MUL

TIRESOLUTION MESH SEGMENTATION OF MRI BRAIN

USING CLASSIFICATION AND DISCRETE CURVATURE

Sami Bourouis, Kamel Hamrouni

Ecole Nationale d’Ing

´

enieurs de Tunis

Laboratoire de Syst

`

emes et de Traitement du Signal : LSTS

Enit, BP-37, Le Belv

´

ed

`

ere 1002 Tunis, Tunisia

Mounir Dhibi

Ensieta E312, 2 rue Francois Verny 29806 Brest Cedex -9, France

Keywords:

Brain segmentation, MRI, Statistical classiﬁcation, Progressive meshes, Mesh segmentation, discrete curva-

tures.

Abstract:

This paper presents a method for brain tissue segmentation and characterization of magnetic resonance imag-

ing (MRI) scans. It is based on statistical classiﬁcation, differential geometry, and multiresolution representa-

tion. The Expectation Maximization algorithm and k-means clustering are applied to generate an initial mask

of tissue classes of data volume. Then, a hierarchical multiresolution representation is applied to simplify pro-

cessing. The idea is that the low-resolution description is used to determine constraints for the segmentation at

the higher resolutions. Our contribution is the design of a pipeline procedure for brain characterization/labeling

by using discrete curvature and multiresolution representation. We have tested our method on several MRI

data.

1 INTRODUCTION

Medical image analysis is a complex task because

medical images usually involve a large amount of

data. The process of generating 3D views to visu-

alize structural information and spatial anatomic rela-

tionships is so difﬁcult, which is usually carried out

in the clinician’s mind. Magnetic Resonance Imaging

provides three-dimensional description of the human

brain which serves as a reference for clinical investi-

gations as well as for functional studies. Geometric

modelling, mesh representation and mesh segmenta-

tion of anatomical structures in 3D images are becom-

ing an increasingly important processing step for vi-

sualization, analysis and interpretation of the brain ac-

tivity. Moreover, geometric measurement of the cor-

tex helps in understanding brain anatomy and func-

tion. In addition, the patient brain surface labeling

provides a map of the brain folds where the neuro-

radiologist and the neurosurgeon can easily track the

features of interest.

The 3D mesh representation is widely used to rep-

resent 3D objects in various applications. It is also

used to modelize and visualize complex objects and

scenes, which cannot be modelled by any geometric

functions. One interesting problem in analysis of sur-

face mesh is the mesh segmentation. The problem

is to cluster vertices of a mesh that are close to each

other and have similar property (e.g., curvature or dis-

tance to a ﬁtting plane) into the same group. Mesh

segmentation has become an important component in

the ﬁeld of medical applications and computer graph-

ics. For example, the need to precisely locate the site

of functional activation within an automatic frame-

work requires the characterization/labeling of the sur-

face mesh.

In the last several years, many algorithms have

been proposed in this growing area, offering a di-

versity of methods and various evaluation criteria.

For example, Mangan and Whitaker (Mangan and

Whitaker, 1999) proposed an approach using the wa-

tershed algorithm for partitioning 3D surface meshes.

In their method, total curvature is computed and used

in segmentation. Initial segments are set up from local

minima of the total curvature. However, in some ob-

jects, local minima of the total curvature is not easy

to ﬁnd. Jaume et al. (Jaume et al., 2002) proposed

an algorithm that matches progressively an atlas la-

beled mesh to the patient brain mesh from the largest

folds to the smallest ones. Then they transfer the la-

bels from the matched mesh to label the patient mesh.

Due to the surface meshes produced at increas-

421

Bourouis S., Hamrouni K. and Dhibi M. (2008).

MULTIRESOLUTION MESH SEGMENTATION OF MRI BRAIN USING CLASSIFICATION AND DISCRETE CURVATURE.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 421-426

DOI: 10.5220/0001078704210426

Copyright

c

SciTePress

ing complexity, the need for an accurate simpliﬁca-

tion of surface meshes is becoming more and more

urgent. This need is not only due to rendering speed

reasons, but also to allow fast transmission of 3D

models in network-based applications. Indeed, efﬁ-

cient simpliﬁcation algorithms may optimize initial

meshes and replace them by nearly indistinguishable

approximations and can hope to automate these tasks.

