Shape analysis of brain ventricles using SPHARM

1,2G. Gerig,1M. Styner,3D. Jones,3D. Weinberger,2J. Lieberman

1Department of Computer Science2Department of Psychiatry

University of North Carolina, Chapel Hill, NC 275993National Institute of Mental Health NIMH

Clinical Brain Disorders Branch, Bethesda, MD 20892email: gerig@cs.unc.edu

Abstract

Enlarged ventricular size and/or asymmetry have beenfound markers for psychiatric illness, including schizophre-nia. However, this morphometric feature is non-specific andoccurs in many other brain diseases, and its variability inhealthy controls is not sufficiently understood. We studiedventricular size and shape in 3D MRI (N=20) of monozy-gotic (N=5) and dizygotic (N=5) twin pairs. Left and rightlateral, third and fourth ventricles were segmented fromhigh-resolution T1w SPGR MRI using supervised classifi-cation and 3D connectivity. Surfaces of binary segmenta-tions of left and right lateral ventricles were parametrizedand described by a series expansion using spherical har-monics. Objects were aligned using the intrinsic coordinatesystem of the ellipsoid described by the first order expan-sion. The metric for pairwise shape similarity was the meansquared distance (MSD) between object surfaces. Withoutnormalization for size, MZ twin pairs only showed a trendto have more similar lateral ventricles than DZ twins. Afterscaling by individual volumes, however, the pairwise shapedifference between right lateral ventricles of MZ twins be-came very small with small group variance, differing sig-nificantly from DZ twin pairs. This finding suggests thatthere is new information in shape not represented by size, aproperty that might improve understanding of neurodevel-opmental and neurodegenerative changes of brain objectsand of heritability of size and shape of brain structures. Thefindings further suggest that alignment and normalizationof objects are key issues in statistical shape analysis whichneed further exploration.

1 Introduction

Quantitative morphologic assessment of individual brainstructures in neuroimaging most often includes segmenta-tion followed by volume measurements. Volume changesare intuitive features as they might explain atrophy or dila-tion of structures due to illness. On the other hand, struc-tural changes like bending/flattening or changes focused ata specific location of a structure, for example thickeningof the occipital horn of ventricles, are not sufficiently re-flected in global volume measurements. Development ofnew methods for three-dimensional shape analysis incor-porating information about statistical biological variabilityaims at tackling this issue.

Davatzikos [1] proposed an analysis of shape mor-phometry via a spatially normalizing elastic transformation.Inter-subject comparisons were made by comparing the in-dividual transformations. The method is applied in 2D toa population of corpora callosa. A similar approach in 3Dhas been chosen by Joshi et al[2] to compare hippocampi.Using the viscous fluid transformation proposed by Miller[3], inter-subject comparisons were made by analyzing thetransformation fields. Theanalysisof transformation fieldsin both methods has to cope with the high dimensionalityof the transformation and the sensitivity to the initial posi-tion. Although the number of subjects in the studied pop-ulations is low, both show a relatively stable extraction ofshape changes (see Csernansky[4]). Quantitative analysisof shape changes provided by the deformation fields, ex-pressed as point-wise changes in size, can be easily inter-preted locally[5, 6]. A difficulty, however, is the inabilityof deformation fields to capture subtle changes that mightbe related to the various scales at which the object’s geo-metric features are manifested.

The approach taken by Kelemen[7] evaluates a pop-

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ulation of 3D hippocampal shapes based on a bound-ary description by spherical harmonic basis functions(SPHARM), which was proposed by Brechbuhler [8, 9].The SPHARM shape description delivers an implicit corre-spondence between shapes on the boundary, which is usedin the statistical analysis. As in the approaches discussedbefore, this approach has to handle the problem of high di-mensional features versus a low number of samples. Fur-ther, the detected shape changes are expressed as changesof coefficients that are hard to interprete.

Golland[10] in 2D and Pizer et al[11, 12] in 3D pro-posed two different approaches of applying shape analysisto a medial shape description. Blum[13] claims that medialdescriptions are based on the idea of a biological growthmodel and a ’natural geometry for biological shape.’ Themedial axis in 2D captures shape intuitively and can be re-lated to human vision (see Burbeck[14] and Siddiqui[15]).Both Pizer and Golland propose a sampled medial modelthat is fitted to individual shapes. By holding the topol-ogy of the model fixed, an implicit correspondence betweenshapes is given and statistical shape analysis can directlt beapplied.

