme dened in [5]. Recently, a new technique based
on the Split-Bregman (SB) method [24] has been introduced to solve the convexied active
contour model in a faster way than [5]. More precisely, this new minimization algorithm is
as least as fast as graph-cuts, is more accurate because it uses isotropic schemes and does
not have memory requirements in 3D. We can notice that the SB method is actually equiv-
alent to the augmented Lagrangian method [49, 51]. In this paper, we propose to apply
the SB method dened in [24] to solve the unsupervised segmentation problem. Finally,
the main contributions of this paper are summarized as follows:
1. Denition of an intrinsic texture descriptor based on the Beltrami representation,
semi-local image information and the metric tensor.
2. Proof of the existence of a minimizing solution for our variational segmentation prob-
lem in the space of functions with bounded variation.
3. Denition of a fast and accurate numerical scheme to solve the texture segmentation
problem.
The outline of this paper is as follows. Section 2 studies the feature extraction problem
for textures. Section 2.1 describes state-of-the-art feature extraction techniques. Then,
Section 2.2 presents the new texture descriptor based on semi-local information in the
Beltrami representation. Section 3 presents the active contour unsupervised segmentation
method. Our segmentation method is then introduced in Section 3.1, followed by a math-
ematical justication in Section 3.2. Section 4 presents the convexication of the active
contour energy and also introduces a fast numerical scheme based on the Split-Bregman
method. In Section 5, experimental results on both texture gray-scale and color images are
shown.
2. Texture features
This section aims at dening a pertinent feature descriptor for textures. The quality of
texture segmentation methods is highly dependent of the quality of texture features. It is
thus essential to dene a texture feature as good as possible for the segmentation task.
2.1. Image features
The rst and most natural texture feature is the image itself. For a scalar image I :

2
, each pixel is characterized by its gray-level value or intensity. If the image is
composed of multiple channels (such as color images), then each pixel is described by

We mean the optimization scheme is faster than the Level Set method.
448 N. Houhou, J.-P. Thiran and X. Bresson
a vector of intensities. When the image is composed of textures, then the pixel intensity
value does not really give pertinent information. In fact, textures cannot be analyzed at the
local pixel scale but need to be analyzed at a higher level scale, scale where information of
textures is pertinent. Besides the intensity information, texture descriptors should consider
the scales and the orientations (or lack of orientation) of the image. A natural approach
for texture segmentation is thus to rst represent the textured image by feature descriptors
and then to apply a vector-valued segmentation scheme. It is clear that the quality of the
segmentation will depend on the extraction of good discriminative features. We hereafter
describe some of these features.
2.1.1. Filter-based features
A very popular class of texture features are the lter-based features of the given image.
Filters such as the gradient lter or the wavelet bank lter [37] have been used for texture
feature extraction and image segmentation (for examples [44, 47]). Texture features gen-
erated by Gabor/Morlet wavelet transform [35] are powerful tools to discriminate textures
of different orientations and scales. The main motivation is based on the fact that simple
cells in the visual cortex can be modeled by Gabor functions [38]. The Gabor functions
are parameterized by a wavelet orientation angle and a scale . Given a certain num-
ber of orientations and scales, the original image can be reconstructed from Gabor lter
responses obtained by convolution of the given image and the set of (, )-parameterized
Gabor functions. Obviously, increasing the number of orientations and scales will improve
the reconstruction quality. However, a good reconstruction can also be achieved by select-
ing only the most relevant lter responses s.a. in [28].
2.1.2. Semi-local information-based feature
As we said, the choice of features is difcult and critical to get an optimal segmentation
result. A recent promising image feature to represent and process textures is the image
intensity patch around the current pixel. The information on a close neighborhood around
the current pixel is extracted and leads to semi-local information at each pixel. The patch
idea as a feature vector was rst introduced for texture synthesis [17,18,36]. Later, Buades
et al. in [9] proposed to denoise images based on patch differences and non-local averag-
ing. Gilboa and Osher in [22] proposed a non-local denoising model based on a variational
framework. Finally, Bresson and Chan in [4] dened a variational unsupervised segmen-
tation method also based on patch differences.
2.1.3. Feature dened in the Beltrami framework
Another way to represent textures is to use the Beltrami representation introduced by
Sochen, Kimmel and Malladi in [50]. Sochen et al. proposed a new geometric represen-
tation of images by considering images as Riemannian manifolds embedded in a higher
dimensional space. For instance, a standard 2 dimensional gray-value image I :
2

