ASSESSING THE VARIABILITY OF INTERNAL BRAIN

STRUCTURES USING PCA ON SAMPLED SURFACE POINTS

Darwin Mart

ınez

1,2

, Isabelle Bloch

and Jos

e Tiberio Hern

andez

TELECOM ParisTech (ENST) - CNRS UMR 5141 LTCI, Paris, France

Universidad Los Andes - Imagine Team, Bogot

a, Colombia

Keywords:

Anatomical variability, MRI images, Brain structures, Principal component analysis.

Abstract:

In this paper we propose to analyze the variability of brain structures using principal component analysis

(PCA). We rely on a data base of registered and segmented 3D MRI images of normal subjects. We propose to

use as input of PCA sampled points on the surface of the considered objects, selected using uniformity criteria

or based on mean and Gaussian curvatures. Results are shown on the lateral ventricles. The main variation

tendencies are observed in the orthogonal eigenvector space. Dimensionality reduction can be achieved and

the variability of each landmark point is accurately described using the ﬁrst three components.

1 INTRODUCTION

Assessing the variability of natural objects has been

the purpose of many works in different domains. Con-

cerning human beings, the analysis of the variabil-

ity of morphological or functional features permits

a better description of the human body and its vari-

ous components, a categorization of populations, and

a study of differences between normal and patholog-

ical cases. The variability can be assessed from sev-

eral sources, such as morphometric measurements or

medical imaging. Conversely, in medical imaging,

variability has also been used to guide segmentation

and recognition applications in several works, for in-

stance based on shape and appearance models (Hill

et al., 1996; Cootes et al., 2001; Larsen and Hilger,

2003).

In this paper, we address the question of shape

variability of normal internal brain structures, ob-

served in 3D magnetic resonance imaging (MRI).

The proposed framework relies on a surface-based

representation of different samples of a given struc-

ture, leading to a matrix of landmark points, which

is further analyzed using principal component analy-

sis (PCA). This approach departs from similar works,

where PCA is typically applied to distance maps char-

acterizing the shapes (see e.g. (Pohl et al., 2004),

among others). As an example, results are illus-

trated on the left ventricle, which presents an impor-

tant inter-subject variability.

This paper is organized as follows. In Section 2

we summarize some related works about the variabil-

ity measurements in human body and in other con-

texts. In Section 3 we address the representation is-

sues. Two surface sampling methods are proposed,

one relying on a regular subdivision of space, and one

based on maximum curvature points. In Section 4

the estimation of the variability in a population us-

ing PCA is described. In Section 5 results on the left

ventricles are provided and analyzed.

2 RELATED WORKS

Studying the variability of a structure from a set of in-

stances of this structure allows us to characterize the

diversity in this set. This estimation can be made us-

ing different techniques based on the analysis of some

features of the structure to be analyzed.

In this paper we focus on human anatomical struc-

tures. However, inspiration can also be found in other

domains of application.

2.1 Variability in Human Body

Many works aim at measuring the differences in the

human body, usually focusing on speciﬁc organs. The

objective can be to describe the normal anatomy along

with its potential variations within a population, or to

172

Martínez D., Bloch I. and Hernández T. (2009).

ASSESSING THE VARIABILITY OF INTERNAL BRAIN STRUCTURES USING PCA ON SAMPLED SURFACE POINTS.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 172-179

DOI: 10.5220/0001790801720179

 SciTePress

assess the presence of some abnormality indicating

some diseases.

In this section, we mention a few examples of

works in this domain, with the aim of illustrating the

variety of addressed problems. This list is of course

far from being exhaustive.

In (Suinesiaputra et al., 2004) the authors model

the myocardial contractility using independence com-

ponent analysis (ICA) in order to identify the dys-

functional myocardium. The authors present a model

of the two contours at distinct times (systole and di-

astole) to create a set of points in the middle of the

contours in order to perform the measurements with

ICA.

In (L

otj

onen et al., 2004) the authors model the

atria, ventricles and epicardium of the heart in a 3D

shape, as a combination of short and long axis car-

diac MR images. For this task the authors use a cor-

respondence voxel by voxel and then construct a set

of landmarks from the surface of the objects which

are used as input to PCA and ICA algorithms in order

to compute the shape variation.

