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CVPR#245

CVPR#245

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Multi-Object Statistics using Principal Geodesic Analysis in a LongitudinalPediatric Study

Anonymous CVPR submission

Paper ID 245

Abstract

A main focus of statistical shape analysis is the descrip-tion of variability of a population of geometric objects. Inthis paper, we present work in progress towards modelingthe shape and pose variability of sets of multiple objects. Asa standard technique, principal component analysis (PCA)is often employed for the computation of such a descriptionof shape variability. Both pose and our medial m-rep shapedescription are though not elements of an Euclidean vec-tor space, but rather elements of a nonlinear Riemanniansymmetric space. For such feature spaces, PCA is incor-rect and can be replaced with principal geodesic analysis(PGA), which correctly captures variability in this curvedfeature space. As for PCA, the initially large set of featuresis reduced to a small set of descriptive features. In this pa-per, we discuss the results of the PGA mean and variabilityin different normalization settings of multiple objects sets.We further discuss the problem of describing the statisticsof object pose and object shape and their interrelationship.

We demonstrate our shape modeling and analysis in anapplication to a longitudinal pediatric autism study withobjects sets of 10 subcortical structures in a populationof 20 subjects. The results show that after the removal ofglobal scale multi-object longitudinal data of the same sub-ject cluster closely in the PGA parameter space. Further-more, the PGA components and the corresponding distribu-tion of different subject groups vary significantly dependingon the choice of the normalization.

1. Introduction

Statistical shape modeling and analysis [6, 12] is emerg-ing as an important tool for understanding anatomical struc-tures from medical images. Statistical shape modeling isconcerned with the construction of a compact and stabledescription of the population mean and variability. Princi-pal Component Analysis (PCA) is probably the most widelyused procedure for generating shape models of variability.

These models can provide shape constraints during imagesegmentation [2], as well as understanding for processes ofgrowth and disease observed in neuroimaging [3].

Clinical applications favor a statistical shape modelingof multi-object sets rather than the one of single struc-tures represented out of their embedding object set. Neu-roimaging studies of mental illness and neurolocal disease,for example, are interested in describing group differencesand changes due to neurodevelopment or neurodegenera-tion. These processes most likely affect multiple structuresrather than single one. A description of the change of the setof objects might help to explain underlying neurobiologicalprocesses affecting brain circuits.

A fundamental difficulty in statistical shape modeling isthe high dimensionality of the set of features with a rela-tively small sample size, typically in the range of 20 to 50in neuroimaging studies. This problem is even more ev-ident for modeling sets of multiple objects, for examplethe set of subcortical brain structures. Statistical model-ing of multi-object sets with their inherent correlations hassignificant advantages for deformable-model segmentation,as nicely demonstrated by Tsai et al. [18] and by Yang etal. [19]. The joint modeling of object shapes defines con-straints that significantly help to stabilize the segmentationprocess. Whereas these two papers describe statistical ob-ject modeling by level-sets, we propose explicit deformableshape modeling with a sampled medial mesh representa-tions called m-rep, introduced by Pizer et al [11].

Deformable shape models represent the underlying ge-ometry of the anatomy and then use a statistical analy-sis to describe the variability of that geometry. Severaldifferent geometric representations other than m-reps havebeen used to model anatomy, such as landmarks[1], densecollection of boundary points [2], or spherical harmonicdecompositions[9]. Another shape variability approach fo-cuses on the analysis of deformation maps ([4, 3, 17, 16].The analysis of transformation fields has to cope with thehigh dimensionality of the transformation, which rendersthe computation of the PCA basically unstable in respectto the training set. Adding or removing a single subject

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from the training set results in a strikingly different princi-pal components.

In most of shape modeling approaches the underlyinggeometry is parameterized as an Euclidean feature space.As Davies and Joshi [5] note the space of diffeomorphismis a curvilinear space and the concept of an Euclidean spacefor many of these approaches is thus a linear simplificationof an actual higher-dimensional curvilinear problem. Formedial descriptions as well as for descriptions of pose pa-rameters, the feature space clearly contains elements of annon-Euclidean vector space. These features need to be pa-rameterized in a nonlinear Riemannian symmetric space. Inthis paper we will discuss the use of curved statistics forthese parameters with modeling major modes of deforma-tions via principle geodesic analysis (PGA)[8], a nonlinearextension of PCA.