can be viewed as a surface with local coordinates (x, y) embedded in
3
by a mapping
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 449
X : (x, y) (X
1
= x, X
2
= y, X
3
= I(x, y)). This manifold-based representation of images
offers two main advantages. Firstly, it allows using efcient tools borrowed from differ-
ential geometry to perform different image processing tasks such as denoising [33, 50]
or segmentation [6, 47]. The second main advantage of this framework is to work with
arbitrary N dimensional images. For example, a color image can be represented in a 5
dimensional space with the mapping X : (x, y) (X
1
= x, X
2
= y, X
3
= R(x, y), X
4
=
G(x, y), X
5
= B(x, y)), where R,G,B stands for red, green and blue. In [47], Sagiv, Sochen
and Zeevi used the Beltrami framework to represent the texture image as a 2-D dimen-
sional manifold embedded in a space of N + 2 dimensions, where N is the number of
Gabor responses. Then, the rst fundamental form (e.g. [34]), also called metric tensor,
of the (N +2)-D texture manifold allowed them to dene an intrinsic edge detector (see
also [48]). The idea of using the metric tensor to intrinsically dene the edges between
different texture regions is natural and well-posed in the context of differential geometry.
Indeed, the rst fundamental form describes the distortion or rate of change of the mani-
fold and so can detect boundary between different parts of the manifold corresponding to
different homogeneous textures. Based on the metric tensor, Sagiv et al. used the geodesic
active contour model [10] to drive the contour toward the boundary between two differ-
ent textured regions by considering the edge detector function or stopping function as the
inverse of the determinant of the metric tensor. To illustrate this idea, let consider the case
of a gray-scale image I :
2
represented in the Beltrami framework by the mapping
X = (x, y, I). The rst fundamental form is dened by:
g

<
X

,
X

>

,
where we have here {, } = {x, y}, which implies:
g
x x
= 1 + I
2
x
,
g
x y
= I
x
I
y
,
g
y y
= 1 + I
2
y
,
where the sufxes stands for partial derivatives. We have:
1
det(g

)
=
1
1 + |I|
2
. (2.1)
Function (2.1) corresponds (up to some constants) to the edge detector function used in
the standard active contour model [10]. Hence, Sagiv et al. generalized the edge detector
to texture images using the metric tensor of the Beltrami representation. Nevertheless,
as we said earlier, the edge detector function has its limitations (sensitive to noise and
initial contour position) and hence, it is not robust enough to segment a wide range of
images. For this reason, Sagiv et al. coupled their edge detector with a region detector,
coming from the vectorial Chan-Vese model [13], to segment complex textures. We observe
that their segmentation model provides promising results, but the region-based part of the
450 N. Houhou, J.-P. Thiran and X. Bresson
model (the vectorial Chan-Vese model) needs some computational time because it works
with many channels. In order to improve the model of Sagiv et al. and decrease the
computational time, we proposed in [27] to dene an intrinsic region detector for textures
in the Beltrami framework, rather than an edge detector. The proposed region detector for
textures is computed once, and used in the active contour model. Thus, our active contour
model did not need several channels, but only one feature image. This helped us to dene
a fast texture segmentation model in [27].
We used the shape operator and its eigenvalues to describe the geometry of textures.
The shape operator is a linear operator which calculates the bending of a surface in differ-
ent directions The eigenvalues of the shape operator correspond to the extremal of bending
of the surface, they are called principal curvatures and they are known to represent the ge-
ometry of the considered smooth manifold. In [27], we observed that the simplest Beltrami
representation of images, i.e. X = (x, y, I), is enough to provide an efcient intrinsic tex-
ture feature to segment many natural images. In this Beltrami representation, the texture
manifold is a 2-D manifold, with two unique principal curvatures (
1
,
2
). The couple
of principal curvatures (
1
,
2
) denes an intrinsic descriptor for homogeneous textures.
We made the assumption that for a given (complex) texture pattern, the couple (
1
,
2
)
follows the same distribution/pdf inside the texture region. This assumption is fullled in
many natural images. Finally, the distribution of (
1
,
2
) (or a combination of
1
,
2
) is
automatically estimated through the segmentation process.
Based on [25, 34], the values of the principal curvatures in our particular mapping can
be expressed as follows:

1,2
=

2
4
2
,
where

= 1,
=
1
Z
3
,
I
x x
(1 + I
2
y
) + I
y y
(1 + I
2
x
) 2I
x y
(I
x
I
y
)
.
,
=
1
Z
3
,
I
x x
I
y y
(I
x y
)
2
.
.
The rst principal curvature
1
(
1

