Invariant Shape Prior Knowledge for an Edge-based Active Contours

Invariant Shape Prior for Active Contours

Mohamed Amine Mezghich, Slim M’Hiri and Faouzi Ghorbel

GRIFT Research Group, CRISTAL Laboratory,

Ecole Nationale des Sciences de l’Informatique, ENSI,

Campus Universitaire de la Manouba, 2010 Manouba, Tunisia

Keywords:

Active Contours, Shape Prior, Phase Correlation, Rigid Transformations, Invariant Descriptors.

Abstract:

In this paper, we intend to propose a new method to incorporate geometric shape prior into an edge-based

active contours for robust object detection in presence of partial occlusions, low contrast and noise. A shape

registration method based on phase correlation of binary images, associated with level set functions of the

active contour and a reference shape, is used to deﬁne prior knowledge making the model invariant with respect

to Euclidean transformations. In case of several templates, a set of complete invariant shape descriptors is used

to select the most suitable one according to the evolving contour. Experimental results show the ability of the

proposed approach to constrain an evolving curve towards a target shapes that may be occluded and cluttered

under rigid transformations.

1 INTRODUCTION

Active contours (Kass et al., 1988; Cohen, 1991; Xu

and Prince, 1997; Malladi et al., 1995; Caselles et al.,

1997; Chan and Vese, 2001) have been widely used

in image segmentation. One can classify them into

two families : The boundary-based approach which

depends on an edge stopping function to detect ob-

jects and the region-based approach which is based

on minimizing an energy’s functionnal to segment ob-

jects in the image. Given that these classical active

contours are intensity-based models, there is still no

way to caracterize the global shape of an object. Es-

pecially in presence of occlusions and clutter, all the

previous models converge to the spurious contours re-

sulted from large local gradient nearby. Many works

incorporate shape prior into the active contour mod-

els. Leventon et al., (Leventon et al., 2000) associ-

ated a statistical shape model to the geodesic active

contours (Caselles et al., 1997). At each step of the

surface evolution, the maximum a posteriori position

and shape are estimated and used to move globally the

surface while local evolution is based on image gra-

dient and curvature. Chen et al., (Chen et al., 2001)

deﬁned an energy’s functional based on the quadratic

distance between the evolving curve and the average

shapes of the target object after alignment. This en-

ergy is then incorporated into the geodesic active con-

tours. Bresson et al., (Bresson et al., 2003) extended

(Chen et al., 2001) approach by integrating the sta-

tistical model of shape proposed by (Leventon et al.,

2000) in the energy functional. Fang and Chan (Fang

and Chan, 2007) introduced a statistical shape prior

into the geodesic active contour to detect partially

occluded object. To speed up the algorithm, an ex-

plicit alignment of the shape prior model and the cur-

rent evolving curve is done to calculate pose parame-

ters. Foulonneau et al., (Foulonneau et al., 2004) in-

troduced a geometric shape prior into a region-based

active contours (Chan and Vese, 2001) based on the

Legendre moments of the characteristic function and

in (Charmi et al., 2010), the authors deﬁned a geomet-

ric shape prior for the region-based active contours

after alignment of the evolving contour and the ref-

erence shape. It’s well know that shape priors based

on contour alignment methods force these approaches

to segment only single object in the image and go

without the contribution of level set, i.e. its ability to

segment multiple objects at once, see (Bresson et al.,

2003). Besides, contour alignment methods are not

adapted to estimate the rigid transformation parame-

ters in the case of objects with holes (which often oc-

curs in medical imagery like MRI brain’s white mat-

ter). This justiﬁes the use of the registration meth-

ods instead of those based on contours alignment.

In this work, we focus on adding a new geometric

shape prior to an edge-based active contours (Malladi

et al., 1995) based on phase correlation. At the begin-

ing, we assume that the shape of reference is known

in advance like the work of (Zhang and Freedman,

454

Mezghich M., M’Hiri S. and Ghorbel F..

Invariant Shape Prior Knowledge for an Edge-based Active Contours - Invariant Shape Prior for Active Contours.