This paper summarizes work in progress towards an ef-ficient and compact analysis of sets of objects. We choosethe sampled medial m-rep representation and a statisticalframework based on Riemannian metrics. Driving appli-cation is a longitudinal neuroimaging study where sets ofanatomical objects have been segmented using highly reli-able user-supervised tools.

2. Methodology

This research is driven by the challenge to describe theshape statistics of a set of 3-D objects. Whereas analysisof single shapes is well advanced and has been describedextensively using a variety of shape parametrization tech-niques, extension to multi-object sets still represent signif-icant challenges. Although it might be straightforward toassume that the shape of abutting objects embedded in volu-metric images are strongly correlated, the research commu-nity does not yet have access to tools for statistical modelingand analysis of sets of objects.

2.1. Estimating Variability of Multi-object Sets:

In linear space, variability of parametrized objects can bedescribed by principle component analysis (PCA) of spher-ical harmonics [9] or point distribution models (PDM) [2].The point to point correspondence established via objectand parameter space alignment in the spherical harmonicconcept or minimum description length optimization in thePDM models guarantees a diffeomorphic mapping. How-ever, the linear principal component analysis (PCA) can-not describe object rotations and the modeling cannot beextended to model points and normals. Extension to non-linear modeling is achieved by principle geodesic analysis(PGA), developed by Fletcher et al. [8]. PGA extends linearPCA into nonlinear space using “curved statistics” and is anatural generalization of PCA for describing the variabil-ity of geometric data that are parametrized as curved mani-

folds. To recall, the intrinsic mean of a collection of pointsx1, · · · , xN on a Riemannian manifold M is the Frechetmean µ = argmin

∑Ni=1 d(x, xi)2, where d(., .) denotes

Riemannian distance on M . Whereas PCA in R3 generateslinear subspaces that maximize the variance of projecteddata, geodesic manifolds are images of linear subspaces un-der the exponential map and are defined as the manifoldsthat maximize projected variance. Principle geodesics canbe found by a recursive gradient descent. In practice, anapproximation of the true solution can be calculated by thelog map and a linear PCA in the tangent space of the map(please see [7] for details). Important is the fact that PGAis not limited to linear statistics of surface points but canbe extended to shape parametrization schemes that includepoint locations, scale, and angle parameters.

Figure 1. Multi-object shape parametrization with m-rep’s: Toprow: Binary voxel objects (left) and overlay of parameterized sur-faces implied by the medial m-rep (right). Bottom row: Mesh ofmedial atoms (left), implied surface mesh (middle) and impliedsolid surface (right).

2.2. Object Representation by a Mesh of MedialSamples:

Medial representations represent an alternative toparametrization of 3-D objects via surfaces. Medial axisrepresentations incorporate the notion of symmetry axis ormanifolds, where the representation is decomposed into theshape and structure of the skeletal sheet but also the widthfunction as a local attribute. Changes in terms of localtranslation, bending and widening can be more naturally ex-pressed by medial than by surface representations. Pizer etal. [11, 10] developed an object representation by a mesh ofmedial atoms. Each atom is characterized as a tuple withposition, radius, and the normal vectors to the boundary:m = {x, r,n0,n1} ∈ M, with M = R3×R+×S2×S2.The object surface can be interpolated from endpoints ofthe sets of medial atoms (see Fig. 1), but this representa-tion also allows a continous interpolation of the whole ob-ject interior and a rim exterior to the object boundary. Fur-thermore, unlike many surface representation schemes, thisrepresentation encodes not only sample points at the bound-

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ary but also normals to the boundary. Since the parametervector of medial atoms includes position, length and angle(between normals), mean and variability of a population ofobject shapes is calculated via the Frechet mean and PGAframework as discussed before.

2.3. Modeling of Sets of Objects with M-reps:

This paragraph summarizes the sequence of steps forbuilding statistical shape models, for shape parametrizationof the set of multiple objects, and for statistical analysis ofgroups of objects.

Segmentation of anatomical objects: Anatomical struc-tures of interest, including left and right hippocampus,amygdala, putamen, caudate, pallide globe and lateral ven-tricles, have been segmented by trained experts using semi-automated procedures. Most structures were segmentedwith the ITK-SNAP tool, which includes implicit level-setevolution and manual editing functions. Our experts wentthrough an extensive training, which is reflected by a veryhigh intra- and inter-reliability 1.

