Y.Y. Yao et al. (Eds.): BI 2010, LNAI 6334, pp. 156–167, 2010. © Springer-Verlag Berlin Heidelberg 2010

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching Method

Mohamed Baghdadi1, Nacéra Benamrane1, and Lakhdar Sais2

1 Department of Informatics, USTOMB, PB 1505, EL’Mnaour 31000, Oran, Algeria baghdadi_mohaamed@yahoo.fr

nabenamrane@yahoo.com 2 Université Lille Nord de France, CRIL - CNRS, Rue Jean Souvraz, SP-18, F-62307, Lens Cedex 3

sais@cril.fr

Abstract. In this paper a modular approach of segmentation which combines the Bayesian model with the deformable model is proposed. It is based on the level set method, and breaks up into two great parts. Initially, a preliminary stage allows constructing the information map. Then, a deformable model, im-plemented with the Generalized Fast Marching Method (GFMM), evolves to-wards the structure to be segmented, under the action of a force defined from the information map. This last is constructed from the posterior probability in-formation. The major contribution of this work is the use and the improvement of the GFMM for the segmentation of 3D images and also the design of a robust evolution model based on adaptive parameters depending on the image. Ex-perimental evaluation of our segmentation approach on several MRI volumes shows satisfactory results.

Keywords: Segmentation, Brain MR Imagery, Deformable model, GFMM, Bayesian model.

1 Introduction

Segmentation of the brain structures in Magnetic Resonance Images (MRI) is the first step in many medical image analysis applications and is useful for the diagnosis and evaluation of the neurological diseases (Alzheimer's, Parkinson's...). Therefore, accu-rate, reliable, and automatic segmentation of the brain structures can improve diag-nostic and treatment of the related neurological diseases. Manual segmentation by an expert is usually accurate but it is impractical for large datasets because it is a tedious and time-consuming process.

Several methods have been developed for tissue segmentation, among these the de-formable models have taken a great importance; indeed they define a powerful tool to accurately recover a structure using very few assumptions about its shape. Two great families of deformable models exist: parametric models and nonparametric models. The first are the oldest and require a parametric or discrete representation. The sec-onds, based on the curves evolution theory, use an implicit representation of the

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 157

model and allow topological changes. The principle of the parametric Active Con-tours (snakes) [6] is to make evolve a contour or a surface towards the borders of the object to be segmented while minimizing certain energy. Two types of energies are useful for the snake. The external energy her role is to guide the active contour on the image, and attracts it towards the position corresponding to a required characteristic. Internal energy, which is associated to the geometry of the contour, its goal, is to avoid the trap of the local minima of external energy, and translates the regularization of the active contour. This type of active contours was widely used in medical im-agery [4] [11]. Indeed parametric active contours are powerful tools for segmenta-tion, however were shown not very efficient to treat the complex forms, presenting convolutions, and to manage the variations of topology, which are frequent in 3D medical images.

The second family is the nonparametric deformable models including geometrical active contours [8] and geodesic active contours [3], which were the subject of very numerous developments and applications in the field of the medical imagery [7]. The nonparametric deformable models adapt better to the medical applications. Moreover, the formalism of the level sets [9] allows implementing these methods efficiently and has the following advantages: (1) an adapted numerical schemes are available, (2) the topological changes are allowed, and (3) the extension of the method to higher dimen-sions is easy. The contour or the surface (3D case) deforms under the action of a nor-mal evolution force, so the choice of the evolution force is very intrinsic, its role is to guide the contour towards the target zones, which depend on the application. In par-ticular on medical images, the structures of interest are often of complex form and of variable topology. That makes difficult the definition of a force which is at the same time general and adaptable to specific structures and pathologies. Several evolution models have been proposed, including some parameters (step size, weighting parame-ters...). The appropriate setting of these parameters strongly influences the performance of the methods.

In this paper, we propose a novel approach based on a robust and adaptive evolu-tion model which enables a volume to be segmented with certain performance. The versatility of our segmentation method is demonstrated by reporting results on brain structures in MR images. The remainder of this paper is organized as follows. Section 2 presents the method of segmentation GFMM which constitutes the kernel of our approach. The approach suggested is detailed in section 3. At last, the results of our experiments are presented and discussed in section 4.

