Image Anal Stereol 2009;28:93-102
Original Research Paper
IMAGE SEGMENTATION: A WATERSHED TRANSFORMATION
ALGORITHM
L AMIA JAAFAR B ELAID1 AND WALID M OUROU2
1 Ecole
Nationale d’Ingénieurs de Tunis & LAMSIN, Campus Universitaire, BP37, le Belvédère, 1002, Tunis,
Tunisia; 2 Institut National de la Statistique de Tunis & LAMSIN, 70 rue Ech-Cham, BP256, 2000, Tunis, Tunisia
e-mail: lamia.belaid@esstt.rnu.tn, mourou.walid@lamsin.rnu.tn
(Accepted March 27, 2009)
ABSTRACT
The goal of this work is to present a new method for image segmentation using mathematical morphology. The
approach used is based on the watershed transformation. In order to avoid an oversegmentation, we propose
to adapt the topological gradient method. The watershed transformation combined with a fast algorithm based
on the topological gradient approach gives good results. The numerical tests obtained illustrate the efficiency
of our approach for image segmentation.
Keywords: image segmentation, mathematical morphology, topological asymptotic expansion, topological
gradient, watershed transformation.
defined in a domain Ω ∈ R2 is to solve the following
PDE problem
−div (c∇u) + u = v in Ω ,
(1)
on ∂ Ω ,
∂n u = 0
INTRODUCTION
Segmentation is one of the most important problem
in image processing. It consists of constructing a
symbolic representation of the image: the image
is described as homogeneous areas according to
one or several a priori attributes. In the literature,
we can find various segmentation algorithms. The
first method appeared during the sixties and then
different algorithms have been constantly developed.
The purpose of this work is to adapt a new method
for image segmentation using the topological gradient
approach (Masmoudi, 2001) and the watershed
transformation (Soille, 1992).
where c is a small positive constant (called the
conductivity), ∂n denotes the normal derivative and
n is the outward unit normal to ∂ Ω. Classical
nonlinear diffusive approaches are based on the fact
that c is a decreasing function of |∇u|, and takes
its values in the interval ]0, c0 [, where c0 is a given
constant depending on the level of noise. Then, in
Jaafar Belaid et al. (2006), the authors propose to
use the topological gradient as a tool for detecting
edges for image restoration by considering only two
values of c: c0 in the smooth part of the image
and a small value ε > 0 on the edges. For this
reason, classical nonlinear diffusive approaches could
be seen as a relaxation of the topological gradient
method. By enlarging the set of admissible solutions,
relaxation increases the instability of the restoration
process and this could explain why the topological
gradient method is so efficient. Then, using the same
idea, the authors generalize in Auroux et al. (2007)
the topological gradient approach for classification
problems and propose an extension to an unsupervised
classification. A natural application of this idea is the
problem of segmentation: since the identification of
the main edges of the image allows us to preserve
them and smooth the image outside the edges,
then if the conductivity c outside edges is large
The goal of topological optimization is to find
the optimal decomposition of a given domain in
two parts: the optimal design and its complementary.
Similarly in image processing, the goal is to split
an image into several parts, in particular, in image
restoration the detection of edges makes this operation
straightforward. In Jaafar Belaid et al. (2006), the
authors show that it is possible to solve the image
restoration problem using topological optimization
tools. The basic idea was based on the topological
gradient approach used for crack detection (Amstutz
et al., 2005): an image can be viewed as a piecewise
smooth function and edges can be considered as a
set of singularities. To solve restoration problems,
diffusive methods were associated to the topological
gradient for edge detection. More precisely, a classical
way to restore an image u from its noisy version v
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JAAFAR B ELAID L
ET AL :
Image segmentation
enough, the regularized image is piecewise constant
and provides a natural segmentation of the image.
However, this efficient technique introduced by some
of the authors in Jaafar Belaid et al. (2006; 2008),
Auroux and Masmoudi (2006), Auroux et al. (2007),
and Auroux (2008), to solve several image processing
problems like restoration, classification, segmentation,
inpainting and enhancement or denoising, presents
a major drawback: the identified edges are not
necessarily connected and then the results obtained
for the segmentation problem can be degraded,
particularly for complex images. So, the main
idea of this work is to take advantage of the
topological gradient efficiency, to detect the main
contours with an interesting computational cost (the
topological gradient algorithms require only three
system resolutions) and to overcome the drawback of
the topological gradient approach by using a method
giving closed contours.
work is the following. The next section is devoted
to the two basic methods considered in this paper.