2
) corresponds to the maximal change of the
normal to the surface and
2
corresponds to the minimal change. For sake of simplicity, and
in order to use the information provided by the two principal curvatures, we considered to
work with the norm of k
1
+k
2
, where vector k
1
(resp. k
2
) has a norm
1
(resp.
2
) and is
oriented by the associated unit principal vector. Since k
1
and k
2
are orthogonal, this leads
to:

t
:=

2
1
+
2
2
, (2.2)
where
t
:
2

+
denes the intrinsic texture descriptor that was used in [27] to seg-
ment real-world textures. The shape operator and its principal curvatures have proven to
be efcient in [27] to segment a large class of textures since it outperforms state-of-the-art
results. However, this texture descriptor is based on the computation of the second deriva-
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 451
tive of I, which makes highly sensitive to noise. In the following section, we introduce a
new texture descriptor based on semi-local image information, which is robust to noise.
2.2. New texture feature based on semi-local image information
In this section, we dene an intrinsic texture descriptor, which is more robust and less
sensitive to noise than the texture descriptor (2.2) used in [27]. To reach more robustness,
we will change the local representation X : (x, y) (x, y, I(x, y)) into a semi-local repre-
sentation. As we pointed out previously, textures are semi-local by nature, which motivates
us to describe them with patches along the same line as non-local means s.a. [9, 21]. Let

x, y
be the square patch of size around the pixel (x, y):

x, y
(I) =

I(x + t
x
, y + t
y
)

, t
x

2
,

2
.
, t
y

2
,

2
.
. (2.3)
We consider the new representation of textures in the Beltrami framework with the follow-
ing mapping:
X : (x, y) (X
1
= x, X
2
= y, X
3
=
x, y
(I)). (2.4)
This mapping includes local information (position in space) and semi-local image informa-
tion (the patch with the neighborhood pixel values around the current pixel). We make
the assumption that for a given (complex) texture pattern, the geometry of the manifold
embedded by the mapping (2.4) will be uniform for the observed texture. This assumption
is actually fullled in many natural images. This implies that the metric tensor of the man-
ifold is actually homogeneous in regions with same texture. We will use this assumption
to dene a new intrinsic texture descriptor, more robust than the one introduced in [27]
and in the previous section. A justication of this observation comes from the denition
of the metric tensor, which measures distances between points on the manifold. If the
manifold is almost at in a certain region of the manifold, then the distance between two
points anywhere on this region is almost equal. The manifold, represented by (2.4), is
almost at in regions with same textures because of the semi-local image information. The
corresponding metric tensor of (2.4) is as follows:
g
x y
=

1 + (
x

x, y
)
2

x, y

x, y

x, y

x, y
1 + (
y

x, y
)
2

,
Finally, the intrinsic texture descriptor, namely F, is dened as:
F = exp

det(g
x y
)

, (2.5)
where > 0 is a scaling parameter. The use of the Gaussian kernel acts as a low-pass lter,
which controls the degree of details. Fig. 1 summarizes the whole process.
For vector-valued images s.a. color images, the extension is straightforward. Let I =
(I
1
, I
2
, , I
k
) be a vector-valued image, where k is the number of channels. Then, the
corresponded semi-local mapping is:
X : (x, y) (X
1
= x, X
2
= y, X
3
=
x, y
(I
1
), , X
2+k
=
x, y
(I
k
)). (2.6)
452 N. Houhou, J.-P. Thiran and X. Bresson
and the metric tensor has the following form:
g
x y
=

1 +

k
j=1
(
x

x, y
)(I
j
)
2

k
j=1
(
x

x, y
(I
j
)
y

x, y
(I
j
)

k
j=1

x, y
(I
j
)
y

x, y
(I
j
) 1 +

k
j=1
(
y

x, y
(I
j
))
2

,
In our experiments, we will segment color images represented in the three primary color
space: red, green, blue (RGB), see Section 5.
3. Unsupervised active contour model based on the Kullback-Leibler distance
In this section, we introduce an unsupervised segmentation model based on the ac-
tive contour model and the Kullback-Leibler (KL) Distance between histograms/probability
density functions (pdfs). We show that the segmentation model is mathematically well-
posed since a minimizing solution exists in the space of functions with bounded variation.
3.1. Proposed segmentation method
We propose to solve the unsupervised image segmentation problem for gray-scale im-
ages composed of an object and the background (two-phase segmentation). An efcient
way to perform an unsupervised segmentation task is to use the Region Competition ap-
proach introduced by Zhu and Yuille in [56]. We will perform a histogram/pdf competition
approach based on the KL distance, which measures the distance between two pdfs. This
segmentation model has been introduced in [27] and has shown promising results. The
method consists in maximizing the KL distance between the pdf inside and outside the
evolving contour, which denes two regions representing the object of interest and the
background.
In what follows, we describe the method with the intrinsic texture descriptor F :
2

