INTRODUCING SHAPE CONSTRAINT VIA LEGENDRE

MOMENTS IN A VARIATIONAL FRAMEWORK FOR CARDIAC

SEGMENTATION ON NON-CONTRAST CT IMAGES

Julien Wojak, Elsa D. Angelini and Isabelle Bloch

Institut Telecom, T´el´ecom ParisTech, CNRS LTCI, 75013 Paris, France

Keywords:

Segmentation, Medical imaging, Shape constraint.

Abstract:

In thoracic radiotherapy, some organs should be considered with care and protected from undesirable radiation.

Among these organs, the heart is one of the most critical to protect. Its segmentation from routine CT scans

provides valuable information to assess its position and shape. In this paper, we present a novel variational

segmentation method for extracting the heart on non-contrast CT images. To handle the low image contrast

around the cardiac borders, we propose to integrate shape constraints using Legendre moments and adding

an energy term in the functional to be optimized. Results for whole heart segmentation in non-contrast CT

images are presented and comparisons are performed with manual segmentations.

1 INTRODUCTION

In thoracic radiotherapy, some organs should be con-

sidered with care and protected from undesirable ra-

diation. Among these organs, the heart is one of the

most critical in this context. It is therefore useful

to have a good knowledge of the heart position and

shape, for applications such as dose estimation and

therapy planning. This information can be provided

by image segmentation. Routine examinations rely on

non-contrast CT scans, in which the heart is often dif-

ﬁcult to distinguish from surroundingstructures based

on only grey level information. Manual segmentation

is tedious and prone to inter-observer variability, thus

calling for automated methods, we which address in

this paper.

Among the existing approaches, in (Ecabert et al.,

2008) a multi-chamber (i.e. complete heart) mesh

model is deformed to segment the heart on high con-

trast and high resolution CT images. Unfortunately,

this method does not apply to non-contrast CT. In

(Moreno et al., 2008), the segmentation is constrained

using fuzzy representations of anatomical knowledge

about the position of the heart in the thorax and with

respect to the lungs. This leads to a good robustness

but to an average similarity index when compared to

manual segmentation of 0.74, which might be too lim-

ited for radiotherapy applications.

In this paper, we propose to integrate shape con-

straints into a variationalmethod,followingtheidea of

(Foulonneau et al., 2006), and based on the model of

(Mory and Ardon, 2007). Shape information is rep-

resented using Legendre moments and integrated as

an additional energy term in the functional to be opti-

mized. Results for whole heart segmentation on non-

contrast CT images are presented and evaluated.

50 100 150 200

20

40

60

80

100

120

140

160

50 100 150 200

5

10

15

20

25

30

35

40

45

50

55

Figure 1: Non-contrast CT thoracic images with manual

heart segmentations.

2 VARIATIONAL

SEGMENTATION BASED ON

GRAY LEVEL INTENSITY

In (Mory and Ardon, 2007), the authors introduced

a fuzzy region competition framework to segment

an image I into two classes (background and fore-

ground) based on the minimization of the following

functional:

209

Wojak J., D. Angelini E. and Bloch I. (2010).

INTRODUCING SHAPE CONSTRAINT VIA LEGENDRE MOMENTS IN A VARIATIONAL FRAMEWORK FOR CARDIAC SEGMENTATION ON

NON-CONTRAST CT IMAGES.

In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 209-214

DOI: 10.5220/0002844502090214

Copyright

c

SciTePress

min

u∈BV

[0,1]

(Ω)

E

TV

g

(u(x),α

i

) = min

u∈BV

[0,1]

(Ω)

Z

Ω

g|∇u|dΩ

| {z }

regularity

+ τ

Z

Ω

u(x)r

α

i

(x)dΩ

| {z }

ﬁdelity to the data

(1)

where u is a membership function in BV

[0,1]

(Ω) (the

space of functions of bounded variations), g is a

weighting function of the regularization term in or-

der to relax the regularization near important con-

tours (for example g =

1

1+|∇

˜

I|

with

˜

I a smooth version

of I), and x denotes the coordinates triplet (x,y,z).