In (Taron et al., 2007) the authors model the left

ventricle as a thin plate spline. An additional esti-

mation of registration uncertainties is included in the

model before applying ICA to capture the variations.

In the work of (Berar, 2007) PCA is used to per-

form a statistical reconstruction of skull and face. Us-

ing PCA in combination with a feature space repre-

sentation then produces a skull facial partial recon-

struction.

In (Nain et al., 2007), the authors present a new

strategy for multiscale shape representation and seg-

mentation of internal brain structures, in particular

caudate nucleus and hippocampus. The shape repre-

sentation is based on conformal mapping and spher-

ical wavelets adapted to a given population. Shape

variations are determined using a spectral partition-

ing of coefﬁcients. The experimental results show the

power of this approach for capturing shape details and

integrating the model in a segmentation framework.

In a different domain, in (Bai et al., 2007) the au-

thors present ICA as a feature to ﬁnd a similarity met-

ric between different sets of fMRI, and the indepen-

dent components are then used to describe the highly

activated regions. In this way, the similarity between

sets is evaluated as the value of the maximum size of

overlap between regions corresponding to activation

maps.

All these works show how different types of fea-

tures are studied in order to keep the most represen-

tative data set, which is enough to evaluate the differ-

ences between elements. The selection of the repre-

sentative set is an important step for posterior assess-

ment.

2.2 Variability in other Image

Applications

Let us now brieﬂy mention a few other examples, in

other application domains.

In (Ekenel and Sankur, 2004) the authors focus on

the problem of feature selection for face recognition

and they propose a method to examine the subsets of

ICA and PCA features in order to increase recogni-

tion performance. For PCA they take the eigenvalues,

while for ICA they made some tests in order to select

the most descriptive/difference features.

In (Yang and Ding, 2002) the authors use a com-

bination of PCA and a symmetry plane (mirror) to

search the faces in a series of images. They divide the

space based on the mirror of the faces and apply PCA

to characterize the important elements in the face and

deﬁne the variations modes of each side.

In (Rogez and Orrite, 2005) the authors use ICA

and PCA in order to compute the variations of the hu-

man ﬁgure, and they model the human body as the

contour of the silhouette. Then they construct a mean

shape and use it as basis to ﬁnd the differences of the

silhouette.

These studies present the analysis of variability

using the combination of PCA and ICA to identify the

sources of variation from global context using PCA

and local context using ICA. The studies show clearly

the difﬁculty to select the correct number of variation

modes in ICA.

In summary, there are two main aspects to con-

sider:

1. Shape representation: a concise representation

scheme has to be determined that will allow a

comparison between different individuals based

on the represented shape.

2. Analysis technique: it should make it possible to

characterize and localize the shape variability in

the studied family, in such a way that a new indi-

vidual can be characterized based on that analysis.

3 SHAPE REPRESENTATION

In our experiments, we use the MRI images of the

IBSR database

, which contains a set of images from

Internet Brain Segmentation Repository. The MR

brain data sets and their manual segmentations were

provided by the Center for Morphometric Analysis at

Massachusetts General Hospital and are available at

http://www.cma.mgh.harvard.edu/ibsr/

ASSESSING THE VARIABILITY OF INTERNAL BRAIN STRUCTURES USING PCA ON SAMPLED SURFACE

POINTS

173

18 normal subjects, and for which a segmentation of

the main anatomical structures is available.

From these segmented images, we select one

structure of interest that will be subject to variabil-

ity analysis. For illustration purposes, we consider

the left lateral ventricle as a typical example (see Fig-

ure 1).

Figure 1: Left lateral ventricles from three different subjects

of the IBSR database, illustrating the variability of the shape

of this structure.

The ﬁrst step in our approach consists in con-

structing the set of elements on which PCA will be

applied. These elements are points of the object un-

der study (i.e. one particular anatomical structure),

observed on several subjects. They should be a good

representation of the shape of the object. In this sec-

tion, we describe the chosen representations.

Usually, a representation is chosen according to

one or several criteria, such as expressiveness of the

representation (information it provides, precision of

the description), possibility of reconstructing the ob-

ject, and complexity of the representation (costs it in-

duces on storage space and computation time).