+
introduced in (2.5). However, the function F can be replaced by any scalar feature
function such as the gray-level value I of the image or the texture feature based on the
shape operator as done in [27]. Let q
in
be the inside pdf, q
out
the outside one, =
in
be the evolving region and
0
\ =
out
it complementary in the image domain
0
. In
this approach, the texture feature F is considered as a random variable. The set of possible
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 453
outcomes is
+
and the pdfs q
in
and q
out
associated with an observation F for a given
region at a xed moment are dened by:

q
in
(F, ) =
1
||

K(F F( x))d x,
q
out
(F, ) =
1
|
0
\ |

0
\
K(F F( x))d x,
where |.| is the area of the given region. The new KL distance is thus as follows:
KL(q
in
(), q
out
())
=

q
in
(F, )
q
in
(F, )
q
out
(F, )

+q
out
(F, ) log
q
out
(F, )
q
in
(F, )

dF
=

q
in
(F, ) q
out
(F, )

logq
out
(F, ) logq
in
(F, )

dF. (3.1)
Functional (3.1) gives a measure of difference between the pdfs dened inside and outside,
for a xed active contour represented by . One naturally wants to maximize Functional
(3.1) in order to determine two regions with two pdfs as disjoint as possible, represent-
ing the object of interest and the background. Maximizing Functional (3.1) involves the
computation of its derivatives w.r.t the evolving domain , which can be done with the
shape derivative tool [1, 29]. Using the shape derivative tool, the Eulerian derivative in
the direction V of the criterion (3.1) is as follows (see Annex A.1):
< KL

, V >
=

1
||

1
q
out
(F, )
q
in
(F, )
+log
q
in
(F, )
q
out
(F, )
.
.

K(F F(s)) +q
in
(F, )
.
dF

+
1
|
0
\ |

1
q
in
(F, )
q
out
(F, )
+log
q
out
(F, )
q
in
(F, )
.
.

K(F F(s)) +q
out
(F, )
.
dF

< V(s). (s) > ds, (3.2)

where is the unit inward normal to the boundary of the evolving region , ds
its length/area element. According to the Cauchy-Schwartz inequality, the fastest way to
decrease energy KL(()) is obtained by choosing
C

= V. , where is an articial
time and C = , which leads to the evolution equation:
C

=

+
1
||

1
q
out
(F, )
q
in
(F, )
+log
q
in
(F, )
q
out
(F, )
.
.

K(F F(s)) q
in
(F, )
.
dF
+

+
1
|
0
\ |

1
q
in
(F, )
q
out
(F, )
+log
q
out
(F, )
q
in
(F, )
.
.

K(F F(s)) +q
out
(F, )
.
dF +

, (3.3)
454 N. Houhou, J.-P. Thiran and X. Bresson
where the last term , > 0 has been added in the evolution equation in order to
regularize the evolving curve. is the curvature of the contour C and it is derived from
the minimization of the curve length

ds.
3.2. Existence of a minimizing solution for the proposed model (3.1)
In this section, we show the existence of a minimizer of the variational problem (3.1)
associated with the length term by the standard method of calculus of variations and the
space of functions with bounded variation. Our energy Functional is composed of two
terms: the rst one is the Kullback-Leibler functional and the second term is a curvature
regularization term.
We assume that the function F is a function F L

(
0
), where
0
is a regular open
bounded set of R
n
corresponding to the image domain. We dene the set of all image
regions in
0
, i.e. the set of regular open bounded sets of
0
.
The image segmentation problem proposed in (3.1) consists in nding a set
which minimizes the following functional:
E() =

q
in
(F, ) log
q
in
(F, )
q
out
(I, )
+q
out
(F, ) log
q
out
(F, )
q
in
(F, )

. .. .
KL()
+

ds
. .. .
L()
, (3.4)
where the non-parametric pdfs q
in
and q
out
are estimated with a Parzen estimation [43]
as follows:

q
in
(F, ) =
1
||

K(F F( x))d x,
q
out
(F, ) =
1
|
0
\ |

0
\
K(F F( x))d x,
where
0
\ is the complement set of in
0
.
We look for a set which minimizes the region functional (3.4). As it is pointed
out in [1, 29], the optimization of the functional (3.4) is difcult to carry out since the
set of regular domains does not have the structure of a vector space. The variation
of the domain is thus done through a family of homeomorphism transformations T (i.e.
one-to-one with T and T
1
continuous), which allows to differentiate E with respect to
and determine a minimization ow in Section 3.1. Thus, one possible approach to prove
the existence of minimizers for (3.4) is to express (3.4) in term of the transformation T
and look for a minimizer T

. However, we decide to choose another approach to prove

the existence of a minimizer which looks easier. This approach consists, in a rst step
in re-writing the functional F with the characteristic function

of the set , dened as

follows:

(x) =

1 if x ,
0 otherwise,
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 455
and then applying the standard method of the calculus of variations to prove the existence
of a minimizer. Functional (3.4) can be expressed w.r.t.