2 The Generalized Fast Marching Method (GFMM)

2.1 Introduction to the GFMM

The Generalized Fast Marching Method [2][5] is a generalization of the Fast March-ing Method (FMM)[6] which is a very fast algorithm allowing to advance a front evolving with a normal velocity. The generalization comes from the fact that the propagation velocity can change sign in space-time, so it can treat the general velocity case without any restriction of sign. In this method the concept of discontinuous solu-tion is used to represent the front. In a very simple way, the front will be represented by the discontinuity of a function θ which will take values 1 and - 1. The function is then the solution of the following equation (in the viscosity sense):

158 M. Baghdadi, N. Benamrane, and L. Sais

θt=c(x, t)|∇θ| (1)

Where c is the propagation velocity. The main idea of this generalized algorithm is as follows. At each stage, we consider two zones: one where the velocity is negative and the other where it is positive. Then, it is sufficient to make evolve the contour using two different Fast Marching (one in each zone).

2.2 The GFMM Algorithm

Given the speed nIc of the front at point I at time tn and the normalized speed n

Ic

defined by:

⎪⎩

⎪⎨⎧ ≤<∈

≡otherwise

,)0()(0ˆ

nI

nJ

nI

nJ

nIn

Ic

ccandccthatsuchIVJexiststhereifc

Where V is a vicinity system defined by:

{ }.1:)( ≤−∈= IJZJIV N

(2)

As in the classical FMM, we define the Narrow Band (NB) which consists of the points I that can be reached by the front. More precisely, the Narrow Band is defined by:

{ },0ˆ),(, <−=∈∃∈= nI

nI

nJ

nI

Nn cetIVJZINB θθθ (3)

{ } { }1,1, −=∩=+=∩= −+nI

nnnI

nn INBNBandINBNB θθ (4)

As in the FMM, for all point I ∈ NB n, we have to compute a tentative value nIu~ repre-

senting the arrival time of the front at point I. To compute this tentative value, we must define, the set of points which are useful for I, namely, the set:

( ) { } ( ).,,)( IUUNBIVJIU n

NBI

nnJ

nI

nnn∈∪=−=∩∈= θθ (5)

For all the points J which are useful for a point I ∈ NBn (J∈ Un (I)), we introduce a

time ),( IJun , which can be interpreted as the time when the front Fn starts to from

point J to point I, this last is used to compute nIu~ . Once we have computed the tenta-

tive values for all points of the Narrow Band, we denote by nt~

the minimum of all this

values. Below is a pseudo code showing the main steps of the method:

Initialization () { n=1; // initialize the field θ 0 as:

⎩⎨⎧

−Ω∈

=elsewhere1

1 00 II

xforθ

(6)

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 159

/* 0Ω is the initial region*/

//Initialize the time for I:

⎩⎨⎧

∞+∈∈

=otherwise

)(),(

0000 NBKandKUIift

KIu (7)

} // end of Initialization Loop ()

{ U_Computation: {

// Compute 1~ −nu on NBn-1 as follows:

/* Let I∈ NBn-1, then we compute1n

Iu − as the

solution of the following second order equation:*/

( )21

2212121

ˆ

~~~

−

−−− Δ==⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂nI

nnn

c

x

z

u

y

u

x

u

(8)

//Such as (similarly for y and z):

[ ]),,1(~),,(~),,,1(~),,(~,0max~

11111

zyxuzyxuzyxuzyxux

u nI

nI

nI

nI

n

−−+−=∂

∂ −−−−−

} //end of U_Computation

{ }.;,~min~ 11 −− ∈= nnIn NBIut

// Truncate nt~

( );),~min(,max 11 ttttt nnnn Δ+= −− (9)

( )nnnn tttttif ~&&)( 1 <Δ+== − {

;1+= nn ;; 11 −− == nnnn uuθθ Go to (U_Computation); } // Initialize the new accepted points

{ } { }n

nI

nnn

nI

nn tuNBINAandtuNBINA ~~,~~, 1111 =∈==∈= −−−−

−−++

(10)

nnn NANANA −+ ∪= // Reinitialize θ n

1

1

1

otherwise

n

n nI

nI

if I NA

if I NAθθ

−

+−

⎧+ ∈⎪

= − ∈⎨⎪⎩

(11)

160 M. Baghdadi, N. Benamrane, and L. Sais

// Reinitialize ),( KIun

⎪⎩

⎪⎨⎧

∞+

∈∈=

−

otherwise

NBKandKUIiftKIuKIu

nnn

nn

,

)()),,(min(),(

1

(12)

n=n+1; go to (U_Computation); }// end of Loop

3 The Proposed Method: Bayesian Generalized Fast Marching Method (BGFMM)

The proposed approach breaks up into two great parts. Initially, a preliminary stage allows constructing the information map. Then, a deformable model, implemented with the Generalized Fast Marching Method, evolves towards the structure to be segmented. Figure 1 presents the various stages of our approach.

Velocity

Map

Posterior

Probability

Map FCMA

Evolution of

the

Level Sets

Final

Segmentation

Fig. 1. Introductory Scheme of the segmentation approach using the BGFMM

3.1 The Choice of the Kernel Method of Segmentation

The Fast Marching Method (FMM) is a powerful method of segmentation character-ized by its rapidity in particular for the 3D volumes, however this method suffer of two major defects: (1) the evolution function must be of constant sign (positive or nega-tive), (2) it does not take into account the curvature term. To treat the first problem we used the Generalized Fast Marching Method (GFMM) proposed in [5]. In this version the evolution function can change sign in space time. Moreover, it preserves the rapid-ity of the FMM. This method is very recent. Indeed, there are no works which uses it for the 3D images segmentation. However the method such as FMM, always suffers from the curvature problem. To treat this pitfall we improved the GFMM (see the fol-lowing section) so that it can treat complex forms of high curvature.

3.2 Improvement of the Classical GFFM

The GFMM is a powerful method of segmentation, but her performance is decreased when treating forms of strong curvature, that can be explained by the fact that the GFMM as the FMM does not know the curvature of the front. To adjust this problem, some modifications were brought to the classical GFMM. We recall that the classical GFMM is based on the use of the one sided derivatives, computed using forward and backward differences.

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 161

],,[~],,1[~],,1[~],,[~

111

111

zyxuzyxuD

zyxuzyxuDnnx

nnx

−−+

−−−

−+=

−−= (13)

To integrate the curvature information, we replace the first order approximations ( xD −

1and xD +

1) of the partial derivative used in GFMM by the following second order

approximations:

2

],,2[~],,1[~4],,[~3

,2

],,2[~],,1[~4],,[~3

111

2

111

2

zyxuzyxuzyxuD

zyxuzyxuzyxuD

nnnx

nnnx

+++−−=

−+−−=

−−−+

−−−−

(14)

When these second order approximations are used, the scheme still works in ex-actly the same way except that we get different polynomial coefficients. The equation to be resolved thus becomes:

( )21

22

222

222

22ˆ

)0,,max()0,,max()0,,max(−

+−+−+− Δ=−+−+−nI

zzyyxx

c

xDDDDDD

(15)

We define then the following vicinity system { }2:)(2 ≤−∈= IJZJIV N . In

this case the useful points are described by:

{ },,)(, 2nJ

nI

nn NBIVJIU θθ −=∩∈∃= (16)

To use the new scheme, the voxels at 2 vu (voxel unit) distance must be useful and have smaller distance values than those at 1 vu distance. If these two conditions are not met, the first order approximations to the derivative can be used instead.

3.3 The Adaptive FCM (FCMA)

The FCM [1] algorithm can be adapted to the case of the fuzzy classification of the N gray levels of an image. In this case, it is not necessary to classify all the voxels of an image, but to simply classify the different gray levels values which we find in this one. In this case the gain will be very important particularly if we consider 3D im-ages. The main stages of the FCMA method are given by:

1. Initialization - Fix the parameters: The number of classes’ c: c (2≤ c≤ M), where M is the

number of the gray levels of the image. The fuzzy degree m ∈[1, ∞[, gen-erally m=2.

- Initialize the vector v of the centres of the classes - Compute the frequency fi of each gray level in the image

2. Repeat: - Compute the distances dik between all the gray levels i, (i=1,2,…,N-1) and

the centres vk of the classes (k=1,2,…, c),and the new centres vk of the c classes :

162 M. Baghdadi, N. Benamrane, and L. Sais

1

2

10

11

0

( )1 ,

( )

Nm

ik ic mii k

ik k Nml i l

i k il

i fd

df

μμ ν

μ

−