The function r

α

i

(x) (α

i

is a set of parameters) could

have different expressions such as the well known

Chan and Vese region competition term r

c

1

,c

2

(x) =

(I −c

1

)

2

−(I −c

2

)

2

where c

1

and c

2

are the mean in-

tensity in each region (Chan and Vese, 2001), or Para-

gios and Deriche geodesic active region term r(x) =

ln

P(I|α

1

)

P(I|α

2

)

(Paragios and Deriche, 1999). The formula-

tion in Equation 1 leads to several interesting proper-

ties:

• using a membership function u instead of a classi-

cal indicator function as in (Chan and Vese, 2001)

leads to a greater stability of the segmentation pro-

cess and to a better control of the regularity of the

ﬁnal contours;

• as the space of solutions is the BV space, the

membership function u converges toward an indi-

cator function, and therefore a simple threshold-

ing at the end of the minimization process pro-

vides the ﬁnal segmentation;

• as the regularization term is convex, the ﬁnal re-

sult is not sensitive to the initialization.

In this work, we use the region competition ap-

proach, with the same formulation as in Equation 1,

leading to a fast and ﬂexible segmentation tool to ex-

tract the lung cavities and internal blood vessels as

illustrated in Figure 2(a-b).

(a) (b) (c)

Figure 2: Segmentation using the region competition for-

mulation: (a-b) lungs and lung vessels (c) the heart is not

well separated from the liver and the aorta.

However, since this approach only relies on in-

tensity discrimination into two classes (or phases),

it cannot segment different structures having similar

gray level characteristics. An example in 2(c) illus-

trates the difﬁculty of separating the heart from adja-

cent structures such as the major blood vessels and the

liver, on non-contrast CT data. In order to overcome

such limitations, we propose to constrain the segmen-

tation using shape information.

3 SHAPE CONSTRAINT

Several methods have been proposed to constrain

shapes in a segmentation functional. In (Gastaud

et al., 2004), a distance between a reference and the

observed shapes is used. In (Leventon et al., 2000),

a PCA analysis is performed on the level sets func-

tions of training segmentations. A review of shape

constraints for level sets segmentation methods can

be found in (Cremers et al., 2007). For the fore-

seen applications for thoracic radiotherapy, we have

to achieve a compromise between the constraint on

the shape characterization and the ﬂexibility of the

representation to cope with inter-patient variability

(patient positioning, heart size,... ). In this context it is

not relevant to compute directly a difference between

a reference shape and the current shape segmentation.

We prefer to control an indirect match by comparing

our segmentation with a generic shape model repre-

sentation that does constrain natural anatomic vari-

ability, and is not sensitive to translation and scale

variability. Such characteristics can be provided by

well-chosen shape moments. For example in (Rose

et al., 2009), Tchebichev moments are used to con-

strain a region growing algorithm, and Foulonneau et

al. (Foulonneau et al., 2006) used Legendre moments

as shape descriptors in an active contour segmentation

framework.Our approach follows a similar scheme as

in this last work.

3.1 Legendre Moments

In (Teague, 1980), Teague introduced moments for

image analysis. He proposes to use Legendre poly-

nomials or Zernike polynomials as kernel functions.

This is motivated by the orthogonality property of

both types of polynomials, which guarantees the non-

redundancy of the description of an image or a shape.

The existence of efﬁcient methods to easily and fastly

compute Legendre moments has guided our prefer-

ence for these moments over Zernike ones. Legen-

dre moments are more sensitive to noise than Zernike

moments, but in our framework we manipulate de-

scriptors only on clean binary masks or membership

functions deﬁned on [0, 1], in a framework of fuzzy

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

210

region competition. Therefore this limitation is not a

drawback for the proposed approach.

Legendre moments are deﬁned by the projection

of a function on a polynomial basis. Let I : [−1, 1]

3

→

R be the image representation. The moment of I of

order p+ q+ r is deﬁned as:

λ

pqr

= C

pqr

Z

[−1,1]

3

P

p

(x)P

q

(y)P

r

(z)I(x, y,z)dxdydz

(2)

where C

pqr

=

(2p+1)(2q+1)(2r+1)

8

and P

p

,P

q

,P

r

are Leg-

endre polynomials, deﬁned by the following two or-

der recursive relation:

P

p+1

(x) =

2p + 1

p+ 1

xP

p

(x) −

p

p+ 1

P

p−1

(x) (3)

(p > 1, P

0

(x) = 1 and P

1

(x) = x).

Legendre polynomials form an orthogonal basis,

with:

2p + 1

2

Z

1

−1

P

p

(x)P

q

(x)dx =

0 if p 6= q

1 if p = q

(4)

Working with a ﬁnite number of moments, an es-

timate

ˆ

I of I is given by:

ˆ

I(x,y,z) =

L

∑

p=0

p

∑

q=0

q

∑

r=0

λ

p−q,q−r,r

P

p−q

(x)P

q−r

(y)P

r

(z)

(5)

where L is the maximum order for which Legendre

moments are computed.