:
F(

) = KL(

) +L(

), (3.5)
where
L(

) =

0
|

|d x, (3.6)
and

q
in
(F,

) =

0
K(F F( x))

( x)d x

d x
,
q
out
(F,

) =

0
K(F F( x))(1

( x))d x

0
(1

)d x
.
(3.7)
(3.6) is the total variation (TV) norm, which will play an important role in the theorem of
existence. It is dened according to:
Denition 3.1 ([19, 23]). Let and u L
1
(
0
). The TV norm of the function u is
dened as follows:
TV(u) =

0
|u|d x = sup

0
udivd x

,
where = { C
1
0
(
0
,
n
)| || 1, on
0
} and C
1
0
(
0
,
n
) are the continuously differen-
tiable real functions on
0
.
Moreover,
Denition 3.2 ([19,23]). A function u L
1
(
0
) is said to have bounded variation in
0
if its
distributional derivative satises TV(u) < . We dene BV(
0
) as the space of all functions
in L
1
(
0
) with bounded variation. The space BV(
0
) is a Banach space, endowed with the
norm:
u
BV(
0
)
= u
L
1
(
0
)
+ TV(u).
We introduce two important theorems that is already used in Eq. (3.6) and will be used in
our theorem of existence.
Theorem 3.1 ([19, 23]). A set has nite perimeter if and only if the characteristic
function

of belongs to BV(
0
). We have
Per() = TV(

) =

0
|

|d x < ,
456 N. Houhou, J.-P. Thiran and X. Bresson
and
Theorem 3.2 ([19, 23]). Let . If {u
k
}
k1
is a bounded sequence in BV(
0
), then there
exists a subsequence {u
k
j
} of {u
k
} and a function u

BV(
0
), such that u
k
j
u

strongly
in L
p
(
0
) for any 1 p < n/(n 1) and
TV(u

) lim inf
k
j

TV(u
k
j
).
We can now state the theorem of the existence of (at least) one minimizer for (3.5):
Theorem 3.3. Our minimization problem
min

BV(
0
)

KL(

) +L(

, > 0, (3.8)
has a solution in BV(
0
).
Proof. The direct method of the calculus of variations [2, 15, 52] is used:
A) Let {

k
}
k1
be a minimizing sequence of (3.8), i.e.
lim
k
F(

k
) = inf

BV(
0
)
F(

).
B) Since

k
is a sequence of characteristic functions of the sets
k
, then

k
(x)
{0, 1} - a.e. in
0
. A constant M > 0 exists such that

L
1
(
0
)
M, k 1. There-
fore,

k
is a uniformly bounded sequence on BV(
0
). Following Theorem 3.2, a subse-
quence of

k
j
that converges to a function

strongly in L
1
(
0
) exists.
C) Taking a minimizing sequence

k
j

, it is easy to showthat lim

k
j

KL(

k
j
) =
KL(

) since q
in
(

) and q
out
(

) in Eq. (3.7) are continuous w.r.t. the BV(

0
) topology.
Thus, according to Theorem 3.2, we deduce that
F(

) lim inf
k
j

F(

k
j
),
which implies that
F(

) = min

BV(
0
)
F(

),
which means that

is a minimizer of F among characteristic functions

of sets
of nite perimeter in
0
. It also implies the existence of at least one set

, given by
{x
0
|

(x) = 1}, which minimizes (3.4).

Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 457
4. Convexication and fast minimization algorithm
4.1. Convexication of the proposed energy functional
The standard approach to minimize energy functional such as (3.1) (associated with
the length term) with active contour is using the Level Set Method (LSM) introduced by
Osher and Sethian in [40]. However, the LSM is a slow minimization technique that only
determines a local minimum of the energy functional. The authors in [5, 12] introduced a
new paradigm to compute a global minimizer of the active contour energy in a fast way.
More precisely they proposed an equivalent convex formulation of energy (3.1) and then
used fast minimization schemes to reach the optimal solution.
We apply this paradigm in this section to quickly minimize (3.1).
The LSM applied to the minimization of energy (3.1) produces the following PDE:


=

V
KL
+

||

||, (4.1)
where V
KL
is the speed provided by Kullback-Leibler distance (3.1). We assume that V
KL
is
xed. Since || > 0, the steady state of (4.1) is the same as:


= V
KL
+

||
. (4.2)
The variational model that provides (4.2) is also given by:
min
[0,1]

0
V
KL
+ ||.
Finally, we change the notation of the function to avoid any confusion with the LSM. We
will seek the minimum of the convex functional F(u) dened by:
min
u[0,1]

0
V
KL
u + |u|, (4.3)
where

0
|u| = TV(u) is the total variation norm of the function u.
4.2. Fast minimization algorithm based on the Split-Bregman method
In [27], we introduced a minimizing algorithm based on [5]. Although the algorithm
provides relatively fast minimization, it has some limitations. First, the algorithm of [5,27]
is based on a splitting/regularization approach (along the same lines as [3]) to minimize
(4.4). In [27], the authors minimized:
min
u,v[0,1]
F(u, v) =

0
V
KL
v + |u| +
1
2
(u v)
2
, (4.4)
458 N. Houhou, J.-P. Thiran and X. Bresson
where the right-most term enforces u v for sufciently small . Minimization with
respect to u corresponds to solving the ROF model [46],which is done using a gradient
projection method [11], and the minimization for v can be solved using an explicit formula.
However, this scheme slows down as the accuracy increases (i.e., as 0). Second,
the approximate enforcement of the constraint u v has the effect of regularizing the
model. The disadvantage of these regularized schemes is that they "smear" the values of
u near the objects boundaries. This makes the results more dependent on the value of
the cutoff parameter and can eliminate ne segmentation details. In [24], the authors
use the Split-Bregman method to overcome regularization effect. Besides, the proposed
method of [24] has the advantage of being a much more efcient solver for (4.4) since the
Projections algorithm has a linear convergence property whereas experiments show that
the Split-Bregman has a quadratic convergence property (it was shown in [24] that the
Split-Bregman method is actually slightly faster than graph-cuts method). The rst step
consists in introducing a new variable d such that the new system to solve becomes:
min
u,d

0
V
KL
u + |d|, s.t. d = u. (4.5)
We apply the Split-Bregman iterative scheme to minimize (4.5) (see [24] for more
details) which leads to the following unconstrained problem:

(u
k+1
, d
k+1
) = arg min
0u1,d
||d|| +V
KL
u +

2
||d u b
k
||
2
2
,
b
k+1
= b
k
+u
k+1
d
k+1
,
(4.6)
starting from b
k=0
= d
k=0
= u
k=0
= 0. The minimizing solution w.r.t. u in (4.6) holds the
optimality condition:
0 = V
KL
+

2
div(d
k
u b
k
),
u = V
KL
+div(d
k
b
k
).
(4.7)
An approximate solution is given by a Gauss-Seidel iterative method, n 0:

i, j
= d
x,k
i1, j
d
x,k
i, j
b
x,k
i1, j
+ b
x,k
i, j
+d
y,k
i, j1
d
y,k
i, j
b
y,k
i, j1
+ b
y,k
i, j
, (4.8)

i, j
=
1
4

u
k,n
i1, j
+u
k,n
i+1, j
+u
k,n
i, j+1
+u
k,n
i, j+1

V
KL
+
i, j

, (4.9)
u
k+1,n+1
i, j
= max

min{
i, j
, 1}, 0

. (4.10)
Finally, the solution of (4.6) w.r.t. d is given by a soft-wavelet thresholding s.t.:
d
k+1
= si gn(u
k+1
+ b
k
) max