This paper applies a technique originally developed formodel-based segmentation, the SPHARM shape represen-tation of object surfaces[8, 7], to analyze brain structures.In particular, we address the clinical research problem ofstudying similarity of brain structures in idential (monozy-gotic, MZ) and non-identical (dizygotic, DZ) twin pairs.The paper is organized as follows. First we discuss theSPHARM description and its use for shape analysis, withspecial emphasis on the issues alignment and normaliza-tion of structures prior to measuring shape difference. Wethen present results of the lateral ventricle study in MZ/DZtwins.

2. Methods

2.1 Spherical harmonics (SPHARM)

In summary, the SPHARM description is a hierarchical,global, multi-scale boundary description that can only rep-resent objects of spherical topology. The basis functionsof the parameterized surface are spherical harmonics. Kele-men[7] demonstrated that SPHARM can be used to expressshape deformations. Truncating the spherical harmonic se-ries at different degrees results in object representations atdifferent levels of detail. SPHARM is a smooth, accuratefine-scale shape representation, given a sufficiently smallapproximation error.

In the next sections, we briefly describe the mathemat-ical properties of spherical harmonic descriptors, and theparameterization computation. Also, we discuss how to es-tablish correspondence between different objects described

by SPHARM. This correspondence is used to compute theMean Squared Distance (MSD) between surfaces, the met-ric for measuring shape difference used herein.

2.1.1 Spherical harmonics descriptors

This section gives a summary of spherical harmonic follow-ing Brechbuhler[8].

Spherical harmonic basis functionsY ml , −l ≤ m ≤ lof degreel and orderm are defined onθ ∈ [ 0;π ] × φ ∈[ 0; 2π) by the following definitions[16] (see Fig. 1 left fora visualization of the basis function):

Y ml (θ, φ) =

√2l + 1

4π(l −m)!(l +m)!

Pml (cos θ) eimφ(1)

Y −ml (θ, φ) = (−1)m Y ml∗(θ, φ) , (2)

whereY ml∗ denotes the complex conjugate ofY ml andPml

describes the associated Legendre polynomials

Pml (w) =(−1)m

2l l!(1− w2)

m2dm+l

dwm+l(w2 − 1)l. (3)

To express a surface using spherical harmonics, thethree coordinate functions are decomposed and the sur-facev(θ, φ) = (x(θ, φ), y(θ, φ), z(θ, φ))T takes the form

v(θ, φ) =∞∑l=0

l∑m=−l

c ml Y ml (θ, φ) , (4)

where the coefficientscml are three-dimensional vectors dueto the three coordinate functions. The coefficientscml areobtained by solving a least-squares problem. Therefore,the values of the basis functions are gathered in the ma-trix z = (zi,j(l,m)) with zi,j(l,m) = Y ml (θi, φi), wherej(l,m) is a function assigning an index to every pair(l,m)and i denotes the indices of thenvert points to be ap-proximated. The coordinates of these points are arrangedin v = (v1,v2, . . . ,vnvert)

T and all coefficients are gath-ered inc = (c 0

0 , c−11 , c 0

1 , . . .)T . The coefficients that best

approximate the points in a least-squares sense are obtainedby

c = (zTz)−1zTv . (5)

Using spherical harmonic basis functions, we obtain ahierarchical surface description that includes further detailsas more coefficients are considered. This is illustrated inFig. 1 right.

2.1.2 SPHARM description from voxel-based objects

The objects of interest are usually manually or semi-automatically segmented by a human expert, resulting in

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Figure 1. Decomposition of objects usingSPHARM description. Left: Visualization ofthe Spherical harmonic basis functions. Theplot shows the real parts of the spherical har-monic functions Y ml , with l growing from 0(top) to 5 (bottom), and m ranging from 0 (left)to l in each row. Right: Description of a lat-eral ventricle (side view) at different degree’s;m = 1,4,8,12 top to bottom.

a voxel representation. The voxel representation has to bepreprocessed to fulfill the precondition of sphere topology,e.g. via a closing operation or a smoothing filter. As weare only interested in its boundary, the voxel representationis converted to a polygonal surface mesh that serves as in-put for the optimization procedure that finds an appropriate(θi, φi) parameterization[8].