The computation can be performed with the fast

and exact Hosny method (Hosny, 2007). First, the

spatial image domain is embedded in the cube U =

[−1,1]

3

. Assuming that the image has X ×Y ×Z vox-

els, the centers of voxels are then given by the coordi-

nates (x

i

,y

j

,z

k

) such that

x

i

= −1+ (i−1/2)∆x i = 1···X and ∆x =

2

X

y

j

= −1 + ( j −1/2)∆y j = 1···Y and ∆y =

2

Y

z

k

= −1+ (k −1/2)∆z k = 1···Z and ∆z =

2

Z

(6)

Voxels on wich the image intensity is constant

are then deﬁned as intervals [U

i

,U

i+1

] ×[V

j

,V

j+1

] ×

[W

k

,W

k+1

] with

U

i

= x

i

−∆

x

/2 U

i+1

= x

i

+ ∆

x

/2

V

j

= y

j

−∆

y

/2 V

j+1

= y

j

+ ∆

y

/2

W

k

= z

k

−∆

z

/2 W

k+1

= z

k

+ ∆

z

/2

(7)

The moment expression computed on the whole

image can then be rewritten as:

λ

p,q,r

= C

p,q,r

L

∑

i=1

M

∑

j=1

N

∑

k=1

I(x

i

,y

j

,z

k

)

Z

U

i+1

U

i

Z

V

j+1

V

j

Z

W

k+1

W

k

P

p

(x)P

q

(y)P

r

(z)dxdydz

(8)

Moreover thanks to the following recursive prim-

itive property of the Legendre polynomes

Z

x

cst

P

p

(y)dy =

P

p+1

(x) −P

p−1

(x)

2p+ 1

, (9)

Legendre moments can be written as:

λ

pqr

=

L

∑

i=1

M

∑

j=1

N

∑

k=1

I

p

(x

i

)I

q

(y

j

)I

r

(z

k

)I(x

i

,y

j

,z

k

) (10)

where I

p

(x

i

) =

2p+1

2p+2

[xP

p

(x) −P

p−1

(x)]

U

i

+1

U

i

and sim-

ilar expressions for I

q

and I

r

. The kernel I

p

I

q

I

r

is in-

dependent of the image intensity and can therefore be

precomputed. Moreover the separability property al-

lows us to compute the 3D moments using three 1D

steps.

To guarantee scale and translation invariances,

Legendre moments must be reformulated in the fol-

lowing way:

Λ

pqr

= C

pqr

Z

[−1,1]

3

P

p

(

x−x

0

A

)P

q

(

y−y0

A

)P

r

(

z−z

0

A

)I(x, y, z)dxdydz (11)

where (x

0

,y

0

,z

0

) are the coordinates of the center of

inertia of I and A is the volume of the shape.

3.2 Discriminating Volumes by a Finite

Number of Legendre Moments

Two similar shapes have the same set of Legendre

moments and two different shapes have two different

sets of Legendre moments. However, since we work

with scale and translation invariant moments, the set

{λ

0,0,0

,λ

0,0,1

,λ

0,1,0

,λ

1,0,0

} is the same for each shape

and should not be used for discriminating between

shapes. In order to illustrate the discriminative power

of the moments, we consider four classes of shapes:

class 1: heart alone, class 2: heart and aorta together,

class 3: heart and liver together, class 4: heart and

liver and aorta together.

Norm 2 Error. To highlight differences between

Legendre moments of the four classes of shapes, we

compute the square ℓ

2

norm between two shapes as

||λ

shape

1

−λ

shape

2

||

2

2

where λ

shape

i

is a vector storing

successive Legendre moments of a shape. Comparing

the measures for a ﬁnite number of moments from a

mask of a reference heart and objects for the other

classes we obtain the following results (Table 1):

Errors at order 5 are inferior to those at order

15. This is due to the fact that small order moments

represent low frequency shape information. The gap

between moment differneces between hearts and be-

tween hearts and other structures is more important

for order 5 (factor 5) than for order 15 (factor 2). It is

due to the fact that at order 5, the difference between

INTRODUCING SHAPE CONSTRAINT VIA LEGENDRE MOMENTS IN A VARIATIONAL FRAMEWORK FOR

CARDIAC SEGMENTATION ON NON-CONTRAST CT IMAGES

211

Table 1: ℓ

2

norm of the difference between sets of Legendre

moments between a reference heart shape and 13 maks of

others hearts, 4 masks of hearts and liver, 4 masks of hearts

and aorta, 4 masks of hearts and aorta and liver.