|u
k+1
+ b
k
|
1
, 0

. (4.11)
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 459
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
5. Experimental results
We applied our texture segmentation algorithm to a set of challenging synthetic and
real-world textural images. The synthetic textural images, Figs. 2(a), 2(d), 2(g) were
generated from the Brodatz data set [7]. The natural textural images, Figs. 3(a), 3(d),
3(g),3(j), 3(m) were taken in the Berkeley data set [39].
The rst step is to extract homogeneous regions from the textured images following
the method proposed in Section 2.2. Figs. 2(b), 2(e), 2(h) show the results of the in-
460 N. Houhou, J.-P. Thiran and X. Bresson
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
(m) (n) (o)
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 461
trinsic texture descriptor applied to the synthetic textures. We observed that the images
of the intrinsic texture descriptor are more "homogeneous" than the original images, and
one would like to apply the well-known Chan-Vese model [14] to segment the textures.
However, although the Chan-Vese model can segment some textures such as the textured
zebra, it will fail for more complicated textures. The intrinsic texture feature is much better
represented by a probability density function, as we proposed in our segmentation model.
The quality of the texture descriptor depends on the patch size. The patch should be
large enough to contain enough discriminative information between the object and the
background. However with a larger patch we note a loss of accuracy in the borders. The
unsupervised segmentation process proposed in Section 3 is then applied on the texture
feature and the fast minimization scheme based on the Split-Bregman algorithm is used
(Section 4.2).
Fig. 2, right column, presents the results obtained for synthetics texture. Our model
has managed to distinguish the different texture classes and can deal with multiple object
and non-convex shapes.
As a comparison with the state-of-the-art techniques, we decided to implement the
efcient texture segmentation model of Savig et al. [47], which uses the vectorial Chan-
Vese model [13] and an edge detector function based on Gabor responses as explained in
Section 2.1. We modied their original model by implementing a dual formulation of their
energy functional as done in [5]. Besides, the selected Gabor features are chosen with
a simple selection criteria dened in [28] in order to have the most relevant collection
of Gabor features. Fig. 3 presents the results obtained with our method on the center
column and the model of Sagiv et al. in the right column. Our method manages to capture
with more regularity and precision the object. We notice that in the case of the tiger
image (Fig. 3(a), the fact that some part (part larger than the size patch) are smooth and
not textured mislead the segmentation. For the images with more regular textures, the
segmentation is satisfactory.
Finally we applied our algorithm to color images (Fig. 4). Results are shown in the
right column.
We notice that our segmentation model needs to choose three parameters: , as
explained in Section 4.2 and for the Parzen estimation in Section 3.1. In all experi-
ments, the mean computing time for both the feature extraction and the segmentation is
around 10 seconds for an image 481321 (including the updated estimation of the pdfs),
whereas, for the same image, the texture segmentation model in [47] adapted to the fast
minimization scheme in [5] takes around 4 minutes.
6. Discussion and conclusion
In this paper, we have proposed an unsupervised segmentation method for a large
class of synthetic and natural textures. Several aspects of this difcult problem have been
considered and lead us to use different theories and tools.
The rst step in the segmentation process is to dene and extract texture features. This
is an important and common step for the texture segmentation task. For instance in [44],
462 N. Houhou, J.-P. Thiran and X. Bresson
(a) (b)
(c) (d)
(e) (f)
Rousson et al. used a non-linear diffusion of the structure tensor from which the statis-
tical information was extracted. Although this method provides interesting segmentation
results, it is limited for highly disordered textures or complex natural images. Besides,
this method requires working with N = 5 image features, unlike our model which requires
one image feature. Other texture segmentation algorithms have been developed on Ga-
bor lters. In fact, Gabor lters are much appreciated for texture segmentation because
these lters can extract complex geometry from images. However, methods based on Ga-
bor lters can be time consuming because they require using many image features which
sometimes do not hold much useful information, see for example [13]. The proposed tex-
ture descriptor developed in this paper shows a good quality to represent complex and
natural textures. The proposed texture descriptor has been developed from the Beltrami
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 463
framework and semi-local image information. The Beltrami framework offers a natural ap-
proach to dene a high-dimensional manifold from the image patches and allows to dene
a texture descriptor based on the intrinsic geometry of this geometrical representation. The
proposed texture feature has provided promising results to segment complex textures with
different orientations and scales. We believe that the patch information, which represents
semi-local information, is a natural and efcient way to extract texture information. From
a numerical point of view, the proposed texture descriptor is easy to implement and an
image of size 256 256 can be processed in few seconds. Besides, the proposed texture
descriptor does not use several channels, like in the case of Gabor lters, but only one
function, which makes possible fast texture segmentation. The limitation of this texture
descriptor lies in the patches size. In our implementation, the size is xed and is the same
for all points in the image. However, the size of textures can change in the image. A so-
lution could be to increase the size of patches but we observed that larger sizes produce
less precision in the segmentation process. We will investigate how to change the size of
patches to match the different textures lying in the images.
In the texture segmentation task, the feature extraction step is followed by the segmen-
tation process, which produces the optimal two-phase partition of the image. In this paper,
we were motivated to have a totally user-free algorithm, i.e. an unsupervised segmen-
tation algorithm. We chose to maximize the Kullback-Leibler (KL) distance between the
probability density function (pdf) of the object of interest and the pdf of the background.
The maximum of the KL produces two regions with the pdfs as disjoint as possible. The
KL distance has proven to be a good distance measure between two pdfs, despite a certain
computational cost. This distance, borrowed from Information Theory, has good theo-
retical properties such as invariance w.r.t. illumination changes, which is an important
property to segment natural images. The KL distance has proven to be an effective tool for
other applications. For instance, Freedman and Zhang [20] used the KL measure for object
tracking. Based on a density probability model, the KL ow allows the matching between
the model and the distribution of the current region in the image. Another application is
given by Wang and Vemuri in [53] for Diffusion Tensor-MRI data, where the authors rede-
ned the KL measure for a positive denite 2d-tensor. The segmentation process is done
by minimizing the KL divergence between an inside average tensor and tensors inside the
active contour and an outside average tensor and the tensors outside de active contour. In
contrast with other segmentation methods based on comparison of pdfs such as [30, 42],
we do not use any prior assumptions on the probability distributions. In [42], Paragios
and Deriche based their segmentation process on the Gaussian distribution hypothesis of
the regions to be segmented. In [30], Jehan-Besson et al. used a prior histogram of the re-
gion of interest to perform the segmentation. Our segmentation model, based on a region
competition approach, performs an unsupervised segmentation task. The unsupervised
segmentation is done through an active contour model, which maximizes the KL distance
while regularizing the contour. Active contour models are known to compute a local mini-
mizer of the energy functional and these models are slow numerically. However, we have
developed a fast numerical minimization algorithm, based on the Split-Bregman method,
which determines a global minimizer of the energy functional.
464 N. Houhou, J.-P. Thiran and X. Bresson
Acknowledgments Nawal Houhou was supported by Swiss National Science Foundation
Grant 205320 101621, Xavier Bresson was supported by ONR N00014-03-1-0071 and
ONR MURI subcontract from Stanford University. Nawal Houhou and Jean-Philippe Thiran
would like to thank Prof. Pierre Vandergheynst for interesting discussions on non-local
representation. Xavier Bresson would like to thank Prof. Michael Ng for his invitation at the
Third International Conference in Scientic Computing and Partial Differential Equation.
Finally the authors would like to thank the referees for their constructive comments and
the helpful suggestions.
Appendix
A.1. Shape derivation tool
We remind hereafter the shape derivative tool proposed by Delfour and Zolesio in [16].
The shape derivative tool basically derives a region-based functional in an elegant and
straightforward way. Let us consider a general region-based functional which depends on
an articial time as follows:
F(()) =