First, we review the topological gradient approach
for edge detection. Then, according to Matheron and
Serra (1998) and Serra (1982; 1988), some preliminary
notions and morphological operators which will
be used in this paper, are described. In section
Results, we propose a watershed algorithm based on
the topological gradient method. Numerical results
are presented and discussed, and some numerical
comparisons with other methods are also given in
this section. We end this paper with some concluding
remarks.
The watershed transformation is one of the oldest
segmentation techniques which was initially due to
Beucher and Lantuéjoul (Beucher and Lantuéjoul,
1979; Beucher, 1990). This technique is well known
to be a very powerful segmentation tool. Gray level
images are considered as topographic reliefs, each
relief is flooded from its minima and when two lakes
merge, a dam is built: the set of all dams define the socalled watershed. Such representation of the watershed
simulates the flooding process. Other processes can be
found in the literature, particularly efficient algorithms
for computing watersheds are described in Beucher
(1990) and Soille (1992). One of the advantages
of the watershed transformation is that it always
provides closed contours, which is very useful in
image segmentation. Another advantage is that the
watershed transformation requires low computation
times in comparison with other segmentation methods.
However, using a standard morphological watershed
transformation on the original image or on its gradient,
we usually obtain an oversegmented image. To
decrease the oversegmentation of watershed based
techniques, several approaches have been proposed
in the literature, we can cite for example techniques
based on markers (Meyer and Beucher, 1990), region
merging methods (Vincent and Soille, 1991), scale
space approaches (Jackway, 1999), methods based
on partial differential equations for image denoising
or edge enhancement (Weickert, 2001), wavelet
techniques combined with a watershed transformation
(Jung and Scharcanski, 2005), etc.
In this section, we use the topological gradient as
a tool for detecting edges for image restoration. First,
we recall the principle of the topological asymptotic
expansion (Masmoudi, 2001; Amstutz et al., 2005).
For a given function v in L2 (Ω), we consider the
following problem: find uρ ∈ H 1 (Ωρ ) such that
−div c∇uρ + uρ = v in Ωρ ,
(4)
∂n uρ = 0
on ∂ Ωρ .
The goal of this work is to deal with the
oversegmentation problem, by proposing a new
method based on a fast topological gradient algorithm
and a watershed transformation. The structure of this
The basic idea is as follows. If we insert a crack in a flat
part of the image, nothing happens. But, if we insert
a crack along an edge (strong gradient), the potential
energy decreases. Then, edge detection is equivalent to
METHODS
THE TOPOLOGICAL GRADIENT
APPROACH
Let Ω be an open bounded domain of R2 and
j(Ω) = J(uΩ) be a cost function to be minimized,
where uΩ is the solution to a given PDE problem
defined in Ω. The initial problem reads as follows: for
a given function v in L2 (Ω), we have to find u ∈ H 1 (Ω)
such that
−div (c∇u) + u = v in Ω ,
(2)
on ∂ Ω ,
∂n u = 0
where c is a positive constant.
For a small ρ ≥ 0, let Ωρ = Ω\σρ be the perturbed
domain by the insertion of a crack σρ = x0 + ρσ (n),
where x0 ∈ Ω, σ (n) is a straight crack, and n a unit
vector normal to the crack. The topological sensitivity
theory provides an asymptotic expansion of j when ρ
tends to zero. It takes the general form
j(Ωρ ) − j(Ω) = f (ρ )G(x0, n) + ◦( f (ρ )) ,
(3)
where f (ρ ) is an explicit positive function going to
zero with ρ and G(x0 , n) is called the topological
gradient at point x0 .
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Image Anal Stereol 2009;28:93-102
look for a subdomain of Ω where the energy is small.