mean min max

P

P

P

P

P

P

ref vs

order

5 15 5 15 5 15

other hearts 0.17 1.21 0.13 1.09 0.26 1.52

heart and liver 0.56 2.40 0.52 2.26 0.58 2.56

heart and aorta 0.61 2.38 0.44 2.13 0.69 2.50

heart and aorta

0.53 2.38 0.45 2.19 0.62 2.60

and liver

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

Figure 3: Examples of mask of: (a) reference heart, (b-c)

two other hearts, (d) heart and aorta, (e) heart and liver.

shapes of different classes is sufﬁciently important to

discriminate between them, and in the same class dif-

ferences of the shape are too small to well discrimi-

nate between them.

In the following experiment we compare moments at

a maximum order 10 in order to well differentiate

shapes and to take advantage of the global represen-

tation of a shape by its moments.

PCA Analysis. In this section, we study the capa-

bility of Legendre moments to efﬁciently discrimi-

nate between correctly segmented hearts and erro-

neous segmentations, by considering again the four

classes of shapes. Inspired by (Poupon et al., 1998),

we performed a PCA analysis on a matrix in which

each raw is an observation, i.e. a segmentation re-

sult, and each column corresponds to an ordered list

of Legendre moments. The PCA analysis shows that

the ﬁrst three modes represent 90% of the variance.

As illustrated in Figure 4, the moments also discrim-

inate efﬁciently the different types of shapes. Indeed

samples from the heart alone are well grouped in the

plane of the ﬁrst two modes and are well separated

from the other types of shapes.

3.3 Introducing Shape Constraint in the

Functional

We propose to introduce an additional term in the seg-

mentation functional to be optimized through a com-

parison between moments of a reference shape and

moments of the current segmented shape, as follows:

Heart

Heart and Liver

Heart and Liver

Heart and Aorta

Heart and Liver and Aorta

first mode

second mode

Figure 4: PCA on Legendre moments. Result on the ﬁrst

two principal axes, “+” heart alone, “o” heart and liver, “*”

heart and aorta, “x” heart and liver and aorta.

min

u∈BV([0,1])

Z

Ω

g|∇u|dΩ+ τ

Z

Ω

r(x)udΩ

+

γ

2

||λ

ref

−λ

cur

(u)||

2

2

(12)

This formulation is quite similar to the one de-

scribed in (Foulonneau et al., 2006). However, using

a membership function u instead of a level set func-

tion leads to a more stable algorithm. Another differ-

ence concerns the order of the used moments. Since

one of the main objectives of the work by (Foulon-

neau et al., 2006) was to provide an algorithm ro-

bust to occlusions, a hard constraint on the shape was

needed and high order moments were computed. In

our framework, the goal is to capture global features

of the shape and allow a small variability between

them. Therefore only quite small order moments are

needed.

Minimization. We perform the minimization of the

functional (12) by a gradient descent method.

To insure that u ∈ BV

[0,1]

(Ω) in (1) we rewrite it in the

same manner as in (Chan et al., 2005) :

min

u

E

TV

g

(u(x)) = min

u

Z

Ω

g|∇u|dΩ+ τ

Z

Ω

u(x)r(x)dΩ

+

Z

Ω

αν

ε

(u)dΩ +

γ

2

||λ

ref

−λ

cur

(u)||

2

2

(13)

where ν

ε

is a regularized approximation of the pe-

nality function ν(a) = max{0, 2|a −1/2|− 1} with

α > ||r(x) + ||||λ

ref

−λ

cur

(u)||

2

2

||

∞

and we obtain the following iterative scheme:

u

(n+1)

(x) = u

n

(x) + dt

div(g

∇u(x)

|∇u(x)|

) −τr(x) −αν

′

ε

(u)

−γ

∑

p,q,r

(λ

ref

p,q,r

−λ

cur

p,q,r

(u))P

p

(x)P

q

(y)P

r

(z)

i

Choise of r. Let us arbitrarily deﬁne the heart region

as the foreground region, (i.e. the region in which we

would like to obtain u(x) = 1) and the rest of the re-

gion of interest as the background region (i.e. the re-

gion in which we would like to obtain u(x) = 0).