DOI: 10.5220/0004692304540461

In Proceedings of the 9th International Conference on Computer Vision Theory and Applications (VISAPP-2014), pages 454-461

ISBN: 978-989-758-004-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

2003; Cremers et al., 2003; Foulonneau et al., 2004;

Chan and Zhu, 2005). Then we will introduce our ap-

proach to select the best shape according to the evolv-

ing contour in situation where the shape of reference

is unknown and many templates are available. The

improved model can retain all the advantages of the

level set approach and have the additional ability of

being able to handle the case of images with multi-

ple objects under partial occlusions and noise. The

remainder of this paper is organized as follows : In

section 2, we will brieﬂy recall the used shape regis-

tration method based on phase correlation. Then, in

section 3, the proposed shape prior will be presented.

We assume that at this stage, the shape of reference

is known in advance. In section 4, we present our

method to choose the suitable shape in case of many

references. Experiments will be presented and com-

mented in order to study the robustness of the model

in section 5. Finally, we conclude the work and high-

light some possible perspectives in section 6.

2 SHAPES REGISTRATION

Before incorporating the prior knowledge into the ac-

tive contours model, the reference shape or the sta-

tistical model must be ﬁrst transformed to best match

the current shape. We adopt the well know method

of phase correlation in Fourier space that is appro-

priate to estimate the translation vector and for es-

timating the rotation angle and the scaling factor,

we use the proposed method of phase correlation in

Fourier-Mellin space. We recall this method which

is based on the Analytical Fourier-Mellin Transform

(AFMT), see (M’Hiri et al., 2012) for a detailed de-

scription. A comparative study with other global reg-

istration methods is presented in (Sellami and Ghor-

bel, 2012). Let f (r, θ) be a polar representation of

the image with the radius r according to the center of

gravity of the image to offset translation and θ the an-

gle according to the horizontal. It was pointed out in

(Ghorbel, 1994) that the crucial numerical difﬁculties

in computing the Fourier-Mellin transform of an im-

age might be solved by using the Analytical Fourier-

Mellin Transform (AFMT) given by

(k, v) =

2π

+∞

2π

f (r, θ)r

σ−iv

−ikθ

dθ,

(1)

where σ > 0 is a ﬁxed and strictly positive real num-

ber. Since no discrete transform exists, three approxi-

mations of the AFMT have been designed : the direct,

the cartesian and the fast algorithm, see (Derrode and

Ghorbel, 2001). Let f

re f

and f

be two binary im-

ages associated respectively with level set functions

re f

and φ. Denote by M

σ,φ

re f

and M

σ,φ

the AFMT

of respectively f

re f

and f

with the same value of σ.

re f

and f

have the same shape if and only if there is

a similarity (α

, β

) ∈ G = (R

∗

, S

) such that

∀(r, θ) ∈ G, f

(r, θ) = f

re f

(

, θ − β

(2)

The action of planar similarities in Fourier-Mellin

space leads to

σ,φ

(k, v) = α

σ−iv

−ikβ

σ,φ

re f

(k, v),

(3)

By calculating the normalized cross-spectrum, only

information on phase difference will be preserved

Φ(k, v) =

∗

σ,φ

re f

(k, v)M

σ,φ

(k, v)

∗

σ,φ

re f

(k, v)||M

σ,φ

(k, v)|

= α

−iv

−ikβ

(4)

Phase correlation of two images represented respec-

tively by f

re f

and f

is deﬁned as

T f m

(α, β) =

+∞

∑

Φ(k, v)α

ikβ

dv,

(5)

We can deduce the images transformation’s parame-

ters (α, β) by estimating (α

, β

) that maximize the

correlation function C

T f m

. Having the parameters of

rigid transformation between the two binary images,

we perform the registration of the image f

re f

accord-

ing to the following formula (Chan and Zhu, 2005)

reg

re f

(x, y) =

α f

re f

(

(x−a)cos θ+(y−b)sin θ

−(x−a)sin θ+(y−b)cos θ

(6)

where (a, b) represents the translation vector, θ the

rotation angle and α is the scaling factor. On the re-

sulting image (Fig.1), the pixels in black (resp. white)

correspond to positive areas (resp. negative) of the

signed distance map which is associated to level set

function. The image on the right of Fig.1 shows the

product function given by

prod

(x, y) = f

reg

re f

(x, y) · f

(x, y),

(7)