The appropriate parameterization of the points of a sur-face description is a key problem. Every pointi of the pointcloud that will be approximated by the surface descriptionis to be assigned a parameter vector(θi, φi). For surfacesof spherical topology, the natural parameter space is the unitsphere with polar coordinates. A homogeneous distributionof the parameter space is essential for the decompositionof the surface. This is also necessary for an appropriateapproximation of corresponding points, as described in thenext section 2.1.3.

A bijective mapping of the surface to the unit sphere iscreated, i. e., every point on the surface has to map to ex-actly one point on the sphere, and vice versa. The mainidea of the procedure is to start with an initial parameteriza-tion. This initial parameterization is optimized so that everysurface patch gets assigned an area in parameter space thatis proportional to its area in object space, while the distor-tion to the quadrilateral mesh is minimized. The proposedparametrization of surfaces is invariant to object scaling asthe whole surface is mapped to the unit sphere and opti-mized for homogeneous distribution of nodes.

2.1.3 SPHARM correspondence

The scheme for establishing correspondence between ob-jects described by SPHARM[8] is a 3D extension of the 2Darc-length shape parameterization (see also Szekely [17]).The first step is a homogeneous distribution of the parame-ter space over the surface, a step done in the parameteriza-tion optimization. In the second step the parameterization isrotated in the parameter space for normalization. This rota-tion is based on the first order ellipsoid, which is computedfrom the first three SPHARM coefficients. The result of therotation satisfies the following properties:

• The parameter locations of the poles of the first orderellipsoid match with the poles of the sphere.

• The parameter locations of the 3 main ridges of thefirst order ellipsoid are moving along the equator, andthe0 andπ meridians of the sphere.

Rotation in parameter space eliminates dependency ofthe parametrization from an arbitrary choice of poles of thesurface parametrization. Correspondence between surfacesis know defined as the point-to-point correspondence estab-lished by points with the same parameter vector(θi, φi).Whereas alignment and rotation in parameter space is thusnormalized on a coarse scale object representation (ellip-soid), the final correspondence is obtained by expansion ofthe object to a higher degree. The quality of the correspon-dence is shown in Fig. 2. The visual comparison demon-strates that normalizing the parameter space on a coarsescale description of objects results a good correspondencedespite the fact that it does not use explicit characteristicsurface features. A quantitative evaluation study againstmanual landmarking is currently in progress. Please notethat objects shown in Fig. 2 have been spatially normalizedas described in later section 2.2.

2.1.4 Mean Squared Distance (MSD) betweenSPHARM objects

The correspondence between objects described bySPHARM allows the computation of distance measuresbetween two objects. The orthogonality of the sphericalharmonic basis functions allows Parseval’s theorem to beused to compute the root Mean Squared Distance (

√MSD)

between two objects directly from their coefficients via adifference calculation. A correction is needed since thesquared spherical harmonic basis functions do not integrateto 1 but to 4π.

MSD=1

4π·

inf∑l=0

l∑m=−l

||cm1,l − cm2,l||2 (6)

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Figure 2. Visualization of the SPHARM corre-spondence. A first order ellipsoid and six leftlateral ventricles are displayed. The surfacenet shows the (θi, φi) parameterization (sameparameters = same homologous points). Theridges on the first order ellipsoid are the equa-tor and {0, π/2, π, 3 · π/2} meridian lines in allobjects. The equator and meridian lines areemphasized in different colors. The poles areat the crossing of the meridian lines.

Error measures other than√

MSD need an appropriatesampling of the spherical parameterization(θi, φi), for ex-ample by an iterative icosahedron subdivision of the spher-ical (θ, φ) parameter space. Using the point-to-point cor-respondence described previously and this sampling, er-ror measures like the Mean Absolute Distance (MAD),Hausdorff-distance or arbitrary quantiles derived from thehistogram of surface distances can be computed straightfor-wardly.

2.2 Object alignment and scaling

As a prerequisite for any shape similarity calculation,shapes have to be normalized with respect to a reference co-ordinate frame. Since we are interested in measuring shapedifferences, a normalization is needed to eliminate differ-ences that are due to rotation, translation and magnification.Normalization of translation and rotation is accomplishedby aligning the SPHARM objects via the first order ellip-soid. This perfectly matches center and axes of the firstorder ellipsoids (see Fig. 3).