Histograms of the foreground and the background

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

212

intensities,obtained from manual segmentations, are

shown in Figure 5. It shows that intensities values

are quite similar in the background and in the fore-

ground. However, the intensity is very homogeneous

in the heart and presents two peaks in the background.

0 50 100 150 200 250 300

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 5: Histograms of foreground (left) and background

(middle) intensities. Right: background intensity probabil-

ity density estimation using Parzen window.

We use this difference to construct the following

relevant data ﬁdelity term r(x):

r(x) = (I −c

1

)

2

−(max((I −c

1

)

2

))(

Z

B

(p(b) −K(I −b))

2

db (14)

where c

1

=

R

Ω

I(x)u(x)dx

R

Ω

u(x)dx

is the empirical estimation

mean of the heart intensity, B is the interval of the im-

age intensities, p is the Parzen estimation of the prob-

ability density function of the intensity of the back-

ground, expressed as:

p(b) =

1

||1−u||

1

Z

Ω

(1−u(x))K((I(x) −b)/σ)dx (15)

where K is a Gaussian window (K(a) =

1

√

2π

exp(−

a

2

2

)) and σ is chosen sufﬁciently small to

distinguish the two modes in the background area.

Choice of τ and γ. CT images are calibrated, whith

known tissue intensity values (for examples compact

bones are known to be around 1000 Hounsﬁeld units

(HU)). This intensity inter-images stability calibra-

tion allows us to pre-set the weight of each term in

the functional in order to insure a good balance be-

tween them. The regularization term, computed on

manual segmentations, is of the order of 10

6

(it corre-

sponds to the surface of the whole heart weighted by

g). The data ﬁdelity term is close to zero. The shape

constraint term at order 10 falls within the range of

values [1, 2]. Finally, by dividing g by 10

6

a good

balance between all terms in the functional (12) is ob-

tained for τ and γ in ]0, 10[. We performed several

experiments for different parameters and ﬁnally we

ﬁxed τ = 1.4 and γ = 0.8 for all tests summarized in

Table 2.

Table 2: Quantitative results: comparison between auto-

matic and manual segmentations. The numbers in paren-

theses are results obtained by (Moreno et al., 2008).

similarity index sensitivity speciﬁcity

Heart 1 0.82 (0.77) 0.96 (0.96) 0.74 (0.64)

Heart 2 0.81 (0.70) 0.89 (0.90) 0.78 (0.58)

Heart 3 0.80 (0.75) 0.94 (0.78) 0.70 (0.72)

Heart 4 0.84 (0.74) 0.76 (0.62) 0.97 (0.92)

Heart 5 0.77 (0.84) 0.81 (0.83) 0.72 (0.84)

Heart 6 0.81 (0.80) 0.93 (0.91) 0.71 (0.71)

Heart 7 0.78 (0.71) 0.84 (0.88) 0.73 (0.60)

Heart 8 0.80 (0.67) 0.92 (0.71) 0.80 (0.62)

Heart 9 0.75 (0.64) 0.83 (0.60) 0.73 (0.70)

4 HEART SEGMENTATION

Masking a Region of Interest and Initializing.

From a pre-segmentation of the lungs, a region of in-

terest (ROI) around the heart is built as the bounding

box of lungs elongated at the bottom to insure that the

heart is completely inside. A mask of the lungs and

the trachea is removed from the ROI.

(a) (b) (c)

Figure 6: Preprocessing. (a) Heart ROI with lungs and

trachea masked out. (b) Example of an initialization of

the hear segmentation. (c) Gradient weighting function g

around the heart.

Moreover tissues at the right of the left lung and

at the left of the right lung are removed. This is illus-

trated in Figure 6.

The initialization is performed semi-automatically. A

point C approximatively at the center of the heart is

marked and the initial value of u(x) is deﬁned by

u(x) = 1 if x is both inside a sphere centered at C

with 4 cm diameter and inside the region of interest,

and u(x) = 0 otherwise.

Tests and Results. Tests on 9 non contrast CT scan

have been performed. For each one, the result of

the segmentation was compared with a manual delin-

eation done by an expert. Similarity, sensitivity and

speciﬁcity indices are computed and reported in Table

2. We also compared our results with those obtained

by the method of Moreno (Moreno et al., 2008). We

globally obtain better results than those obtained by

Moreno et al. Differences between the results of the

two methods are illustrated in Figures 7(b) and 8. Ex-

INTRODUCING SHAPE CONSTRAINT VIA LEGENDRE MOMENTS IN A VARIATIONAL FRAMEWORK FOR

CARDIAC SEGMENTATION ON NON-CONTRAST CT IMAGES

213

cept for the heart 4 (Figure 8 column 2), the speci-

ﬁcity index is generally higher than the sensitivity. It

means that our automatic segmentation often provides

a larger region than the manual one (it is mainly due

to the fact that a small part of the aorta is often in-

cluded in the segmentation of the heart, as illustrated

in Figure 7(a).