M-rep model computation: The segmented objects arerepresented as binary voxel objects (see Fig. 1). The lat-eral ventricles had to be excluded due to topology differ-ences across subjects. We have applied our shape pro-cessing pipeline which includes parametrization by spher-ical harmonic representations [9], point distribution mod-eling (PDM) with homology obtained via subidivion sam-pling, object pose normalization, and alignment of surfaceparametrizations via the ellipsoid of the first harmonics[2,14]. We then used the modeling scheme developed byStyner et al. [13] to construct sampled medial models frompopulations of objects. An explicit error term ε defines themaximum reconstruction error of each object to the model,and the same error term is used to determine the minimumsampling of each medial mesh model. As a result, the fiveleft and right subcortical structures are modeled as compactmedial mesh structures (see Fig. 1).

M-rep shape parametrization of all instances: The m-rep models are deformed into the original segmentations ofeach anatomical object using the ”Binary Pablo” tool de-veloped by Pizer et al.[10]. Driven by a local image matchfunction at object boundaries, m-rep models are deformedto optimally fit the binary voxel segmentations. This pro-cess is applied individually to each of the 10 anatomical ob-ject in each of the 20 image datasets. The correspondenceacross the datasets is established by this deformation proce-dure of the m-rep models.

M-rep shape pose normalization: The normalization ofthe m-rep shape pose is based on a procedure similar to Pro-crustes analysis. In the standard Procrustes analysis, a rigidor similarity transformation is applied to each instance so

1See - this anonymous link - for a detailed description of protocols andreliability results.

that the total sum-of-squared Euclidean distances betweencorresponding points is minimized. In contrast, in the m-reppose normalization the sum-of-squared geodesic distancesbetween corresponding medial atoms is minimized, as de-scribed in detail in [8]. The procedure consists of two steps.First, the translational part of each instance is minimizedby centering each m-rep object set. That is, each instanceof multiple models is translated so that the overall averageof all its medial atoms’ positions is at the origin. Then,in an iterative process, each instance is normalized for ro-tation and optionally also scaling by an optimization thatminimizes the sum-of-squared geodesic distances betweencorresponding medial atoms. The optimization is computedvia gradient descent employing a numerical approximationof the gradient by central differences.

In this paper we are discussing two types of pose normal-ization in the context of multiple object sets. The first oneapplies the above procedure to all objects jointly. We callthis the global pose normalization of the objects. After thisglobal pose normalization the individual objects will likelyhave residual pose variation relative to the global pose. Wethus additionally perform an object specific normalizationcalled local pose normalization. The resulting pose-changeparameters from the global to the local normalization canthen also be studied using the PGA.

Figure 2. Visualization of the 10 selected deep brain structures plusthe lateral ventricles in 2 pediatric subjects at age 2 and 4 years(top row: front view, middle row: left side view, bottom row: topview). The inter-subject shape differences clearly are larger thanthe longitudinal differences, which seems quite small.

3. Results3.1. Motivation and Clinical Data

Driving clinical problem is the need for a joint analysisof the set of subcortical brain structures. There is strong ev-idence that the morphology and size of anatomical shapesmight show a strong correlation between objects that arepart of a circuit or share common functionality and so the

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joint analysis might reveal additional insight on top of theanalysis of individual structures. The image data used inthis paper is taken from an ongoing clinical longitudinalpediatric autism study. This study includes autistic sub-jects (AUT) and matched typically developing healthy con-trols (TYP) with baseline at age 2 and follow-up at age4. Through this longitudinal design, we cannot only studycross-sectional differences (see Fig 8) between groups butalso growth (see Fig 2 and 3) and even group differences be-tween growth patterns. For the preliminary analysis shownhere, we have selected 5 subjects each from the TYP andAUT groups. For eight of these subjects, we had longitudi-nal data with successful scans at 2 and 4 years of age.

As illustrated in Figure 2, the longitudinal shape changesfrom age 2 to age 4 are considerably smaller as the cross-sectional differences at either age. The overall longitudinalvolumetric changes (about 10%) are though of a similar or-der of magnitude as the cross-sectional volumetric changes.Our main goal is to study both longitudinal and cross-sectional shape changes using principal geodesic analysisin different pose normalization settings(e.g., as in Fig. 3).

Figure 3. Visualization of the implied boundary of the m-rep meanstructures computed with separate PGA’s for the groups of 2 (left)and 4 (middle) year old subjects alongside the Euclidean distancemap from the 2 to the 4 year old group (right).