So our goal is to minimize the energy norm outside
edges
Z
j(ρ ) = J(uρ ) =
Ωρ
k∇uρ k2 ,
THE WATERSHED TRANSFORMATION
One aim of this work is to show how the use of
mathematical morphology operators can be very useful
in image segmentation. Particularly, we show how the
watershed transformation contributes to improve the
numerical results for image segmentation problems.
We describe briefly in this section the basic notions
and operators we use.
(5)
by considering v0 , the solution to the adjoint problem
−div(c∇v0 ) + v0 = −∂u J(u) in Ω ,
(6)
on ∂ Ω .
∂n v0 = 0
We obtain in the case of a crack σρ (n) with a
boundary condition ∂n u = 0 on ∂ σρ (n), the following
topological asymptotic expansion
j(ρ ) − j(0) = ρ 2 G(x0 , n) + ◦(ρ 2) ,
Let u(x, y) with (x, y) ∈ R2 , be a scalar function
describing an image I. The morphological gradient of
I is defined in Beucher et al. (1993) by
(7)
δD u = (u ⊕ D) − (u ⊖ D) ,
with
G(x0 , n) = −π c(∇u0 (x0 ).n)(∇v0(x0 ) · n)
where (u ⊕ D) and (u ⊖ D) are respectively the
elementary dilation and erosion of u by the structuring
element D.
− π |∇u0 (x0 ) · n| .
2
The topological gradient could be written as
G(x, n) = hM(x)n, ni ,
The morphological Laplacian is given by
(8)
where M(x) is the symmetric matrix defined by
M(x) = −π c
∇u0 (x)∇v0
(x)T
∆D u = (u ⊕ D) − 2u + (u ⊖ D) .
+ ∇v0 (x)∇u0
2
− π ∇u0 (x)∇u0(x)T .
We note here that this morphological Laplacian
allows us to distinguish influence zones of minima
and suprema: regions with ∆D u < 0 are considered
as influence zones of suprema, while regions with
∆D u > 0 are influence zones of minima. Then
∆D u = 0 allows us to interpret edge locations, and will
represent an essential property for the construction of
morphological filters. The basic idea is to apply either
a dilation or an erosion to the image I, depending on
whether the pixel is located within the influence zone
of a minimum or a maximum. For a detailed treatment
of this topic, the reader is referred to Serra (1988).
The Catchment basin C(M) associated to a
minimum M is the set of pixels p of Ω such that a water
drop falling at p flows down along the relief, following
a certain descending path, and eventually reaches M.
The catchment basins of an image I correspond then to
the influence zones of its minima, and the watershed
will be defined by the lines that separate adjacent
catchment basins.
Topological gradient algorithm
–
Initialization : c = c0 .
–
Calculation of u0 and v0 the solutions of the direct
(Eq. 4) and adjoint (Eq. 6) problems .
–
Computation of the 2 × 2 matrix M and its lowest
eigenvalue λmin at each point of the domain Ω.
–
Set
c1 =
–
(11)
(x)T
For a given x, G(x, n) takes its minimal value
when n is the eigenvector associated to the lowest
eigenvalue λmin of M. This value will be considered
as the topological gradient associated to the optimal
orientation of the crack σρ (n). We note that from a
numerical point of view, cracks are simulated by a
small value of c. It is more convenient for numerical
implementation. The restoration algorithm consists
in inserting small values of c (cracks) in regions
where the topological gradient is smaller than a given
threshold α < 0. These regions are the edges of the
image. The algorithm is as follows.
(10)
ε if x ∈ Ω, λmin < α < 0, ε > 0
c0 elsewhere.
Several algorithms have been proposed for the
computation of watersheds and the most commonly
used are based on an immersion process analogy.
Let us express this immersion process more formally
according to Soille (1992): we consider hmin and hmax
the smallest and the largest values taken by u. Let
Th = {p ∈ Ω, u(p) ≤ h} be the threshold set of u at
level h. We define a recursion with the gray level
(9)
Compute u1 , the solution of Eq. 2 with c = c1 .
We refer the reader to Jaafar Belaid et al. (2008),
for some theoretical and numerical comparisons with
conventional restoration methods.