For our application, we need a representation of

3D objects that has a reasonable size, while preserv-

ing the information quality. This issue is particularly

important in medical imaging. The MRI images used

in this work have a spatial resolution of about 1mm

and contain about 8.10

points. Even if a particular

anatomical structure only occupies a restricted part of

this volume, the amount of data to be processed in-

creases rapidly. This is ampliﬁed here by the fact that

several images of different subjects are used in each

computation step. This advocates for a sampling of

the objects in order to derive the representation.

Since we are dealing with simply connected ob-

jects, which are adequately represented by their sur-

face, we choose a surface sampling approach. The

gain in the representation, with respect to volumetric

sampling, depends on the shape of the object (high for

a spherical object, low for an almost linear one), but

is in all cases appreciable.

In this paper, we propose to use two types of sam-

pling. The ﬁrst one relies on a systematic and regular

decomposition of the space, while the second one is

shape-dependent and relies on high curvature points.

We denote by S

the surface of a given structure in

image i (i = 1..M), and by x

the jth sampled point on

( j = 1..N). The matrix X = ((x

)) will be the input

of PCA (see Section 4).

3.1 Description based on Regular Grid

This approach relies on a simple regular subdivision

of the bounding box of S

. The landmark points x

are

then deﬁned as the intersections between the planes

deﬁning the subdivision and S

. This is illustrated in

Figure 2.

Figure 2: Regular grid subdivision of the bounding box of a

structure and induced surface sampling.

These points are then projected on the other exam-

ples of the structures S

, for i = 2..M, using the ICP

method (Besl and McKay, 1992). This guarantees a

good correspondence between x

and x

for all j and

all examples S

and S

. This correspondence is re-

quired in order to be sure that the variability analysis

is based on similar features.

3.2 Curvature-based Description

The second approach we propose is to build the de-

scription from remarkable points on the surface, de-

rived from curvature information. This approach is

similar to the one developed in (Moreno et al., 2007;

Moreno et al., 2008) for constrained registration of

lung images. It is motivated by the fact that points

with high curvature are particularly relevant for de-

scribing a shape and distinguishing between different

shapes. We argue that they are also relevant for de-

scriptions aiming at variability analysis.

The proposed procedure is as follows. The Gaus-

sian and mean curvatures (deﬁned as the product and

the average of the principal curvatures, respectively)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

174

are computed at all points of S

. This computation

is based on a ﬁnite difference approximation of the

derivatives.

Then two ordered lists of points are built, with

decreasing absolute value of Gaussian (respectively

mean) curvature.

Landmark points are selected starting from the

highest curvature points, by picking alternatively

from the two ordered lists. An additional constraint

is added in order to have an almost uniform distri-

bution on the surface: a point is selected only if its

distance to already selected points exceeds some pre-

set threshold. The distance is computed conditionally

to the surface (i.e. geodesic distance), using a prop-

agation method, like the classical chamfer algorithm

(Borgefors, 1986), adapted to the geodesic case (the

masks deﬁning the local distances and used for prop-

agation are restricted to the surface points at each po-

sition during the propagation).

In order to deal with potentially ﬂat areas, zero

curvature points are considered as well, so as to guar-

antee the uniformity constraint. The results on one

example are displayed in Figure 3.

Figure 3: Distribution of points based on curvature calculus.

Finally, the selected points x

, j = 1..N are used

to ﬁnd the corresponding points of the other exam-

ples S

, i 6= 1, again using the ICP method. It should

be noted that the projected points do not necessarily

correspond to the highest curvature points of S

. How-

ever, they are close to those points (see Figure 4), and

provide an appropriate representation of the shape.

The fact that similar landmark points are used in all

objects is important for the next steps, and this is guar-

anteed by the proposed approach.

The obtained landmarks provide an homogeneous

description of the shape, thanks to the distance con-

straint, and rely on signiﬁcant points, thanks to the

curvature constraint. The correspondence between

these landmarks in all shape examples provides the

necessary information for variability analysis.

Figure 4: Landmark points computed on S

and their cor-

responding points obtained using ICP on another example

4 VARIABILITY MEASURES

BASED ON PCA

The choice of principal component analysis (PCA) as

a technique to model the variability among elements

of the same type is driven by its powerful feature of

proposing an order using the variance of the represen-

tative data of the examples of the analyzed set.

PCA is an orthogonal linear transformation be-

tween one data coordinate system to a new coordi-

nate system such that the new system takes advantage

of the variance associated to each axis. The compo-

nents are ordered according to the variance value and

the projections of the data on the ﬁrst components ex-

press the largest variations.