50 100 150 200

5

10

15

20

25

30

35

40

45

50

55

(a) (b)

Figure 7: (a) Automatic segmentation (in red) includes

small part of the aorta (green: manual segmentation). (b)

A 3D view of a whole heart segmentation.

Figure 8: Examples of segmentation results. First row:

original image, second row: image superimposed with seg-

mentations (green expert segmentation, magenta Moreno et

al segmentation, red our automatic segmentation).

5 CONCLUSIONS

We have adapted a fuzzy region competion frame-

work for the segmentation of the heart in non-contrast

CT images by adding a shape constraint. Shape infor-

mations was encoded with Legendre moments. Since

we work with CT images (which are calibrated), we

use hard a priori on the image intensity. The ini-

tialization is performed semi-automatically using a

spherical approximation of the heart. Several tests

on clincal cases provide satisfying results. In par-

ticular, the shape constraint allows us to achieve a

good separation between the heart and surrounding

organs (liver, aorta), improving the initial fuzzy re-

gion competition model. When compared to another

method (Moreno et al., 2008) using structural knowl-

edge (but no shape information) the results are also

improved. This framework could be extended in a se-

quential way to segment other organs in the thorax

like the aorta.

ACKNOWLEDGEMENTS

This work was partially funded by the Medicen Pˆole

de Comp´etitivit´e within the Miniara project.

REFERENCES

Chan, T. F., Esedoglu, S., and Nikolova, M. (2005). Finding

the global minimum for binary image restoration. In

ICIP, pages I: 121–124.

Chan, T. F. and Vese, L. A. (2001). Active contours with-

out edges. IEEE Trans. Image Processing, 10(2):266–

277.

Cremers, D., Rousson, M., and Deriche, R. (2007). A re-

view of statistical approaches to level set segmenta-

tion: Integrating color, texture, motion and shape. In-

ternational Journal of Computer Vision, 72(2):195–

215.

Ecabert, O., Peters, J., and etal (2008). Automatic model-

based segmentation of the heart in CT images. IEEE

Trans. Medical Imaging, 27(9):1189–1201.

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

Afﬁne-invariant geometric shape priors for region-

based active contours. IEEE Trans. Pattern Analysis

and Machine Intelligence, 28(8):1352–1357.

Gastaud, M., Barlaud, M., and Aubert, G. (2004). Combin-

ing shape prior and statistical features for active con-

tour segmentation. IEEE Trans. Circuits and Systems

for Video Technology, 14(5):726–734.

Hosny, K. M. (2007). Exact Legendre moment compu-

tation for gray level images. Pattern Recognition,

40(12):3597–3605.

Leventon, M. E., Grimson, W. E. L., and Faugeras, O. D.

(2000). Statistical shape inﬂuence in geodesic active

contours. In CVPR, pages I: 316–323.

Moreno, A., Takemura, C. M., Colliot, O., Camara, O., and

Bloch, I. (2008). Using anatomical knowledge ex-

pressed as fuzzy constraints to segment the heart in

CT images. Pattern Recognition, 41(8):2525–2540.

Mory, B. and Ardon, R. (2007). Fuzzy region competi-

tion: A convex two-phase segmentation framework.

In Scale Space and Variational Methods in Computer

Vision, pages 214–226.

Paragios, N. and Deriche, R. (1999). Geodesic active con-

tours for supervised texture segmentation. In CVPR,

pages II: 422–427.

Poupon, F., Mangin, J. F., Hasboun, D., Poupon, C., and

Magnin, I. (1998). Multi-object deformable templates

dedicated to the segmentation of brain deep structures.

In Medical Image Computing and Computer-Assisted

Intervention - MICCAI 1998, pages 187–196.

Rose, J.-L., Revol-Muller, C., Charpigny, D., and Odet, C.

(2009). Shape prior criterion based on Tchebichef mo-

ments in variational region growing. In ICIP.

Teague, M. R. (1980). Image analysis via the general theory

of moments. Journal of the Optical Society of Amer-

ica, 70(8):920–930.

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

214