By construction, the function f

prod

is negative in

the areas of variability between the two binary images

(occlusion, clutter, missing parts etc.) whereas in pos-

itive regions, the objects are similar. Thus, in what

Figure 1: Left : f

reg

re f

, Middle : f

, Right : f

prod

InvariantShapePriorKnowledgeforanEdge-basedActiveContours-InvariantShapePriorforActiveContours

455

follows, we propose to update the level set function φ

only in regions of variability between shapes to make

the evolving contours overpass the spurious edges and

recover the desired shapes of objects. This property

recalls the Narrow Band technique used to acceler-

ate the evolution of the level set functions (Malladi

et al., 1995). In many works like those of (Foulonneau

et al., 2004; Foulonneau et al., 2006) and (Leventon

et al., 2000; Fang and Chan, 2006; Fang and Chan,

2007), all the pixels in the image, called n, are in-

voked in the process of incorporating the shape prior

which may increase the calculus complexity (O(n

))

of the model. In our work, only pixels of the region

of variablity, called k, are invoked. Generaly k << n,

hence the calculus complexity (in our case O(k

)) is

reduced. This remark will be more developed in sec-

tion 5.

3 THE PROPOSED SHAPE PRIOR

Geometric active contours are iterative segmentation

methods which use the level set approach (Osher and

Sethian, 1988) to determine the evolving front at each

iteration. Working with this approach makes it possi-

ble to manage topology changing of the contour like

splitting and merging, and consequently the segmen-

tation of an arbitrary number of objects in the image.

In (Malladi et al., 1995), the level set method is used

to model the shape of objects with an evolving front.

The evolution’s equation of the level set function φ,

which is the embedding function associated to the ac-

tive contour, is

+ F|∇φ| = 0,

(8)

F is a speed function of the form F = F

+ F

(K)

where F

is a constant advection term equals to (±1)

depending of the object inside or outside the initial

contour. The second term is of the form −εK where

K is the curvature at any point and ε > 0, is a constant

real. To detect objects in the image, the authors pro-

posed the following function which stops the level set

function’s evolution at the object boundaries

g(x, y) =

1+|∇G

∗ f (x,y)|

, p >= 1

(9)

where f is the image and G

is a Gaussian ﬁltre with

a deviation equals to σ. This stopping function has

values that are closer to zero in regions of high image

gradient and values that are closer to unity in regions

with relatively constant intensity. Hence, the discrete

evolution equation is

n+1

(i, j)−φ

(i, j)

∆t

= −g(i, j) F(i, j) |∇φ

(i, j)|,

(10)

It’s obvious that the evolution is based on the stop-

ping function g which depends on the image gradi-

ent. That’s why this model leads to unsatisfactory re-

sults in presence of occlusions, low contrast and even

noise. To make the level set function evolves in the

regions of variability between the shape of reference

and the target shape, we propose the new stopping

function as follows

shape

(x, y) =



0, i f ψ(x, y) >= 0,

sign(φ

re f

(x, y)), else,

(11)

where ψ(x, y) = φ(x, y) · φ

re f

(x, y), φ is the level set

function associated to the evolving contour, while φ

re f

is the level set function associated to the shape of ref-

erence after registration. As it can be seen, the new

proposed stopping function only allows for updating

the level set function in the regions of variability be-

tween shapes. In these regions g

shape

is either 1 or -1

because in the case of partial occlusions, the function

is equals to 1 in order to push the evolving curve in-

ward (deﬂate) and in case of missing parts, this func-

tion is equals to -1 to push the contour towards the

outside (inﬂate). This property recalls the Balloon

snake’s model proposed by Cohen in (Cohen, 1991)

in which the direction of evolution (inﬂate or deﬂate)

should be precised from the beginning. In our work,

the direction of evolution is hundled automatically

based on the sign of φ

re f

. The total discrete evolu-

tion’s equation that we propose is

n+1

(i, j)−φ

(i, j)