3.2. Principle Geodesic Analysis

PGA performs a compression of the multi-object shapevariability to a small set of major eigenmodes of deforma-tion. We assume that the first few modes describes mostof the shape variability whereas the reminder might mostlyrepresent individual noise. The quality of this compressioncan be evaluated with the criteria compactness, sensitivityand specificity as discussed in [13]. As a preliminary test,we followed the standard procedure of projecting the multi-object sets into the shape space of the eigenmodes λi. Thisleads to a set of weights in the shape space that describe thedeviation of individual shapes from the mean shape. In ourcase, each weight vector represents a multi-object shape set.

We applied PGA to the whole set of objects and com-bined the four subject groups, which ensures projectioninto the same geometric domain for all subjects. Whereasthe analysis of single shapes usually follows the typical se-quence of global linear pose normalization and analysis ofthe residual shape change, a processing of sets of objects re-

quires an extended concept. The individual objects withinobject sets after global pose normalization can have differ-ent relative positions, e.g. they can slide against each otheror even show relativ rotation. We thus chose to apply PGAin different global and local pose normalization settings. Inthe first PGA, the sets of objects are aligned purely by aglobal process (see Fig. 4). This global pose normalizationincludes global translation, rotation and scaling. The re-sulting PGA captures both variability in shape as well as inresidual local pose.

Figure 4. Eigenmodes of deformation by principle geodesic anal-ysis (PGA). The top and middle rows show the first eigenmode at−3 · λ − 1, mean and +3 · λ1 for sagittal (top) and coronal views(bottom).

3.3. Differences in global scaling normalization

In a second step, we varied the global normalization pro-cedure by disabling scaling normalization and applied againPGA. All objects thus were left in their original size andonly rotation and translation was normalized. In order tocompare the differences in the two global PGA’s (with andwithout scaling normalization), we plotted the values of thefirst two major eigenmodes (λ1, λ2) of deformation. Theconnections between corresponding longitudinal pairs areindicated by additional arrows, which allow the identifica-tion of the pairs and the qualitative evaluation of correlationbetween PGA modes and longitudinal changes.

Figure 5 shows this major eigenmode λ1 vs λ2 plot inthe pure global normalization settings. The λ1 axis in thePGA without global scaling normalization seems to char-acterize mainly differences between age 2 and 4 (Fig. 5top), as indicated by the parallel alignment of the connect-ing arrows to the λ1 axis. After scaling normalization (Fig.5 bottom) no coherent alignment of the arrows within theλ1 vs λ2 plot is visible. This suggests that the main ef-fect of correlated longitudinal age change is reflected in thescaling normalization and thus the overall size of the objectsets. Also, corresponding longitudinal pairs cluster quite

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well in the plot including scaling normalization. These ob-servations support our hypothesis that shape changes dueto growth are considerably smaller than shape differencesbetween subjects.

Figure 5. λ1 vs λ2 plot in pure global normalization settings.The top plot shows global rotation and translation normalization,whereas the objects in the bottom plot additionally were scalingnormalized. The arrows indicate corresponding subject pairs. Theplot range of the top plot is larger than for the bottom plot.

3.4. Differences in local scaling normalization

As mentioned before, the individual objects within ob-ject sets still have relative residual pose differences afterglobal pose normalization. We thus analyzed in a next stepthe PGA shape space of objects with a global, followed bya local pose normalization. In the local pose normalization,the individual objects are normalized either with or withoutscaling. Figure 6 shows the λ1 vs λ2 plot in the global pluslocal normalization settings. The top plot shows global nor-malization followed by local normalization including scal-ing. It is noteworthy that the use of scaling in the globalnormalization is irrelevant, as the local scaling operationsupersedes the global one. This pose normalization settingis similar to one commonly used in the shape analysis ofsingle objects with full Procrustes alignment. The bottomplot of Figure 6 shows the objects with global scaling nor-malization but no local scaling normalization. This setting

is somewhat similar to scaling normalization with brain sizevolume, another common scaling normalization used in sin-gle object shape analysis.

Figure 6. λ1 vs λ2 plot in the combine global and local normal-ization settings. The top plot shows global rotation and translationnormalization followed by local rotation, translation and scaling.The bottom plot has global scaling normalization enabled, but doesnot normalize local scaling.