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[
Image segmentation
image a Gaussian noise (σ = 20). The reconstructed
image is shown in Fig. 1c: the topological gradient
method for the restoration process was applied with
c0 = 1, ε = 10−3 , and α = −70. Finally, we give in
Fig. 1d, the edges of the reconstructed image. To obtain
the restored image, the topological gradient algorithm
requires only 3 system resolutions for calculating u0 ,
v0 and the restored image u1 given respectively by
Eqs. 4, 6, and 2. For a better edge preservation,
one has to threshold the topological gradient with
a small enough coefficient. In the other case, if
the thresholding coefficient is set to a large value,
then the edges obtained will be thick, leading to
an oversmoothing and a loss of an important edge
information and then a degradation of the restored
image. Finally, to speed up the computations, a spectral
method based on the discrete cosine transform has
been used for the resolution of the direct and adjoint
problems. Since the coefficient c is equal to a constant
c0 except on edges, then the discrete cosine transform
is a good preconditioner for the conjugate gradient
method. The complexity of the restoration algorithm
is O(N log N) where N is the number of pixels of
the image. Some comparisons about the computation
times with other classical methods are presented in
Jaafar Belaid et al. (2008).
h increasing from hmin to hmax , in which the basins
associated with the minimum of u are successively
expanded. We consider Xh the union of the set of basins
computed at level h. A connected component of the
threshold set Th+1 at level h + 1 can be either a new
minimum, or an extension of a basin in Xh . Finally,
by denoting by minh the union of all regional minima
at level h, we obtain the following recursion which
defines the watershed by immersion
Xhmin = Thmin ,
∀h ∈ [hmin , hmax − 1] , Xh+1 = minh+1 ∪ IZTh+1 (Xh) ,
with IZTh+1 =
ET AL :
izTh+1 (Xhi ), where k is the number of
i=1
minima of I, and izTh+1 (Xhi ) is defined by
izΩ (Yi ) = {z ∈ Ω, ∀k 6= i, dΩ (z,Yi ) ≤ dΩ (z,Yk )} .
(12)
The set of the catchment basins of a gray level image I
is equal to the set Xhmax . At the end of this process, the
watershed of the image I is the complement of Xhmax
in Ω.
It is well known that the main problem of this
method is that the images we consider are often
noisy, which implies that we have a lot of local
minima and this leads to an oversegmentation. We
propose in this paper, a new method for decreasing
the oversegmentation of standard watershed based
techniques. Our method is based on the topological
gradient approach. The topological gradient has here
the interesting property to give more weight to
the main edges, it provides a more global analysis
of the image than the Euclidean gradient or the
morphological gradient, so results are less sensitive
to noise as we show it in the numerical applications
section.
SEGMENTATION USING A CLASSICAL
WATERSHED TRANSFORMATION
The goal of this section is to present numerical
tests for the segmentation problem using mathematical
morphology tools. The approach used in this work
is based on the watershed transform. One should
remark that we can either define the watershed of the
function u or of its gradient: the difference between
the two definitions is that in the first case we obtain
the influence zones of the processed image, while
the second case gives the image edges. In both
cases, the watershed gives an oversegmentation and
to avoid this drawback, a markers technique can be
used. We propose in this section to give numerical
results based on this classical method according to
the work of Beucher (1990), in which the author has
proposed to use both influence zones and minima of
the filtered image as marker criteria. Fig. 2 illustrates
this approach. We have considered the same original
image as previously. Fig. 2a shows the watershed of
the image and Fig. 2b shows the watershed of its
gradient. The oversegmentation is clearly seen. We
should mention here, that the numerical tests given by
Fig. 2b give a segmented image with 1905 regions for
a computational time of 550 s. This oversegmentation
can be in a first step corrected by applying a
morphological filter. Fig. 2c shows the watershed of
RESULTS
TOPOLOGICAL GRADIENT AND EDGE
DETECTION
We consider in this section the problem of
denoising an image and preserving features such
as edges. According to the previous section, the
topological asymptotic analysis provides the location
of the edges as they are precisely defined as the most
negative points of the topological gradient. Fig. 1
shows the results obtained by the topological gradient
algorithm. The image processed given by Fig. 1a is a
256 × 256 gray level image and represents some rice
grains. Fig. 1b is obtained by adding to the original
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Image Anal Stereol 2009;28:93-102
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Fig. 1. Restoration and edge detection processes by the topological gradient approach: original image (a), (b)
is the noisy image obtained by adding a Gaussian noise (σ = 20), (c) is the restored image, and (d) shows the
detected edges.