∆t

−(w g(i, j) + (1 − w) g

shape

(i, j)) F(i, j)|∇φ

(i, j)|,

(12)

where w is a weighting factor between the image-

based force and the knowledge-driven force. To il-

lustrate the ability of the proposed shape prior to con-

strain geometrically an active contour, we show the

evolution of the contour under the inﬂuence of the

proposed shape prior term only (i.e. w = 0). We

present in Fig.2 an example of successive evolutions

between several shapes of different topologies. We

have chosen for initial curve a green square. The ﬁrst

shape of reference is a tree leaf. An intermediate step

in this evolution is shown by the ﬁrst row and the ﬁ-

nal curve is presented by the last column. This last

conﬁguration of the contour is used as an initial curve

for the next experiment by taking the image of the

left and right ventricles of the heart as a reference and

then ﬁnally in the same way by taking the shape of a

pen and a ring as shapes of reference.

4 CASE OF MANY REFERENCES

In presence of many templates, we have to choose the

VISAPP2014-InternationalConferenceonComputerVisionTheoryandApplications

456

Figure 2: Curve evolution under the proposed shape prior.

most suitable one according to the evolving curve. In

(Fang and Chan, 2006), a new statistical shape prior is

proposed for robust object detection. This prior model

is capable to handle multiple shape states of the ob-

ject. A Gaussian Mixture Model (GMM) is used to

estimate the data distribution in the feature subspace

and a Bayesian classiﬁer is used to assign the cur-

rently detected object to the most similar shape clus-

ter. A shape prior is then constructed by using the sta-

tistical properties of that cluster to constrain the sub-

sequent curve evolution. This model requires a pre-

liminary step which consist in aligning the training

data. Besides PCA must be applied. In (Charmi et al.,

2009), the authors proposed a geometric approach to

add prior information to the snake model in case of

many references. A set of complete and locally sta-

ble invariants to Euclidean transformations, based on

Fourier transform of the contour, is used to deﬁne new

force which makes the snake overcome some well-

known problems. Motivated by (Charmi et al., 2009),

we propose in this section to use as criterion the dis-

tance between a complete family of similarity invari-

ant features based on the AFMT suggested in (Ghor-

bel, 1994; Derrode and Ghorbel, 2001). This family

can be easily rewritten and applied to any strictly pos-

itive value σ in the following way, ∀(k, v) ∈ (Z, R):

(k, v) = M

(0, 0)

−σ+iv

−ikArg(M

(1,0))

(k, v),

(13)

Completeness is recognized as an important criterion

for full shape discrimination and reconstruction from

features. Since this invariant set is convergent for

square summable functions, it can be shown in (Ghor-

bel, 1994) that the following function deﬁnes a true

mathematical distance between shapes :

, I

) = (

+∞

−∞

∑

k∈Z

(k, v) − I

(k, v)|

dv)

(14)

f and g represent two gray-scale objects. Due to nu-

merical sampling and approximation, we never have

exactly zero and the value of the distance is used for

the quantiﬁcation for the similarity between objects,

regardless of their pose, orientation and size in the

image.

5 EXPERIMENTAL RESULTS

In this section, the proposed model with shape prior

will be applied to the segmentation problem. Conse-

quently the model will evolves under both data and

prior terms. In order to reduce the computational

complexity and to have a good estimation of the pa-

rameters of the rigid transformation as in (Foulonneau

et al., 2004; Fang and Chan, 2007), we ﬁrst evolve the

active contour without shape prior until convergence

(i.e. w = 1). This ﬁrst result provides an initializa-

tion for the model with prior knowledge. To promote

the convergence to the target shape, we generally give

more weight to prior knowledge (i.e w <= 0.5). In the

next experiment, we compare our model to that pro-

posed by Fang and Chan in (Fang and Chan, 2007).

The shape of reference is provided by image (a). Im-

ages (b) and (c) represent repectivelly the results ob-

tained by our model and the model of Fang and Chan.