In recent years, the problem of surface simpliﬁca-

tion has received increasing attention. Several dif-

ferent algorithms have been formulated for simpli-

fying surfaces. Schroeder et al. (Schroeder et al.,

1992) proposed an algorithm ”decimation of triangle

meshes” to reduce the number of triangles in a trian-

gle mesh, while preserving the original topology and

forming good approximation to the original geometry.

Another algorithm has been proposed by Rossignac

and Borrel (Rossignac and Borrel, 1993). It is one

of the few works which is able to process arbitrary

polygonal input. Others algorithms have been pub-

lished that simplify models by iteratively contracting

edges. The essential difference between these algo-

rithms lies in how they choose an edge to contract.

Some notable examples of such algorithms are those

of Hoppe (Hoppe, 1996) and Ronfard et al. (Ronfard

and Rossignac, 1996).

This work is initially motivated by the need to

describe and characterize 3D data in the analysis of

functional imaging data. We propose in this paper

a pipeline steps for the problem of 3D mesh seg-

mentation of the brain in magnetic resonance images.

Our algorithm uses three stages to segment MRI im-

age. Initially, data is classiﬁed into three main classes

in order to extract white matter (WM), gray matter

(GM), cerebral-spinal ﬂuid (CSF). Then, we generate

a discrete representation of each tissue class and we

create a mutiresolution representation for each one. In

the third stage, we segment the obtained meshes using

a criteria based on discrete curvature. The proposed

framework is presented in ﬁgure 1.

2 DISCRETE REPRESENTATION

OF BRAIN TISSUES

The proposed framework uses a combination of sta-

tistical classiﬁcation, isosurface extraction, multires-

olution representation and brain labeling. These steps

address various problems in computer graphics, sci-

entiﬁc visualization, analysis and interpretation of the

brain activity and many others. In the next sections

we explain in details our method and present some

results.

Figure 1: The general principle of the method.

2.1 Expectation Maximization

In statistical classiﬁcation, each object is represented

by d features, and the goal of classiﬁcation becomes

ﬁnding compact and disjoint regions for classes in a

d-dimensional feature space. Such decision regions

are deﬁned by decision rules that are known or can

be trained. In the case of unsupervised classiﬁca-

tion, the Expectation Maximization (EM) algorithm

(Dempster et al., 1977), is an efﬁcient iterative proce-

dure to compute the Maximum Likelihood (ML) esti-

mate in the presence of hidden data f. In short, the EM

alternates between two steps: an expectation (E) step

and a maximization (M) step. In the E-step, the miss-

ing data are estimated given the observed data and

current estimate of the model parameters. In the M-

step, we compute the maximum likelihood estimates

of the parameters by maximizing the expected likeli-

hood found in the E-step. The parameters found in

the M-step are then used to begin another E-step, and

the process is repeated.

Let us consider the mean parameter µ

k

and the

variance parameter σ

k

of the intensity distribution

of the k-th tissue class grouped in θ

k

such as θ

k

=

{µ

k

,σ

k

}. We denote also π

k

the prior probability of

each class k and γ

k

i

the posteriori probability calcu-

lated in each voxel i for each class k. In the expec-

tation step, we calculate the posteriori probability ac-

cording to the following formulas:

γ

k

i

= P(x

k

/y

k

,θ

k

) =

π

k

i

f

k

(y

i

/θ

k

)

∑

K

l=1

π

l

i

f

l

(y

i

/θ

l

)

(1)

In the maximization step, we estimate data driven pa-

rameters by:

π

k

=

∑

K

l=1

γ

k

i

N

µ

k

=

∑

K

l=1

γ

k

i

y

i

∑

K

l=1

γ

k

i

σ

2

k

=

∑

K

l=1

γ

k

i

(y

i

−µ

k

)(y

i

−µ

k

)

∑

K

l=1

γ

k

i

(2)

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

422

2.2 K-means Clustering

K-means clustering is a popular clustering algorithm

because it is simple and usually converges to a reason-

able solution. Here, we apply the k-means algorithm

in order to deﬁne classes that represent statistical dis-

tributions of intensity values in the voxels and to ex-

tract different tissues. K-means works as follows:

• Obtaining the initial k means input,

• Assigning each measurement vector in a sample

container to its closest mean among the k number

of means,

• Calculating each cluster’s mean from the newly

assigned measurement vectors (updates the cen-

troid (mean) of k clusters),

• Repeating step 2 and step 3 until it meets the ter-

mination criteria.