In order to normalize for magnification, an appropriatescaling method has to be defined. The choice of the scalingmethod depends on the task and the type of objects. Weinvestigated two possibilities:

A No scaling correction: The computation of shape dif-ferences without any scale normalization reveals dif-ferences between small and large objects even thoughthey might have the same shape properties. Thus, the

differences will reflect mixed values of both the shapedifferences and the size differences.

B Uniform scaling to unit volume: Creating a shape dif-ference measure that is orthogonal in its nature to thevolume measure has the potential to reveal informationadditional to size. The volume measurements can beincorporated later into a multivariate statistical analy-sis as an additional orthogonal feature (see later Fig 8).

The effects of the two different scalings applied to thedriving clinical problem are illustrated in Fig. 3.

Alignment

No scaling Unit Volume scaling

Figure 3. Object alignment and scaling. Toprow: Two left lateral ventricles are aligned toperfectly match the center and axis of the firstorder ellipsoid; left: objects, middle: first or-der ellipsoids, right: aligned ellipsoids. Bot-tom row: Pairs of right lateral ventricles (MZtwin pair) unscaled (left) and scaled to unitvolume (right). This example shows thatshapes are quite similar and that the scalingcorrected for an existing size difference.

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3. Results

This section presents SPHARM applied to a populationof lateral ventricles, a fluid filled structure in the center ofthe human brain that is divided into a left and a right part lo-cated in the respective brain hemispheres. The image data ispart of a mono/dizygotic twin study and consists of 20 twinsubjects (10 pairs, 5 monozygotic and 5 dizygotic pairs).The original brain images were provided by D. Weinberger,NIMH Neuroscience in Bethesda, Maryland. The segmen-tation method used a single gradient-echo channel (T1w,256x256x128, 240mm FOV, 1.5mm slice distance) withmanual seeding for Parzen-window based non-parametricsupervised statistical classification. Manually-guided three-dimensional connectivity was used to extract the left andright lateral ventricles. The segmented structures were post-processed using a closing operation with a spherical struc-turing element of radius of two voxels to provided simplyconnected 3D objects as required for surface parametriza-tion.

Figure 4. Three-dimensional rendering of theskin surface (transparent) and the lateral ven-tricles. Lateral ventricle of left and right brainhemispheres are shown in green and orange.

Fig. 5 displays the lateral ventricles of all twin pairs. Thestatistical shape analysis presented below aimed to distin-guish monozygotic (MZ) twins from dizygotic (DZ) twinsand from unrelated pairs. The three subject groups arematched for age, gender and handedness. The populationsize of each group is very small (5 MZ, 5DZ, 10 unrelated),so the observed effect must be quite large for the statisticalanalysis to yield a significant result. A previous study wasperformed by Bartley et al[18] on the same datasets withthe goal of distinguishing the populations. They comparedcortical gyral patterns and the total brain volumes. Bothmeasures show significant differences between the MZ andDZ populations.

Figure 5. Visualization of the lateral ventriclesof all twin pairs (same color for pairs) scaledwith the individual volume (correct relativesize). Top two rows: Left ventricles. Bot-tom two rows: Right ventricles. Each block:Top row: MZ twins. Bottom Row: DZ twins.

3.1 Volume similarity analysis

We studied the twin pair’s similarity using signed vol-ume difference but also normalized absolute volume differ-ence: ∆volT1,2 = |volT1 − volT2 |/(volT1 + volT2). Theformer uses volumes normalized by individual size of intra-cranial cavity, whereas the latter is a relative measure andindependent of overall brain size. We will only discuss therelative measure as the absolute measure presented signifi-cantly more population noise. As shown in Fig. 6, there isa trend in both brain hemispheres between the two popula-tions, but no significant conclusions can be drawn since thevolume measurement distributions are overlapping. Thep-values for discriminating the two population are at 0.15 and0.16, which is non-significant at a5% significance level (seeTab. 1).