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Fig. 2. Classical segmentation technique using morphological operators: (a) is the watershed of the original
image and (b) is the watershed of its gradient, (c) is the segmentation result of the filtered image, (d) shows the
minima of the original image and (e) are the minima of the filtered image, (f) are the new influence zones, (g)
shows the image (e) superposed on the influence zones, and finally (h) is the segmented image.
in Fig. 2g the new influence zones superimposed on
minima obtained in Fig. 2e. Finally, Fig. 2h shows
the contours obtained after the segmentation process
using this approach. Some criticisms can be expressed
according to this approach: one can remark that the
detection of the rice grains is not perfect. In fact, some
of them are badly detected and others are completely
eliminated. These drawbacks can provide either from
the choice of the marker criterion used to detect the
grain contours or the choice of the influence zones
as markers. It can also come from the choice of
the watershed transform. However, some additional
morphological operators can be used for overcoming
such drawbacks, see for example Decencière et al.
(2005).
the filtered image: the number of regions segmented
is attenuated (868 regions) for a computational
time of 775 s, but the segmentation result remains
unacceptable. However, as the oversegmentation is due
to the fact that we obtain a lot of minima, and the
use of morphological filters can only suppress some
of them, then another way to act on these minima
is to apply the swamping approach, by imposing
markers for new minima. Fig. 2d shows the minima
of the original image and Fig. 2e shows the minima
of the filtered one. Clearly, the number of minima is
attenuated. To obtain the wanted final contours which
derived from the watershed of the gradient modulus,
we have to compute the watershed of the swamping
of the gradient modulus of the filtered image. Fig. 2f
shows the new influence zones obtained which will be
used as markers. For better visualization, we present
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JAAFAR B ELAID L
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WATERSHED
ET AL :
Image segmentation
image). We recall here that the topological gradient
algorithm is solved with a O(N log N) complexity and
that the watershed algorithm runs in a linear time with
respect to the number N of pixels of the image.
We propose in this part a new algorithm for the
segmentation problem which combines the topological
gradient approach with a watershed transformation.
Our goal is to improve the segmentation results by
considering the second kind of watershed transforms
(the watershed of the image gradient) previously
defined, using a topological gradient instead of
the morphological gradient classically used with
watersheds. As mentioned previously, the topological
gradient is much less sensitive to noise and small
variations of the image, than the Euclidean and
morphological gradients. This is due to the fact that the
topological gradient evaluates in a global way whether
a pixel is a part of an edge or not, compared to the
Euclidean gradient which has more local properties.
On the other hand, as the morphological gradient
corresponds in a certain way to the modulus of the
Euclidean gradient, then it will be easy to conclude that
the topological gradient provides the best identification
of the main edges of the processed image, and the
oversegmentation obtained in the previous numerical
results, will be clearly attenuated. Fig. 3 shows the
three gradients: Fig. 3a is the topological gradient
and Figs. 3b-c are respectively the Euclidean gradient
and the morphological gradient. The edges defined
by the Euclidean and the morphological gradients are
very accentuated in comparison with those defined
by the topological gradient. This thickness of edges
provides a loss of edge information. This was our first
motivation.
Unfortunately, as shown in Fig. 4a, some unwanted
crest lines remain at the end of the segmentation
process. To better illustrate these unwanted regions,
we give in Fig. 4d the detected contours after
the segmentation process: we have more segmented
regions than what we should obtain. In order to
improve our segmentation process and to attenuate the
number of these unwanted crest lines, we propose to
accentuate the smoothing on both sides of an edge. The
idea until now was to choose c ≈ 1, but this choice
is not so appropriate for the segmentation problem.
We propose then, an improvement of the previous
algorithm as follows we consider the following partial
differential equation
−div c(ε )∇uρ + uρ = v in Ωρ ,
(13)
∂n uρ = 0
on ∂ Ωρ ,
with c(ε ) = ε on the edges and c0/ε elsewhere, ε > 0
and c0 is a given positive constant.