It is visually clear that in region of variability (occlu-

sion), the proposed approach gives a better result. By

Figure 3: (a) : The reference, (b) : Result obtained by our

model, (c) : Result obtained by Fang and Chan model, (d)

: Segmentation without shape prior, (e) and (f) : Segmenta-

tion with the proposed model.

the second row, we handle a situation where the object

to be detected is no longer connected (due to missing

parts or hole). In such situations, methods based on

contours alignment does not allow to estimate the pa-

rameters of the rigid transformation. Thanks to regis-

tration by phase correlation, our model can handle this

case and consequently the detection of the target ob-

ject. In order to illustrate how the calculus complex-

ity can be reduced by updating the evolving level set

function only in the regions of variability between the

reference and the target objects, we compute the exe-

cution time for the image (b) until convergence with

InvariantShapePriorKnowledgeforanEdge-basedActiveContours-InvariantShapePriorforActiveContours

457

Table 1: Execution time depending on the image size.

128x128 256x256 512x512

Our model 9.743 30.671 151.998

Fang’s model 10.157 41.153 299.3

different sizes (128x128, 256x256 and 512x512). We

recall that for the model proposed by (Fang and Chan,

2007), the proposed shape prior is as follows :

n+1

(i, j) = φ

(i, j) + ∆t(φ

∗

(i, j) − φ

(i, j)),

(15)

where φ

∗

is the shape model. We set the same values

for the weighting factor and time step and we consider

that the shape of reference (the shape model) is given.

The table below presents the needed execution time

(in seconds) to have satisfactory results. It’s clear that

for an important image size, our approach needs less

iterations to reach good results. In Fig. 4, the case

of real image with several objects under partial oc-

clusions and different types of noise (Gaussian, Salt

and Pepper and Speckle) is considered. Segmentation

without shape prior fails to detect the familiar objects

(second image of ﬁrst row). However, using the shape

prior, the proposed model succeeds in segmenting the

desired objects (third image of ﬁrst row). For the sec-

ond row, a rotation of −90

◦

with differents kind of

noise are applied. Results seems to be satisfactory.

In what follows, our model is applied to medi-

cal images obtained from Brain Web Simulated Data

Base

. We focus on the segmentation of white matter

of slice 56 that contains holes. The ﬁrst image repre-

sents the reference segmentation. We have chosen to

initialize the model with a green curve. The obtained

contours without the constraint of shape are presented

by the third image. Starting with this result and after

the registration step, ﬁnal segmentation based on prior

knowledge and image-based information is presented

by the last image. Table 2 shows the value of RMSE

(Root Mean Square Error), execution time until con-

vergence and the time for parameters estimation (in

seconds) of the rigid transformation for different val-

ues of w. The image size is 256 x 256. We set the total

number of iterations equals to 500. The prior knowl-

edge is introduced at the iteration 400 for different

values of w. For the second part of the experiments,

we assume that the reference shape is unknown and

we rely on the set of invariants presented in section

4 to choose the most appropriate template. Thus the

proposed algorithm is as follows :

1. Segmentation of the target object using the ac-

tive contours model (Malladi et al., 1995) without

shape prior.

http://mouldy.bic.mni.mcgill.ca/brainweb/

Figure 4: Several object detection under partial occlusion

and noise.

Figure 5: Segmentation of brain’s white matter.

Table 2: Variation of the RMSE and the execution time de-

pending on w.

w RMSE Execution time Motion estimation

1 0.549 24.563 -

0.8 0.237 26.264 0.430

0.4 0.230 27.009 0.470

0.1 0.222 24.798 0.375

0 0.216 27.161 0.526

2. When the evolving curve becomes stable, we

compute the set of invariant descriptors associated

with the target object.

3. We compare this set of descriptors to those associ-

ated with the available templates computed at an

off-line step, then we select the reference shape

presenting the smallest Euclidean distance.

4. Then, we perform the registration step and we

compute the proposed shape prior.

5. Finally, we evolve the model under both image

and prior forces.