The landmarks x

, for i = 1..M and j = 1..N, are

organized in a matrix X where each row X

is built

from the coordinates of the landmarks computed on

(i.e. each row has 3N components).

In order to minimize the variation induced by the

differences in the position (as illustrated in Figure 5)

each shape is centered: the center of gravity of each

set (x

), j = 1..N is computed and subtracted from

. It follows that all shapes are registered, with their

new center of gravity equal to the origin of the space.

Moreover, this also achieves the zero mean distribu-

tion constraint imposed when applying PCA. For sake

of simplicity, we still denote by X the matrix obtained

after this registration step. Figure 6 illlustrates the set

of centered structures.

The next steps consist of a classical application

of PCA: the covariance matrix of the centered data

C = XX

is computed. Then its eigenvectors V and

eigenvalues D are computed. The eigenvalues are

sorted in decreasing order. We denote by D

the vec-

tor of ordered eigenvalues and V

the corresponding

matrix of eigenvectors. The columns of V

are then

the principal components.

By analyzing the variance of each component and

its contribution to the global variation we can select

how many components should be taken into account

ASSESSING THE VARIABILITY OF INTERNAL BRAIN STRUCTURES USING PCA ON SAMPLED SURFACE

POINTS

175

to represent the variability. Usually, only a few com-

ponents are sufﬁcient, thus leading to a reduced di-

mensionality.

Then we compute the projection of the centered

data on the new orthogonal space for each of the se-

lected components. We denote by P

= V

X the kth

projection where V

is the kth component of V

. Fi-

nally, we reconstruct the objects in the original space

by computing

= P

−1

X for each of the selected

components. By adding again the mean value that

was subtracted during the registration step, we ob-

tain a comparable set of data. For instance, if the ﬁrst

component is selected, then P

is the projection of the

data onto the ﬁrst principal component and

, which

is an M × 3N matrix, contains the projection of the

ﬁrst principal component for each of the M samples.

In our experiments with 3D data, we obtain, for

each landmark and for each component of variation

mapped onto the original space, the direction and

magnitude of data variation in that direction.

5 EXPERIMENTS AND RESULTS

In this section, we illustrate the proposed approach on

the analysis of the left lateral ventricle (as illustrated

in Figure 1). We used the manual segmentations

available for the 18 examples of the IBSR database.

The images in the data base are globally registered

and have been “positionally normalized” into the Ta-

lairach orientation (rotation only). The left lateral

ventricles of the data base are shown in Figure 5. As

mentioned in Section 4, a further registration based

on a translation matching all centers of gravity is also

applied. The centered shapes are displayed in Fig-

ure 6. This registration guarantees position normal-

ization and avoids interpreting the difference in abso-

lute position as a variability factor.

Figure 5: The 18 left lateral ventricles of the IBSR database.

The general scheme of the processing is illustrated

in Figure 7. From the segmented objects, sampled

points are computed using the two methods described

in Section 3. Then PCA is applied on the resulting

Figure 6: Set of centered structures.

matrices, as described in Section 4. The results are

analyzed based on the reconstruction using the main

components and the variance.

Figure 7: Experiment ﬂowchart.

The results depend on the following parameters:

• density of samples;

• choice of the distribution (regular, based on cur-

vature);

• the number of main components in the reconstruc-

tion.

In order to test the shape representation ap-

proaches presented in section 3, both set of results

are presented (curvature and regular grid based mod-

els) below. The ﬁgures 8 to 12 have been obtained

with a selection of landmark points based on curva-

ture information. On the ﬁrst example S

, 136 points

are selected according to the highest curvature and

uniformity criteria, and the corresponding points are

then computed on all other 17 examples. This re-

sults in a 18 × 408 matrix. The initial surfaces contain

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

176

about 2400 points in average, which makes a density

of 0.06.

Applying the proposed approach to the left lateral

ventricle leads to the following results:

• The principal component analysis made it possi-

ble to explain 75.67% of the variation of the left

lateral ventricle (Figure 8) with only three compo-

nents.

Figure 8: Variance of each of the principal components and

its contribution to the total variation.