5

NR MZ DZ

0.02

0.04

0.06

0.08

0.1

Left Vol Diff, mm^3

NR MZ DZ

0.02

0.04

0.06

0.08

0.1

Right Vol Diff, mm^3

Figure 6. Plot of pairwise relative ventriclevolume difference ∆volT1,2 between MZ andDZ twin groups. Results of left and right ven-tricles are shown in the left and right figures.No significant conclusions can be drawn.

3.2 Shape analysis via SPHARM

The SPHARM coefficients of the objects were deter-mined and normalized with respect to rotation and transla-tion using the first order ellipsoid. Using the objects notnormalized for scaling, our analysis yields no significantdifference between the two populations as shown in Fig. 7and Table 1. There is just a trend showing a smaller volumedifference in MZ pairs versus DZ pairs. A significant differ-ence between MZ and DZ groups is observed for the lateralventricles in the right brain hemisphere after normalizing(isotropic scaling) the objects for individual volumes. Thep-value is at 0.019 (see Tab. 1), which suggests significanceat the5% level.

MZ/DZ MZ/Other DZ/Other

Volume Left 0.151 0.333 0.486Volume Right 0.167 0.377 0.500MSD unscaled Left 0.201 0.030 0.295MSD unscaled Right 0.145 0.042 0.419MSD scaled Left 0.106 0.046 0.825MSD scaled Right 0.019 0.009 0.471

Table 1. Table of p-values for discriminatingthe 3 populations of MZ and DZ twin pairs andunrelated pairs. Bold numbers are significantat the 5% level.

A closer analysis of the results plotted in Figure 7 revealsinteresting new information about the structure of brain ven-tricles in genetically identical MZ twin pairs, non-identicalDZ twins, and non-related but age and gender matchedpairs. We are well aware that we have to be cautious withconclusions due to the small sample size. Before size nor-

malization (upper row), the left and right ventricles showthe same trend, namely that MZ twins are more similar thanDZ twins. Group tests were not significant which is in partdue to the small sample size. After size normalization withindividual volumes, the right ventricles reveal a very inter-esting result. MZ twin pairs show a very small shape dif-ference and very low variability, suggesting that the shapesafter normalization are very similar (see also Fig. 5). Thisshows that differences of right ventricles in MZ twin pairsare mostly due to a global scaling (see Fig. 6) and that thereare only minor residual shape differences. Surprisingly, thisstrong and significant effect is not found for the left ven-tricles, where even after size normalization there is only atrend showing more similar ventricle shapes in MZ as com-pared to DZ. The statistics further illustrates that DZ twinpairs didn’t differ from unrelated pairs, both before and af-ter volume normalization.

No scaling

NR MZ DZ

2.5

5

7.5

10

12.5

15

17.5Left MSD, no scaling, mm

NR MZ DZ

2.5

5

7.5

10

12.5

15

Right MSD, no scaling, mm

Unit Volume scaling

NR MZ DZ

0.5

1

1.5

2

2.5

3

Left MSD, vol scaling, mm

NR MZ DZ

0.5

1

1.5

2

2.5

Right MSD, vol scaling, mm

Figure 7. Plot of MSD shape difference be-tween twins of the left (left column) and right(right column) lateral ventricles. Top row:Plots for shape difference when no scalingnormalization is applied. Bottom row: Plotswhen objects are isotropically scaled to unitvolume. The vertical axis displays the MSDshape difference in mm.

The splitting of size difference and residual shape dif-ference between pairs allows us to use both measures in a

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combined analysis, as there might be an interaction betweenvolume and shape differences leading to a better group dis-crimination or an improved understanding of morphologicdifferences. Figure 8 illustrates the two-dimensional featurespace with relative volume difference (vertical axis) andMSD shape difference of size-normalized objects (horizon-tal axis) for the left and right ventricles. The three groupsunrelated subjects (NR), MZ twins and DZ twins are indi-cated by triangles, squares and diamonds. Interpretation isdifficult due to the small sample size and the non-Gaussiandistribution of samples, suggesting that multi-variate non-parametric statistical techniques would be required for aquantitative analysis. A visual comparison shows that dis-crimination between MZ and DZ twin pairs is better for theright ventricles than for the left ventricles. Discrimination,however, would be mostly governed by the MSD shape dif-ference measure rather than the volume difference measure.