In what follows (Fig.6, second row), we have

occluded some objects from the COIL database

(Columbia University) and for every object, we seek

for the best template from the available ones (ﬁrst

row). For each shape, we compute its associated in-

variant descriptors to choose the appropriate template.

Table 3 summarizes the obtained results and Fig.7

shows the ﬁnal segmentation without and with our

proposed method.

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458

Table 3: Shape distance between every occluded object and

the set of the available references.

(a) (b) (c) (d)

(a) + occlusion 0,471 1,376 1,698 0,924

(b) + occlusion 1,460 1,113 2,088 1,445

(d) + occlusion 1,519 1,854 1,933 1,419

Figure 6: Selected objects from the COIL database

(Columbia University), First row : the original objects, sec-

ond row : objects after partial occlusions and missing parts.

Figure 7: Detection of the familiar objects without (ﬁrst

row) and with the proposed method (second row).

For this example, the selected objects were ex-

tracted from the background. Hence we directly com-

pute the value of shape descriptors on the gray-scale

images. Then we select the template of every tar-

get image. In real situation, we have to extract the

target object from the background using the active

contours model in order to compute the set of invari-

ant shape descriptors. The following experiment il-

lustrates the proposed algorithm. Fig.8 presents sev-

eral images of reference. The objects of interest were

isolated from the cluttered background of the image

using the edge-based active contours (Malladi et al.,

1995). We propose to detect the true edges of the tar-

get shapes (1) and (2) which are partially occluded

(see Fig.9, images (a) from ﬁrst and second rows).

Table 4 shows the Euclidean distance between the tar-

get object’s invariants and those associated with the

available shapes of reference.

Table 4: Shape distance between the two target images and

the set of available reference objects.

(a) (b) (c) (d) (e)

Target 1 0.426 0.988 1.208 1.020 0.805

Target 2 1.308 1.509 1.160 1.364 1.483

Figure 8: The reference images and the associated objects

of interest.

Figure 9: Object detection (b) without shape prior, (c) with

shape prior.

Figure 10: Different human states under partial occlusions.

Once the reference shape with small distance is

selected, its associated level set function is used to

constrain the contour evolution towards the true con-

tours of the object of interest. For the last experiment,

we consider the segmentation of human shape under

different states and partial occlusions. Fig.10 shows

the considered human shape states. The used images

were obtained from

. We note that unlike the work

http://homepages.inf.ed.ac.uk/rbf/CAVIAR/

InvariantShapePriorKnowledgeforanEdge-basedActiveContours-InvariantShapePriorforActiveContours

459

of (Fang and Chan, 2006), the use of invariant fea-

tures allows to avoid the registration step at the learn-

ing stage and the computation of PCA on the training

data in order the estimated the appropriate number of

clusters in a low dimensional feature subspace. The

ﬁnal detection results are given by Fig.11.

Figure 11: Detection of partially occluded human shapes in

video images. First row shows the detection results obtained

by the traditional edge-based active contours and the Sec-

ond row shows the result obtained by our proposed method.

6 CONCLUSIONS

New method of geometric active contours with shape

prior is presented in this research. This approach uses

the registration by phase correlation and a set of in-

variant descriptors to deﬁne prior knowledge. Exper-

iments have shown the ability of the new added term

to improve the robustness of the detection process in

presence of missing parts and partial occlusions of the

target objects. The addition of shape prior has not

increased signiﬁcantly the execution time given that

the proposed approach does the registration only once

and it is done by the Fast Fourier Transform unlike

(Foulonneau et al., 2006; Charmi et al., 2008) where

at each iteration shape descriptors are calculated for a

given order which has to be set empirically. In fact, a

small order gives unsatisfactory results and a big one

increase signiﬁcantly the execution time. As future

perspectives, we are working on applying our model

in the context of medical application where the shape

of reference is given by medical atlas in order to aid in

the diagnosis. Also, we plan to extend this approach

to more general transformations such as afﬁne trans-

formations.

REFERENCES

Bresson, X., Vandergheynst, P., and Thiran, J. (2003). A pri-

ori information in image segmentation : energy func-

tional based on shape statistical model and image in-

formation. In Proc. of IEEE Conference on Image

Processing.