−=

−=

=

⋅ ⋅⎛ ⎞

= =⎜ ⎟⎝ ⎠

∑∑

∑

(17)

3.4 The Preliminary Stage

At the beginning, we construct the fuzzy map using the result of the FCMA, which gives us for each gray level I its membership degree to the various classes. The mem-bership degree of the gray level I to the class k can be interpreted as the probability of being in the presence of the class

kω knowing I. From this fact we will thus have the

following equality:

)()( IUIP kk =ω (18)

In the case of the brain MR images, we generally use Gaussian distributions to rep-resent the various classes of tissues. Each distribution is associated to a class la-beled kω and described by a parameter vector

kΓ made of three components. The first

two components are the parameters of the distribution (mean and standard deviation) the third component is the prior probability of the class kω , noted )( kk P ωπ = .The mixture of N distributions thus is described by the vector { },1, nkk ≤≤Γ=Γ whose

parameters are calculated as follow: After the application of the FCMA algorithm, we label each voxel x of which the gray level equals to I to his class, using the posterior probability.

nqqIpIpifx qkk ≤≤∀>∈ 1,),()( ωωω (19)

Then, we calculate the prior probability kπ of each class starting from the proportion of the voxels belonging to the class k, also the other parameters of the various classes (variance, mean). We denote T the set of the classes corresponding to the segmenta-tion target. The complementary set including the classes corresponding to the back-ground, or to the outside of the structure, is denoted B. The classes mainly represented inside the initial volume of the segmentation are assigned to T. More precisely, the intensity distribution over the whole image can be explicitly written as:

)()()()(

11k

n

k

ek

ikk

n

kk IPIPIP ωππωπ ⋅+=⋅= ∑∑

== (20)

Where ikπ and e

kπ are respectively the proportions (or prior probabilities) of class kω inside and outside the initial surface segmenting the object of interest. Once the sets T and B have been determined the estimated intensity distributions inside and outside the structure to be segmented can be defined as:

⎪⎩

⎪⎨⎧

⋅=

⋅=

∑∑

∈

∈

Bk kke

Tk kki

k

k

IPIp

IPIp

ω

ω

ωπ

ωπ

)()(

)()(

(21)

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 163

And the prior probability of a voxel to be inside the structure of interest is given by:

∑∈

=Tk

kT

kωπη

(22)

3.5 The Evolution Model

The propagation equation that we employ uses the region based information of the posterior probability map and described by the following formula:

SFSF *= (23)

Where: S is the propagation sign, of which the goal is to guide the contour towards the borders of the structure to be segmented (target of segmentation). SF is the stopping factor its role is to stop the evolution of the contour at the desired location. The computation of the propagation sign S: This term favours a contraction or a local expansion of the whole surface all along the process. The problem can be ex-pressed as the classification of each point of the current interface, if a point belongs to the object of interest then the surface should locally extend; if it does not, the surface should contract. We perform this classification by maximizing the posterior segmen-tation probability )( Iwp , where w denotes the membership class of the considered

point. According to Bayes rule, the maximization of the posterior distribution )( Iwp is equivalent to the maximization of )()( wIpwp ⋅ , where )( wp is the

prior probability of class w and )( wIp is the conditional likelihood of intensity. S

is computed as the following way:

⎩⎨⎧

<∈−∈−>∈−∈+

=0)()(1

0)()(1

IBwPITwPif

IBwPITwPifS

(24)

And by using the Bayes rule, we finally obtain:

( ( ) (1 ) ( ))T i T eS Sign p I p Iη η= − − (25)

The computation of the stopping factor SF: Taking its values in the interval [0, 1]. Indeed, If its value is close to 1, the propagation velocity of the contour is high, and inversely, if its value is close to 0, the propagation velocity of the contour is low (stopping of the evolution).

We imagine the zones of the true borders of the object of interest i.e. the zones where the contour must be stop like a transition bridge between the interior and the exterior of the segmentation target. We need thus to calculate for each voxel its prob-ability to being on this bridge which is denoted Pbridge. Let x be a voxel of the current interface, and ω the estimated class of x. The posterior probability of x being on the transition bridge, given I and ω, is described by:

⎪⎩

⎪⎨⎧

∈∈

∈∈=

BifIByP

TifITyPIxP

bridge

bridge

bridge ωω

ω)(

)(),( (26)

164 M. Baghdadi, N. Benamrane, and L. Sais

The voxel y is a neighbouring voxel of x in the normal direction located outside the volume defined by Ө (Ө in reference to the GFMM). Thus if x is more likely to be inside the object to be segmented (i.e. if ω∈T), then the posterior probability of x being on the transition bridge is the probability of y to be located outside the object to be segmented. Under the assumption that the intensity values are independent vari-ables, if we denote I' the intensity of the voxel y, we have:

)'()( IByPIByP ∈=∈ (27)

Then Bayes rule leads to:

)'(

)'()()(

IP

TyIPTyPIByP

∈⋅∈=∈

(28)

The equation (26) can then be written as:

⎪⎪⎩

⎪⎪⎨

⎧

∈⋅−+⋅

⋅

∈⋅−+⋅

⋅−

=Bif

IpIp

Ip

TifIpIp

Ip

IxP

eTii

eT

eTii

eT

bridge

ωηη

η

ωηη

η

ω

)'()1()'(

)'(

)'()1()'(

)'()1(

),(

(29)

We return now to the GFMM, to be able to calculate Pbridge, and for each point x belonging to F (x ∈ F+, or x∈ F-), we detect the set of points EP such as:

{ })'()()(',' xxandxVxxEP θθ −=∈= . If EP contains more than one element, we choose x’∈ EP verified:

⎩⎨⎧

∈∀′′′∈∀′′′

)(<)(,),(>)(

)(>)(,),(>)(

IPIPifEPqIPIP

IPIPifEPqIPIP

eiii

eiee (30)

where I, I’ and I’’ are the gray levels of the voxels x ,x’ and q respectively, Then we calculate Pbridge(x) using the formula (29). The data consistency term SF at point x belonging to the interface is defined as a decreasing function of Pbridge(x), and given by:

)),(( ωIxPgSF bridgeb=

(31)

The decreasing function gb is given by the following formula:

⎪⎩

⎪⎨⎧

−<−

=∈∀elsex

xifxxgx b 2

2

)1(3

5.031)(],1,0[

(32)

4 Experiments and Results

In order to test the BGFMM for the segmentation of the brain structures several series of experiments were made on varied data bases, also quantitative measurements were performed on a data base comprising a manual segmentation used as reference(ISBR). For all the series of the experiments we have initialised the number of class for the FCMA method to 5 for detecting the five regions (gray matter, white matter, cerebro-spinal fluid and the two other classes constituting of voxels corresponding to the par-tial volume (WM-GM) and (GM-CSF).

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 165

Fig. 2. Results of the segmentation of WM (first line), GM (second line) and CSF (third line of the image). The left images represent the 3D view of the results.

Fig. 3. Segmentation of (a) the whole brain and (b) the ventricles using the BGFMM

First, the method was applied to the segmentation of a particular class of tissue (white matter, grey matter, cerebro-spinal fluid). For this purpose, we have initialized the surface with a small cube located inside the tissue. An example of the results is visible on figure 2 (In order to improve the visibility of the results a white mask was superimposed on the surface of the tissues). We have also evaluated the method by computing the overlapping index DSC (Dice Similarity Coefficient) between the manual segmentation of the three tissues provided with the IBSR base, considered as ground truth, and our results. The values of similarity obtained for the 18 volumes of this base treated for each tissue are presented on the curves of figure 4(A). In another series of experiments, we have segmented the whole brain, and the ventricles. For the whole brain, the segmentation was initialized with a cube 100×80 ×80, and the classes

(A)

(B)

166 M. Baghdadi, N. Benamrane, and L. Sais

Fig. 4. Values of DSC for the segmentation of: (A) white matter, grey matter, and CSF. (B) whole brain, and the ventricles.

of interest (grey matter + white matter) automatically determined as described in section (3.4). An example of the results is visible on figure 3. The quantitative results are presented on figure 4(B).

The method gives satisfying results for the segmentation of the three brain tissues (WM, GM, CSF) of the whole of the bases, although more or less marked according to subjects. Also for the brain and the ventricles, in particular the furrows of the brain and the contours of the ventricles are well segmented.

It remains that the segmentation of the ventricles is not always perfect expressed by a low value of DSC. This can be explained by the fact that the ventricles are struc-tures of small size what is reflected on the values of the DSC. Also the characteristics of the base have an influence on the results because the dynamics of the intensity is reduced even very reduced for certain volumes of the ISBR base.

5 Conclusion

We have proposed in this paper a new approach based on the Deformable model and on the Bayesian model for the segmentation of 3D medical images. This new model was applied more specifically to brain MRI volumes for the segmentation of the brain structures. This approach is divided into two parts. Initially, a preliminary stage al-lows constructing the information map. Then, a deformable model, implemented with the Generalized Fast Marching Method (GFMM), evolves towards the structure to be segmented. Our contribution consists of the use and the improvement of the GFMM for the segmentation of 3D images and the design of a robust evolution model based on adaptive parameters. In a future work we expect to enrich this evolution model with some a priori knowledge (expert, anatomical atlas, form models, space rela-tions...) to improve the performances of the method and also to extend it to more difficult and complicated applications.

Segmentation of 3D Brain Structures Using the Bayesian Generalized Fast Marching 167

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