0 0.5 1 1.5 2 2.5MSD

0

0.02

0.04

0.06

0.08

0.1

Vol

Left MSD vs Vol diff

0 0.5 1 1.5 2MSD

0

0.02

0.04

0.06

0.08

Vol

Right MSD vs Vol diff

Figure 8. Plot of combined MSD shape differ-ence and volume difference between twins ofthe left (left column) and right (right column)lateral ventricles. MZ Twins are shown as redsquares, DZ Twins as blue diamonds and un-related subjects (NR) as black triangles. Lat-eral ventricles were scaled to unit volume forthe MSD shape difference measure.

4. Conclusions

This paper presents shape analysis using surfaceparametrization by spherical harmonics. The hierarchicalnature of the SPHARM shape representations allows ob-ject alignment by an intrinsic coordinate system derivedfrom a coarse scale representation, which is feasible for ob-jects depicting three distinctly different major axis. Previ-ous work[8, 7] and results presented in this paper demon-strate that this alignment results in a good initial point-to-point correspondence of surfaces if shapes are derived froma homogeneous shape population. A quantitative analy-sis of the quality of 3D correspondence is the subject ofa current study. The method presented herein can be seen

as complimentary to the seminal work by Toga et al.[5,6] as it does not express shape variability in Talairachstereotaxic space but an an intrinsic object-centered coordi-nate system. The shape distance metric used herein is equiv-alent to the mean square distance (MSD) between denslysampled corresponding surface points, however obtained bya quick calculation in parameter space. Alternative shaperepresentation schemes providing dense sets of correspond-ing points could be used as well. The MSD is known to besensitive to outliers. Future work will replace the MSD met-ric by statistical analysis of quantiles derived from surfacedistance histograms as discussed in the text.

The MZ/DZ twin study demonstrates that shape mea-sures reveals new information additional to size measure-ments which might become relevant for improved under-standing of structural differences in normal populations butalso in comparisons between healthy controls and patients.Twin studies offer the advantage to reduce natural biolog-ical variability by choosing subjects with identical genes.This study clearly demonstrates that significant differencesbetween MZ and DZ pairs could not be found by volumemeasurements but only by analyzing shape. Scaling by vol-ume showed that there is only a minor residual shape dif-ference for the right but not for the left ventricle, and thatthis lateralized shape effect results in a significant groupdifference between MZ and DZ twin pairs. We have noobvious explanation for this finding but hope to get moreinsight through close collaboration with experts in neuro-biology and neurodevelopment. Global scaling reveals aneffect otherwise hidden due to the fact that any size changewould also be reflected in the shape difference measure.The choice of scaling might depend on the application do-main and on the type of shape effects to be studies. Alter-native choices would include the size of the longest axis,an affine transformation, or a Procrustes fit between sets ofcorresponding points. A follow-up study currently analyzesdifferences between MZ twins discordant for schizophre-nia to reveal insight into hypothesized morphologic changesdue to illness. Analysis of shape changes similarly to thecase study presented here might also become important inlongitudinal assessments of morphologic change due to de-velopmental or degenerative processes.

A weakness of shape analysis by SPHARM is the non-intuitive nature of the set of coefficients describing shape.Shape difference findings, even found to be statistically sig-nificant, do not easily reveal the type and localization ofthe effect. Localized shape differences between groups areonly qualitatively accessible by 3D surface renderings la-beled with results of local test statistics (e.g.[4]) or by cineloops displaying a morphing between mean shapes. Theimportant issue of providing an intuitive shape descriptionexpressed in terms of natural language, e.g. change of widthor curvature) are currently addressed by developing an alter-

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native shape representation technique based on 3D medialrepresentations (skeletons)[19, 20].

Acknowledgements

Daniel Weinberger and Douglas Jones, NIMH,Bethesda, kindly provided the twin MR datasets andrelated material. Gabor Szekely and Christian Brechbuhlerare acknowledged for helping us with transfer of softwareand analysis tools. Further, we would like to thank theMMBIA reviewers for careful reading and for providingvaluable suggestions.

References

[1] C. Davatzikos, M. Vaillant, S. Resnick, J.L Prince,S. Letovsky, and R.N. Bryan, “A computerizedmethod for morphological analysis of the corpus cal-losum,” J. of Comp. Ass. Tomography., vol. 20, pp.88–97, Jan./Feb 1996.