In the context of Medical Imaging, each class is

typically associated to a particular type of tissue and

can therefore be used as a form of image segmenta-

tion. Fig.2 illustrates the classiﬁcation of an MRI data

into three classes (k=3): GM, WM and CSF. Parame-

ters are estimated using EM.

Volumetric Data

Classification

Volumetric Data

Classification

Figure 2: k-means clustering.

2.3 Generating Mesh

Representing the surface as explicit geometry is ef-

ﬁcient when used with the conventional computer

graphics approaches for shading and viewing. Fur-

ther, it greatly reduces the necessary data storage and

provides a data structure that can be measured. In our

case, the initial volumetric MRI data is transformed

into a triangulation using a standard isosurface ex-

traction algorithm -Marching Cubes- (Lorensen and

Cline, 1987). Marching cubes provides an accu-

rate three-dimensional polygonal representation that

can be used for other image processing. A cube is

bounded by eight pixels located in two adjacent slices.

Each vertex is coded as either inside or outside the ob-

ject relative to the surface-deﬁning threshold. Based

on the conﬁguration of vertices that lie inside and out-

side the object, the cube is triangulated. The triangles

indicate where the surface passes through the cube.

However, the output from the marching cubes usu-

ally contains multiple small ”useless” meshes. Since

these meshes are physically disconnected from each

other. In order to reduce aliasing artifacts in images,

we use the algorithm proposed in (Whitaker, 2000).

Brieﬂy, the algorithm is as follows: The binary vol-

ume is used as a set of constraints in an iterative relax-

ation process of an initial estimate of the surface. The

surface is allowed to deform under curvature ﬂow. A

set of constraints is introduced in its movement which

ﬁts the surface to the volume in such a way as to min-

imize surface area, producing a smooth estimation.

3 MESH SEGMENTATION

A surface’s behavior can be described by dividing the

surface into distinct regions of elliptic behavior and

hyperbolic behavior. The regions of elliptic behavior

can be classiﬁed into convex and concave regions by

considering the direction of the surface normal.

3.1 Mesh Simpliﬁcation

Marching Cubes is very efﬁcient but usually produces

more than the necessary number of polygons needed

to represent an object accurately. It records all details

associated with the original data regardless of whether

these details are insigniﬁcant. The result contains an

enormous number of extremely small triangles that

prevent interactive render of models. Therefore, the

problem is how to create a new set of surfaces, which

are less complex than the original one without mod-

ifying the overall geometric information. To solve

this problem, we have resorted to mesh simpliﬁca-

tion techniques. Indeed, if we segment the surface at

a high resolution, the segmentation will be too com-

plex. A way to accelerate this process and preserve

topology is to create a low-resolution representation

of the model by using the PM algorithm. Thus, mesh

segmentation at a low resolution deﬁnes constraints

for more complex features detected at the higher res-

olution.

The PM algorithm (Hoppe, 1996) is attractive be-

cause it simpliﬁes a model by minimizing an energy

function. Indeed, the method provides four visual

beneﬁts. It uses many polygons to represent areas of

high curvature. It aligns long edges along directions

of low curvature. It adjusts the aspect ratios of trian-

gles to match the local curvature. Finally, it places

edges and vertices near sharp features of the original

MULTIRESOLUTION MESH SEGMENTATION OF MRI BRAIN USING CLASSIFICATION AND DISCRETE

CURVATURE

423

model. An iteration of this algorithm uses an edge

collapse operation. This operation (Fig. 3) selects

an edge and replaces it with a single vertex. This re-

moves one vertex, three edges, and two faces. The

edge collapse operation is attractive because it allows

the new vertex to be placed in a manner that helps

preserving the location and the shape of the original

surface.

Figure 3: Edge collapse operation .

3.2 Curvature-based Segmentation

In surface analysis, we compute gaussian curvature

and mean curvature, and later use them to classify the

surface type of vertices. The major complication is

that curvature cannot be directly evaluated for trian-

gle meshes because it is mathematically deﬁned for

smooth surfaces only. However, discrete differential-

geometry operators have been developed which can

estimate curvatures on triangulated manifolds.

many approximation schemes for curvature esti-

mates meshes have been developed such as (Max,

1999; Hamman, 1993). We apply here some op-

erators, which are derived recently by Meyer et al.

(Meyer et al., 2002), to estimate curvature informa-

tion at each vertex in the mesh. We denote k

H

and k

G

,

the Mean and the Gaussian curvature operators. In

our algorithm, we calculate these operators according

to the following formulas:

Figure 4: 1-ring neighbors and angles opposite to an edge.