In comparison with the previous section, the
topological gradient and the general algorithm remain
unchanged since the new algorithm proposed is based
on the same restoration algorithm but according to
Eq. 13. The edge set is still given by the thresholding
λmin , but from numerical point of view, by choosing
large values of c, our partial differential equation is
nearly equivalent to ∆u = 0 which will provide a really
smooth image outside edges leading to a considerable
attenuation of segmented regions.
Since the topological gradient is the best tool for
preserving the most important edges and eliminating
all the other insignificant ones, then the first idea
was to replace the morphological gradient by a
topological gradient in order to minimize the set of
minima of I, leading to better segmentation results.
Our algorithm is then composed of two different
and separate steps: the first one consists of detecting
the main edges of the image using the topological
gradient restoration process. Then, the second step
consists of applying the watershed algorithm using
the topological gradient determined in the first step,
instead of the morphological gradient classically
used in watershed algorithms. Fig. 4 illustrates this
segmentation process. The first results obtained are
very promising. Fig. 4a shows the segmented image
obtained with this approach. It should be noted that
we obtain 587 homogeneous regions and that our
new algorithm requires around 1500 CPU seconds.
The computation times are then more or less similar
between the two methods (this is due to the fact that
the computation time may depends of the processed
The numerical results of this approach are
illustrated in Fig. 4, by considering three values of
the coefficient c. Fig. 4b is obtained with c = 25.
It should be noted that we obtain 153 homogeneous
regions and our algorithm requires around 340 CPU
seconds. Fig. 4c is obtained with c = 50 and gives
only 120 homogeneous regions with a computational
time of 290 seconds. Finally, for a better illustration
of the segmented regions, we represent in Figs. 4d-f
the detected contours after our segmentation process.
Moreover, to better illustrate the efficiency of our
method, we present other numerical results by
considering more complex images. The two real
images presented in this paper have various difficulties
(curves, circles, straight lines, etc.). The first image
is a 302 × 280 fruit-basket gray level image and the
second one is a 399 × 246 gray level image which
represents a road scene. We present in Fig. 5 the two
original images and the reconstructed edges by the
topological gradient approach, after adding a Gaussian
noise (σ = 20) to the original images.
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Image Anal Stereol 2009;28:93-102
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Fig. 3. The topological gradient (a), the Euclidean gradient (b), and the morphological gradient (c).
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Fig. 4. Segmentation results using a watershed transformation applied to the topological gradient: the images
(a), (b) and (c) are obtained by the proposed approach with c = 1, c = 25 and c = 50 and the images (d), (e) and
(f) show respectively the detected contours after the segmentation process.
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Fig. 5. Examples of real images: (a) is a fruit-basket original image, (b) shows the detected edges according to
the topological gradient approach, (c) is a road scene original image, and (d) shows the detected edges using the
topological gradient approach.
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Fig. 6. Segmentation results using a watershed algorithm combined with the topological gradient approach: (a)
is the segmented fruit-basket image with c = 1, (b) is the same segmented image with c = 50, (c) is the segmented
road scene image with c = 1, and (d) is the same segmented image with c = 50.
Fig. 6 shows the segmentation result of the two
previous images, according to our new algorithm: a
coupled method for image segmentation based on
a watershed algorithm and the topological gradient
approach. The results obtained show clearly the
efficiency of the topological gradient for extracting
features from complex images and for reducing
drastically the oversegmentation.
object. This was the first idea of classical snakes and
active contour models proposed by Kass et al. (1987).
Our numerical tests are based on a level set method
proposed by Caselles et al. (1997) and given by the
following evolution equation
The results shown in Figs. 6a-c are obtained with
c = 1, we can attenuate the number of identified
regions after the segmentation process by choosing
larger values of the coefficient c, but this will depend
of what we want to segment and what we want to
obtain at the end of the segmentation process: if one
would obtain a maximum of details then it suffices
to take c = 1, otherwise, one should consider larger
values of c which gives a considerable attenuation of
the segmented regions. Figs. 6b-d show the numerical
results of the segmentation process with c = 50. We
can clearly observe an attenuation of the segmented
regions such that the main edges of the two images are
conserved.