• The ﬁrst principal component represents 44.9% of

the variance in the set of shapes. Thus, with this

only component we can express almost the half of

the variance of the whole set. Figure 9 represents

the ﬁrst mode of variation projected in the origi-

nal data space. Each point indicates the mean of

the landmark, while the line segments indicate the

direction and magnitude. These segments do not

exhibit strong structures, thus expressing each lo-

cal variation of the shapes.

Figure 9: First Component.

• The second principal component represents

18.94% of the variance. It is illustrated in Fig-

ure 10. It shows a kind of torsion between the

different shapes (the segments have a quite con-

sistent organization).

• The third principal component represents 11.83%

of the variance, and is shown in Figure 11. The

Figure 10: Second Component.

orientation of the segments on the right part of

the ﬁgure can be interpreted as a variance of the

global curvature of the shape.

Figure 11: Third Component.

Figure 12 presents the results of the three ﬁrst

components projected in the original space. It illus-

trates the variation of all landmarks in the data set and

the representation and contribution to the variance of

each landmark to the total variation. It also shows that

all landmarks have a similar contribution to the vari-

ability and that it is difﬁcult to assign the variation to

a particular subset of landmarks. This result was ex-

pected because PCA estimates the differences of a set

of samples (anatomical structures in this case) against

the mean values of the samples.

Figure 12: Three ﬁrst components.

The zoomed area in Figure 12 shows the inﬂuence

of each component (blue: ﬁrst component, green: sec-

ond component, red: third component). The inter-

ASSESSING THE VARIABILITY OF INTERNAL BRAIN STRUCTURES USING PCA ON SAMPLED SURFACE

POINTS

177

section of the three components corresponds to the

mean value of the landmark, and each little point on

the segment line represents the value of one projected

sample. This illustrates the range of variation of the

landmark and the set of points on each of the segment

lines shows an interesting distribution. We think that

it is an important characteristic that has to be evalu-

ated in the future because it can give us more speciﬁc

information about the variability in a population.

The results illustrated in the ﬁgures 13 and 14

have been obtained with a selection of landmark

points based on the regular grid division. For this ex-

ample (S

) 216 points are selected according of the

division by 10 of each side of the bounding box. The

point density over the surface is about 0.09% , which

is similar to the density we had in the curvature-

based point distribution. The corresponding points

are computed on all the other 17 examples, leading

to a 18 × 648 matrix to compute PCA.

Figure 13: Variance contribution using a point distribution

based on a regular grid.

The type of distribution (regular grid vs. curva-

ture based) does not affect representatively the vari-

ance contribution of the components, as shown in Fig-

ure 13. This ﬁgure is similar to Figure 8.

Figure 14: Three ﬁrst components on regular grid. Zoom in

two nearest points.

Although, regular grid representation is attractive

due to its simplicity and regularity, the model presents

some disadvantages: ﬁrst, it can miss some impor-

tant points (hidden points) for the shape representa-

tion and second, there is oversampling of points in

some shape sectors due to element form as is pre-

sented in Figure 14. The hidden points and oversam-

pling are risky for shape representation, and that is

why it is recommendable to use the curvature based

model and the geodesic distance on the surface, in

spite of its simplicity and computational performance

(1 / 10, regular grid / curvature).

6 CONCLUSIONS

In this paper, we have proposed a method for the study

of shape variability of anatomical structures in the hu-

man brain using information extracted from magnetic

resonance images. The main features of the method

are:

• representation of the surface of the structures, for

which two methods have been proposed for land-

mark selection, as well as a correspondence be-

tween points of different examples;

• construction of a data set with zero mean in each

sample in order to reduce the inﬂuence of position

variation between the structures;

• use of PCA as a technique of analyzing variabil-

ity;

• appropriate visualization of the results.

The experimental results show the interest of the pro-

posed approach for variability analysis and data re-

duction. In the illustrated example for the left lateral

ventricle, we could express the major contribution of

variability with only three components.

Further work aims at analyzing in more detail

the inﬂuence of the parameters of the method, and

comparing the results with those obtained using in-

dependent component analysis (Cardoso and Comon,

1996). Experiments on other brain structures will be

carried out as well. A potential application could be

to compare the observed variations in different popu-

lations (normal and pathological ones for instance).

ACKNOWLEDGEMENTS

This work has been partly supported by an ECOS-

Nord grant, Darwin Martinez is supported by a

COLCIENCIAS-Bonpland grant and Universidad de

los Andes project CIFI-56 of 2008.

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178

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