(

k

H

=

1

2A

∑

j∈N(i)

(cot(α

i j

) + cot(β

i j

))(x

i

− x

j

)

k

G

=

2π−

∑

] f

j=1

Θ

j

A

(3)

Where α

i j

and β

i j

are the two angles opposite to the

edge in the two triangles sharing the edge (x

i

,x

j

) as

Table 1: Possible combinations of surface types according

to discrete curvature criteria.

Mean(x

i

) convex convex concave concave

Gauss(x

i

) hyperb elliptic hyperb elliptic

Color green yellow blue red

depicted in ﬁgure. 4. Θ

j

is the angle of the j-th face

at the vertex x

i

and ] f denotes the number of faces

around this vertex.

If the value of k

G

is negative, then we have a

hyperbolic behaviour, otherwise it is an elliptic one;

then, the regions of elliptic or hyperbolic behaviour

is convex if the k

H

value is negative, otherwise it is

concave. This mechanism is illustrated in the ﬁgure

5.

Figure 5: Curvature-based criteria for surface characteriza-

tion.

4 EXPERIMENTAL RESULTS

We have performed a series of experiments on brain

MR images. Resulting segmentations are depicted in

Figures 6, 7 and 8. Each label corresponds to a dif-

ferent color. The two curvatures used are mean and

gaussian. By combining the operators k

H

and k

G

, we

can encode surface types with different color accord-

ing to table 1. In our case, this leads to four cases (

table 1).

Our segmentation consists of regions of the same

type of curvature according to Table 1 by encoding

surface types with colors. Figure 6 shows the labeling

of the cortical surface into elliptical convex regions

(yellow), elliptical concave regions (red), hyperbolic

convex regions (green), and hyperbolic concave re-

gions (blue). These segmentations are performed in

the case of multiresolution representation as shown

in ﬁgures 7 and 8). An inherent difﬁculty in the in-

terpretation of these results is that there is no def-

inition of what is correct. Some papers use expert

opinion from neurologists on where sulci and gyri ex-

ist. However, visual results can be interpreted given

the idea that gyri are the surfaces top of the brain

folds (ridges), and that sulci are the surfaces within

the brain folds (basins). For example, it will be easy

to distinguish between elliptic and hyperbolic forms.

The ﬁnal labels resulting from our segmentation are

depicted graphically in ﬁgures 7 and 8. Each label

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

424

corresponds to a different color in all obtained res-

olutions. According to these results, we notice that

segmentation at low-resolution (10% of the original

data) leads to an extraction of the major surface fea-

tures/characteristics, whereas when the algorithm is

applied to the surface at high resolution (100%), it

does not necessary produces more relevant structure

information but includes small features, which do not

contribute very much to the overall shape. In addi-

tion, low-resolution description can be used to deter-

mine constraints for the segmentation at higher reso-

lution. For instance, these ﬁrst results are encourag-

ing but further investigation is required to extend the

algorithm to a large range of data.

Figure 6: Segmentation of the cortical surface according to

table 1.

5 CONCLUSIONS

In this paper, an automated approach for brain MRI

segmentation and discrete mesh characterization is

proposed. We have currently explored and outlined

the importance of multiresolution representation to

simplify processing, segment meshes and accelerate

medical analysis. Our technique has a great interest

in the study of structural and functional characteris-

tics of the brain. It is also relatively computationally

efﬁcient. At this stage, we have only applied the ap-

proach to a few experimental cases and we have pre-

sented some preliminary results to demonstrate its po-

tential: the method gives satisfying results for mesh

labelling in the case of multiresolution representation.

Even if they have not yet been compared to manual

or other automatic segmentation results, we think that

they are encouraging and faster than manual proce-

dures. However, there are some future works to do.

Clinical validation remains to be done, which will re-

quire additional work. Future validations will com-

pare our segmentation with manually labelled data

and other segmentation results. Finally, the same

framework can be used and extended to segment and

quantify abnormal brains.

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MULTIRESOLUTION MESH SEGMENTATION OF MRI BRAIN USING CLASSIFICATION AND DISCRETE

CURVATURE

425

Figure 7: Segmentation of ventricle surface at different resolutions according to table 1.

100% of original data

50%

20%

10%

3%

100% of original data

50%

20%

10%

3%

Figure 8: Segmentation of cortical surface at different resolutions according to table 1.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

426