COMPARISON WITH AN ACTIVE
CONTOURS MODEL
Finally, due to a large number of approaches
used for the segmentation problem, it clearly appears
that it is important to compare our experimental
results with methods already proposed in the literature.
Particularly, we propose to compare our method
with an active contour model based on the level set
approach, which is well known to be an efficient
method, extensively used in many applications over the
last decade. The basic idea in active contour models is
to evolve a curve subject to constraints from a given
image, in order to detect different objects in that image.
To achieve this goal, we start with a curve around
the object to be detected, the curve moves toward its
interior normal and has to stop on the boundary of the
100
∂u
∇u
= g (|∇I|) div
+ α |∇u|
∂t
|∇u|
+ h∇g, ∇ui, in (0, ∞) × Ω,
with boundary and initial conditions given by ∂ u/∂ n = 0
on (0, ∞) × ∂ Ω and Φ(0, x, y) = Φ0 (x, y) in Ω, α ≥
0 is a positive given coefficient, g (|∇I|) is an edge
detector function defined in our numerical tests by
g(s) = 1/(1 + s2), and φ0 represents the initial level set
function. We refer the reader to Aubert and Kornprobst
(2001) for more details about geodesic active contours
and level set methods. Fig. 7 shows the numerical
results obtained. We should mention that in order
to obtain the final contour illustrated in Fig. 7e, the
algorithm requires 500 iterations and a computational
time of 78 seconds. We present in Fig. 7 some solutions
as time evolves: during the evolution, the initial curve
is shrinking and stopping as soon as it is close to an
object boundary and spilling in order to detect the
others objects.
One can easily detect some drawbacks of this
method. Fig. 7e shows that the interior of the disk is
not segmented: once the curve has detected a contour,
it stops. We represent in Fig. 7f, the same image
segmented using our new approach. Our algorithm
requires 38 CPU seconds and the drawback seen before
is overcome. We also tested this method on a real
151 × 151 gray level image. Fig. 8 shows the ability
of the model to capture the main object of the image,
which can be important for object tracking applications
for example. On the other case, Fig. 8f is the result
of the segmentation process by our algorithm, and
shows more homogeneous regions detected. This can
be interpreted by an oversegmentation in comparison
Image Anal Stereol 2009;28:93-102
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Fig. 7. Segmentation results of a synthetic image using an active contour model: different iterations are displayed
from (b) to (e) , and (f) is the segmented image using our new approach.
with active contours model to capture an object, but
on the other hand, these different regions extracted can
be particularly helpful for color processing exploration
for example. We can conclude that the results obtained
by the two methods are just different and depend of the
objects we want to detect and the kind of applications
we want to explore.
Inspired by the work of Meyer and Beucher
(1990), we propose in a forthcoming paper
to perform our results using marker criteria.
More precisely, as the edges are detected in
regions where the topological gradient is the most
negative, then it suffices to extract some points
which belong to the edge set and then use these
points as a selected marker set. We also intend to
extend this work to color image segmentation and
three dimensional segmentation.
CONCLUSION
We have presented in this work a new approach
for the segmentation problem taking advantage of
the topological gradient approach and the watershed
transformation. The numerical results obtained are
very promising and the proposed algorithm has many
advantages:
ACKNOWLEDGMENTS
We are grateful to the unknown reviewers for
their valuable comments. We also thank Professor Jean
Serra for his helpful suggestions.
–
First, the algorithm cost is interesting.
–
Second, as the topological gradient provides a
global analysis of the image then the almost
unwanted contours due to the noise added to a
given image can be significantly reduced by our
approach.
Amstutz S, Horchani I, Masmoudi M (2005). Crack
detection by the topological gradient method. Control
Cybern 34:119-38.
Third, the experimental results show that the
oversegmentation problem, which usually appears
with the watershed technique, can be attenuated,
and the segmentation results can be performed
using the topological gradient approach.
Auroux D, Masmoudi M (2006). A one-shot inpainting
algorithm based on the topological asymptotic analysis.
Comput Appl Math 25:1-17.
–
–
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