[2] S. Joshi, M Miller, and U. Grenander, “On the geome-try and shape of brain sub-manifolds,”Pattern Recog-nition and Artificial Intelligence, vol. 11, pp. 1317–1343, 1997.

[3] G.E. Christensen, R.D. Rabbitt, and M.I. Miller, “3dbrain mapping using a deformable neuroanatomy,”Physics in Medicine and Biology, vol. 39, pp. 209–618, 1994.

[4] J.G. Csernansky, S. Joshi, L.E. Wang, J. Haller,M. Gado, J.P. Miller, U. Grenander, and M.I. Miller,“Hippocampal morphometry in schizophrenia viahigh dimensional brain mapping,”Proc. Natl. Acad.Sci. USA, vol. 95, pp. 11406–11411, September 1998.

[5] A.W. Toga and P.M. Thompson, “The role of im-age registration in brain mapping,”Image and VisionComputing, vol. 19, pp. 3–24, 2001.

[6] K.L. K.L. Narr, P.M. Thompson, T. Sharma, J. Mous-sai, R.E. Blanton, B. Anvar, A. Edris, R. Krupp,J. Rayman, M. Khaledy, and A.W. Toga, “3d map-ping of temporo-limbic regions and the lateral ventri-cles in schizophrenia,”Biological Psychiatry, vol. 50,pp. 84–97, 2001.

[7] G. Kelemen, A.and Szekely and G. Gerig, “Elas-tic model-based segmentation of 3d neuroradiologicaldata sets,” IEEE Trans. Med. Imaging, vol. 18, pp.828–839, October 1999.

[8] C. Brechbuhler, G. Gerig, and O. Kubler,“Parametrization of closed surfaces for 3-D shape

description,” CVGIP: Image Under., vol. 61, pp.154–170, 1995.

[9] C. Brechbuhler, Description and Analysis of 3-DShapes by Parametrization of Closed Surfaces, 1995,Diss., IKT/BIWI, ETH Zurich, ISBN 3-89649-007-9.

[10] P. Golland, W.E.L. Grimson, and R. Kikinis, “Sta-tistical shape analysis using fixed topology skeletons:Corpus callosum study,” inInf. Proc. in Med. Imaging,1999, pp. 382–388.

[11] S. Pizer, D. Fritsch, P. Yushkevich, V. Johnson, andE. Chaney, “Segmentation, registration, and measure-ment of shape variation via image object shape,”IEEETrans. Med. Imaging, vol. 18, pp. 851–865, Oct. 1999.

[12] M. Styner and G. Gerig, “Medial models incorporat-ing shape variability,” inInf. Proc. in Med. Imaging,2001, pp. 502–516.

[13] T.O. Blum, “A transformation for extracting new de-scriptors of shape,” inModels for the Perception ofSpeech and Visual Form. 1967, MIT Press.

[14] C.A. Burbeck, S.M. Pizer, B.S. Morse, D. Ariely,G. Zauberman, and J Rolland, “Linking object bound-aries at scale: a common mechanism for size andshape judgements,”Vision Research, vol. 36, pp. 361–372, 1996.

[15] K. Siddiqi, B. Kimia, S. Zucker, and A. Tannenbaum,“Shape, shocks and wiggles,”Image and Vision Com-puting Journal, vol. 17, pp. 365–373, 1997.

[16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, andB. P. Flannery,Numerical Recipes in C, CambridgeUniv. Press, 2 edition, 1993.

[17] G. Szekely, A. Kelemen, Ch. Brechbuhler, andG. Gerig, “Segmentation of 2-D and 3-D objects fromMRI volume data using constrained elastic deforma-tions of flexible Fourier contour and surface models,”Medical Image Analysis, vol. 1, no. 1, pp. 19–34,1996.

[18] A. Bartley, D. Jones, and D. Weinberger, “Geneticvariability of human brain size and cortical patterns,”Brain, vol. 120, pp. 257–269, 1997.

[19] P. Yushkevich and S. Pizer, “Coarse to fine shape anal-ysis via medial models,” inInf. Proc. in Med. Imaging,2001, LNCS 2082, pp. 402–408.

[20] G. Gerig and M. Styner, “Shape versus size: Improvedunderstanding of the morphology of brain structures,”in Medical Image Computing and Computer-AssistedIntervention, 2001, accepted for publication.

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