The high degree of difference between the PGA valuesin the two different local scaling settings shown in Figure6can be due to two different factors: a) inter-subject variabil-ity of the residual scaling factors after global normalizationor b) instability in the computation of the PGA directions.We are currently working in evaluating the stability of thePGA, but our earlier studies indicate that this cannot be thesole reason for the discrepancy. Also, the arrangement ofthe groups look quite different in the two plots. In neitherplot, there seems to be a clustering according to age, butclustering according to diagnosis is strikingly different.

3.5. Group discrimination

In the two pose normalization settings with global scal-ing normalization enabled but without local scaling normal-ization(see Figs 5 and 6) there seem to be a separation of thegroups according to diagnosis TYP vs AUT (characterizedby + and ◦). It is though too premature to draw any conclu-sions due to the small sample size of the study and the lim-

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(−3 ·√

λ1,+3 ·√

λ2) (+3 ·√

λ1,−3 ·√

λ2)

Figure 7. Two selected object sets at opposing locations of theseparation between autistic and control subjects in the λ1 vsλ2 plot using the global rotation/translation/scale and local rota-tion/translation pose normalization setting. The left figure is lo-cated in the area with autistic datasets and the right figure in thearea with control datasets.

ited knowledge about the stability of the PGA eigendefor-mations. The preliminary results shown in this paper thoughdemonstrate the potential of multi-object shape analysis.

We focus in this section on the pose normalization set-tings with global scaling but no local scaling (see Fig 6 bot-tom). Approximately, the 45◦ line separates autistic sub-jects from control subjects in our study. We thus recon-structed 2 selected points in the λ1 vs λ2 plot located or-thogonal to the 45◦ line (see Fig 7). These points exemplifythe differences between the 2 groups in this plot. Addition-ally we computed the full means of the diagnosis groups atage 2 and 4 and compared their Euclidean distance maps(see Fig 8). The differences between groups are distributedacross multiple objects and major differences can be seenin both figures in the hippocampus, caudate and amygdala.The fact that this separation of groups cannot be seen in theplot including local scaling normalization suggests that theresidual local scaling factors themselves should be an objectof interest for our next step in this study.

4. DiscussionWe have discussed work in progress towards extend-

ing statistical analysis of anatomical shape from singlestructures to multi-object sets. In regard to the driv-ing applications in neuroimaging, a joint analysis of size,shape and pose of objects and their interrelationships seemsstraightforward and might answer questions about corre-lation among objects, for example objects that are part ofbrain circuits or are known to be functionally connected.

Key issue addressed in this paper are the extraction of asmall set of key features representing the set of objects andcalculation of mean and variability via Riemannian met-ric. The joint analysis of multiple objects even amplifiesthe fundamental problem of small sample size and high di-mensionality of features.

The current results suggest that after the removal of

Distance map age 2, Autism vs Controls

Distance map age 4, Autism vs Controls

Figure 8. Euclidean distance maps between the mean object setsof the autism and control group each at age 2 and age 4 with posenormalization including global scaling but no local scaling. Thedistance maps are shown relative to the control means. The col-ormap of the distances ranges from red (-2.0mm, AUT > TYP)over green (0mm) to blue (+2.0mm, AUT < TYP).

global scale multi-object longitudinal data of the same sub-ject cluster closely in the PGA parameter space and thus thatlongitudinal shape changes is considerably smaller than theshape variability across subjects. Also, the PGA compo-nents and the corresponding distribution of different sub-ject groups vary significantly depending on the choice ofthe normalization. Furthermore, a separation between theautism and control groups both at age 2 and age 4 are visi-ble in the PGA analysis after pose normalization with globalscaling and without local scaling normalization.

Several open issues remain and need to be addressed notonly by our group but by the international research com-munity. The study of object pose, interrelationship betweenabutting objects, shape changes, and correlation of changesof shape and pose within sets of objects offers challengingbut exciting research projects. In regard to the m-rep objectparametrization as used here, we still need to demonstratethe quality and stability of correspondence, as well as therobustness, sensitivity and specificity of PGA-based com-pression of features. Applications in neuroimaging furtherrequire hypothesis testing schemes that will have to com-bine shape and pose features with clinical variables, and thathave to properly address the problems of nonlinear model-ing and multiple comparison testing. Encouraging progressis shown by recent work of Terriberry et al. [15].