f (x, ())d x, (A.1)

that we want to optimize w.r.t . The Gteaux derivative of the functional F in the direc-
tions V is given by:
< F

(), V >= lim

0
F(()) F()

, (A.2)
where F

() :=
F

(()). As it has been shown in [1], the Gteaux derivative can be
expressed as:
< F

, V >=

f

(x, , V)d x

f (s, )(V(s). (s))ds, (A.3)
where is the unit inward normal to the boundary of the evolving region , ds its
length/area element. In the case where we have f (x) = f (x, G
1
(), G
2
()) and
G
1
() =
H
1
()
H
2
()
, G
2
() =
H
3
()
H
4
()
,
we obtain:
f

=
f
G
1
G
1

+
h
G
2
G
2

=
f
G
1

G
1
H
1
H
1

+
G
1
H
2
H
2

.
+
f
G
2

G
2
H
3
H
3

+
G
2
H
4
H
4

.
.
Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour 465
A.2. Gteaux derivative of functional KL
Consider
KL(q
in
(), q
out
())
=

q
in
(I, ) q
out
(I, )

logq
out
(I, ) logq
in
(I, )

dI, (A.4)
where
q
in
(I, ()) = G
1
() =

()
K(I(x) I( x))d x

()
d x
=
H
1
()
H
2
()
,
q
out
(I, ()) = G
2
() =

0
\()
K(I(x) I( x))d x

0
\()
d x
=
H
3
()
H
4
()
,
h() = G
1
() log

G
1
()
G
2
()

+G
2
() log

G
2
()
G
1
()

.
Consequently,
h

=
h
G
1

1
H
2

()
K(I(x) I(s))(V.N)ds

H
1
H
2
2

()
(V.N)ds
.
+
h
G
2

1
H
4

()
K(I(x) I(s))(V.N)ds

H
3
H
2
4

()
(V.N)ds
.
.
We can deduce that
h

=
1
||

1
q
out
(I(x), )
q
in
(I(x), )
+log
q
in
(I(x), )
q
out
(I(x), )

()
(K(I(x) I(s)) +q
in
(I(x), ))(V.N)ds

+
1
|
0
\ |

1
q
in
(I(x), )
q
out
(I(x), )
+log
q
out
(I(x), )
q
in
(I(x), )

()
(K(I(x) I(s)) q
out
(I(x), ))(V.N)ds

.
Finally, the Gteaux derivative of functional (A.4) is:
< KL

, V >=

h

(x, , V)d x.
466 N. Houhou, J.-P. Thiran and X. Bresson
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