Caselles, V., Kimmel, R., and Sapiro, G. (1997). Geodesic

active contours. In Int. J. of Comp. Vis.

Chan, T. and Vese, L. (2001). Active contours without

edges. In IEEE Trans. Imag. Proc.

Chan, T. and Zhu, W. (2005). Level set based shape prior

segmentation. In CVPR.

Charmi, M., Derrode, S., and Ghorbel, F. (2008). Fourier-

based shape prior for snakes. In Pat. Recog. Let.

Charmi, M., Derrode, S., and Ghorbel, F. (2009). Using

fourier-based shape alignment to add geometric prior

to snakes. In ICASSP.

Charmi, M., Mezghich, M., M’Hiri, S., Derrode, S., and

Ghorbel, F. (2010). Geometric shape prior to region-

based active contours using fourier-based shape align-

ment. In IST.

Chen, Y., Thiruvenkadam, S., Tagare, H., Huang, F., Wil-

son, D., and Geiser, E. (2001). On the incorpora-

tion of shape priors into geometric active contours. In

IEEE Workshop on Variational and Level Set Methods

in Computer Vision.

Cohen, L. (1991). On active contour models and balloons.

In Graphical Models Image Process.

Cremers, D., Sochen, N., and Schnorr, C. (2003). To-

wards recognition-based variational segmentation us-

ing shape priors and dynamic labelling. In Interna-

tional Conference on Scale Space Theories in Com-

puter Vision.

Derrode, S. and Ghorbel, F. (2001). Robust and ef-

ﬁcient fourier-mellin transform approximations for

gray level image reconstruction and complete invari-

ant description. In Computer Vision and Image Un-

derstanding.

Fang, W. and Chan, K. (2006). Using statistical shape priors

in geodesic active contours for robust object detection.

In ICPR.

Fang, W. and Chan, K. (2007). Incorporating shape prior

into geodesic active contours for detecting partially

occluded object. In Pattern Recognition.

Foulonneau, A., Charbonnier, P., and Heitz, F. (2004). Con-

traintes gomtriques de formes pour les contours actifs

orients rgion : une approche base sur les moments de

legendre. In Traitement du signal.

Foulonneau, A., Charbonnier, P., and Heitz, F. (2006).

Afﬁne-invariant geometric shape priors for region-

based active contours. In IEEE Trans. Patt. Anal.

Mach. Intell.

Ghorbel, F. (1994). A complete invariant description for

grey-level images by the harmonic analysis approach.

In Pattern Recognition Lett.

Kass, M., Witkin, A., and Terzopoulos, D. (1988). Snakes :

active contour models. In Int. J. of Comp.Vis.

Leventon, M., Grimson, E., and Faugeras, O. (2000). Sta-

tistical shape inﬂuence in geodesic active contours. In

Proc. of IEEE Conference on Computer Vision and

Pattern Recognition.

Malladi, R., Sethian, J., and Vemuri, B. (1995). Shape mod-

eling with front propagation: A level set approach. In

IEEE Trans. Patt. Anal. Mach. Intell.

VISAPP2014-InternationalConferenceonComputerVisionTheoryandApplications

460

M’Hiri, S., Mezghich, M., and Sellemi, M. (2012). Con-

tours actifs avec a priori de forme bas ´e sur la

transform ´ee de fourier-mellin analytique. In Revue

Traitement du Signal.

Osher, S. and Sethian, J. (1988). Fronts propagating

with curvature-dependent speed: algotithms based on

hamilton-jacobi formulation. In J.of Computational

Physics.

Sellami, M. and Ghorbel, F. (2012). An invariant similarity

registration algorithm based on the analytical fourier-

mellin transform. In EUSIPCO.

Xu, C. and Prince, L. (1997). Gradient vector ﬂow: A

new external force for snakes. In IEEE Proc. Conf.

on Computer Vision and Pattern Recognition.

Zhang, T. and Freedman, D. (2003). Tracking objects using

density matching and shape priors. In International

Conference on Computer Vision.

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