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5. AcknowledgementsAnonymous

References[1] F. Bookstein. Shape and the information in medical images:

A decade of the morphometric synthesis. In MMBIA, 1996.1

[2] T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham. Ac-tive shape models - their training and application. In Com-puter Vision and Image Understanding, pages 38–59, 1995.1, 2, 3

[3] J. Csernansky, S. Joshi, L. Wang, J. Haller, M. Gado,J. Miller, U. Grenander, and M. Miller. Hippocampal mor-phometry in schizophrenia via high dimensional brain map-ping. Proc. Natl. Acad. Sci. USA, 95:11406–11411, Septem-ber 1998. 1

[4] C. Davatzikos, M. Vaillant, S. Resnick, J. Prince,S. Letovsky, and R. Bryan. A computerized method for mor-phological analysis of the corpus callosum. J. of Comp. As-sisted Tomography, 20:88–97, Jan./Feb 1996. 1

[5] B. Davis, P. Lorenzen, and S. Joshi. Large deformation min-imum mean squared error template estimation for computa-tional anatomy. In ISBI, pages 173–176, 2004. 2

[6] I. Dryden and K. Mardia. Multivariate shape analysis.Sankhya, 55:460–480, 1993. 1

[7] P. Fletcher. Statistical Variability in Nonlinear Spaces: Ap-plication to Shape Analysis and DT-MRI. PhD thesis, TheUniversity of North Carolina at Chapel Hill, 2004. 2

[8] P. Fletcher, C. Lu, S. Pizer, and S. Joshi. Principal geodesicanalysis for the study of nonlinear statistics of shape. InIEEE Transactions on Medical Imaging, volume 23, pages995–1005, 2004. 2, 3

[9] A. Kelemen, G. Szekely, and G. Gerig. Elastic model-basedsegmentation of 3d neuroradiological data sets. In IEEETrans. Med. Imaging, volume 18, pages 828–839, 1999. 1,2, 3

[10] S. Pizer, T. Fletcher, Y. Fridman, D. Fritsch, A. Gash,J. Glotzer, S. Joshi, A. Thall, G. Tracton, P. Yushkevich, andE. Chaney. Deformable m-reps for 3d medical image seg-mentation. International Journal of Computer Vision IJCV,55(2):85–106, 2003. 2, 3

[11] S. Pizer, D. Fritsch, P. Yushkevich, V. Johnson, andE. Chaney. Segmentation, registration, and measurement ofshape variation via image object shape. In IEEE Trans. Med.Imaging, volume 18, pages 851–865, 1999. 1, 2

[12] C. G. Small. The statistical theory of shape. Springer, 1996.1

[13] M. Styner, G. Gerig, J. Lieberman, D. Jones, and D. Wein-berger. Statistical shape analysis of neuroanatomical struc-tures based on medial models. Medical Image Analysis (ME-DIA), 7(3):207–220, Sept 2003. 3, 4

[14] M. Styner, J. A. Lieberman, D. Pantazis, and G. Gerig.Boundary and medial shape analysis of the hippocampus inschizophrenia. In MedIA, pages 197–203, 2004. 3

[15] T. Terriberry, S. Joshi, and G. Gerig. Hypothesis testing withnonlinear shape models. In G. E. Christensen and M. Sonka,

editors, Information Processing in Medical Imaging (IPMI),number 3565 in Lecture Notes in Computer Science LNCS.Springer Verlag, July 2005. 6

[16] P. Thompson, J. Giedd, R. Woods, D. MacDonald, A. Evans,and A. Toga. Growth patterns in the developing brain de-tected by using continuum mechanical tensor maps. Nature,404:190–193, 2000. 1

[17] P. Thompson, M. Mega, and A. Toga. Brain Mapping: TheDisorders, chapter Disease-Specific Brain Atlases. Aca-demic Press, 2000. 1

[18] A. Tsai, A. Yezzi, W. Wells, C. Tempany, D. Tucker, A. Fan,E. Grimson, and A. Willsky. Shape-based approach to curveevolution for segmentation of medical imagery. IEEE Trans-actions on Medical Imaging, 22(2):137–154, Feb 2003. 1

[19] J. Yang, L. H. Staib, and J. S. Duncan. Neighbor-constrainedsegmentation with level set based 3d deformable models.IEEE Transactions on Medical Imaging, 23(8):940–948,Aug 2004. 1

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