measuring biological shape using geometry-based shape ... · measuring biological shape using...

Measuring biological shape using geometry-based shape transformations

C. Davatzikos*

Departments of Radiology and Computer Science, Johns Hopkins University, School of Medicine, JHOC 3230, 601 N. Caroline St., Baltimore MD 21287, USA

Received 30 August 1999; revised 11 April 2000; accepted 27 June 2000

Abstract

This paper presents a methodology for shape analysis of anatomical structures. A template shape is used as a unit, and a shapetransformation mapping the template to a particular structure is used to quantify the shape of the structure with respect to the template.The geometric characteristics of the boundaries of the template and the individual structure are used to define point correspondences betweenthe template and the structure. These correspondences are then used to determine the shape transformation, which is based on an elasticadaptation of the template. Regional inter-subject and inter-population shape differences are then identified by comparing the correspondingshape transformations point-wise. Validation using simulation experiments is also presented.q 2001 Elsevier Science B.V. All rightsreserved.

Keywords: Biological shape analysis; Geometry-based shape transformations; Deformation-based morphometry; Shape correspondence

1. Introduction

Brain disorders are often accompanied by structuralchanges in the brain, and by cognitive effects which maybe related to them. The precise quantification of such struc-tural changes is important for expanding our understandingof the mechanisms of a disease. It may also be important inthe early diagnosis and effective treatment of brain diseases,before the damage becomes evident through its behavioraleffects. Characterizing brain structure requires advancedmorphometric tools, capable of capturing subtle details ofcomplex shapes, such as those of brain structures. The needfor such techniques has motivated this work, whose primarygoal is to develop mathematical methodologies for repre-senting brain morphology. Quantitatively representing andanalyzing brain morphology is also important in under-standing the functional organization of the human brain,in that it enables the correlation of structural characteristicswith functional measurements, either using in-vivo func-tional images or by utilizing lesion/deficit data [1].

The majority of the literature on morphological analysisof the brain has focused on volumetric analysis, which isbased on measuring the volume of a structure of interest orof a number of partitions of the structure. Several partition-ing schemes have been proposed in the literature [2–6].The main advantage of volumetric analysis is its simplicity.

However, in addition to the fact that such analysis onlymeasures volume and not other shape characteristics ofstructures, it is limited in many other respects. Specifically,it relies on an arbitrary partitioning of a structure intosubregions; such partitioning is often unreliable since itlargely depends on the shape of the structure in each indi-vidual brain. For example, in a study of sex-related differ-ences in the corpus callosum [2] the anterior–posteriorextent of the corpus callosum was divided into five regionsof equal length, and the area of each partition wasmeasured separately; this method has been adopted byseveral investigators in this field [4,7–9]. The resultingpartitioning of the callosal area then depends largely onthe curvature of the corpus callosum, as shown schemati-cally in Fig. 1a.

Other partitioning schemes methods have, in part, over-come this limitation [10,5], by partitioning the callosumalong its long axis. However, all the aforementioned volu-metric analysis methods are limited by the need for an a-priori partitioning of a structure into sub-regions. However,the location and shape of a potential region of interest (e.g.an abnormal region) is, in general, not known in advance.Therefore, in a-priori partitioning methods, that region ofinterest might be split into two or more parts each belongingto different partitions, or it might be included in a muchlarger partition (see Fig. 1b); in either case, a regional differ-ence may not be revealed by the volumetric analysis.

More sophisticated, shape-based techniques have beenused. In particular, Fourier analysis [11–13] or eigenmode

Image and Vision Computing 19 (2001) 63–74

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* Tel.: 11-410-955-4510; www.http://nilab.rad.jhu.eduE-mail address:[email protected] (C. Davatzikos).

analysis [14–16] do not require a-priori knowledge of theregion of interest, and they rely on shape and not merelyvolumetric characterization of structures. However, thetypically small number of parameters used in theseapproaches and their global nature limits their applicabilityto morphological characteristics of a wide spatial extent,and not to highly localized ones.

Landmark-based morphometrics have been the firstattempt during the past decade to obtain a local representa-tion of shape. Bookstein’s work in this field [17,18] has beenbased on the Procrustes fit of a number of landmarks,followed by the appropriate statistical analysis of theProcrustes residual vectors. This approach has contributedconsiderably to the analysis of subtle characteristics ofanatomical shapes, and it has been applied in severalmorphometric studies including the study of the shape ofthe corpus callosum [19,20] and of the ventricles [21] inschizophrenia, and in craniofacial surgery [22]. It is notwithout limitations, however. Specifically, it is based onthe assumption that a large number of landmarks can bereliably identified in the three-dimensional (3D) space,using cross-sectional images. This is not always the case,particularly with brain structures of complex shape. Evenwhen landmarks can be reliably defined, however, a largeamount of human effort is required, which limits the abilityof a study to utilize large image databases [1,23] and there-fore limits its statistical power.

One approach to quantitative morphological analysis ofthe brain, which has been recently recognized by severalinvestigators as a powerful way to obtain a detailed andlocalized structural characterization of the brain [24–32,16,33], is studying the properties of ashape transforma-tion that maps a template (e.g. a brain atlas) to an individualbrain. The principle of this approach, which has its roots inD’Arcy Thompson’s seminal work [34], is that the morpho-logical brain characteristics of two individuals or popula-tions can be compared by comparing the correspondingtransformations. The main advantage of morphologicalanalysis approaches based on shape transformations istheir high spatial resolution. Specifically, since these meth-ods are based on a point-wise morphological characteriza-tion of anatomical structures, they are independent from theshape and spatial extent of a region of interest, which reme-dies the problem of Fig. 1b. Moreover, they don’t require

the a-priori knowledge of the location or approximate loca-tion of the region of interest; regions that have significantshape characteristics can be determined a posteriori, as, forexample, in Fig. 14. Finally, since spatial transformationmethods map morphological characteristics from a popula-tion to a common reference system, they readily accommo-date comparisons and pooling of data from different subjectsand investigators.

Since the foundation of this approach is the determinationof a spatial transformation that maps one brain to another,several methods to do this have been developed during thepast decade. The most widely used approach is finding atransformation that maximizes some measure of similarityin the image intensities of the spatially transformed templateand the target image; such techniques have been developedby several groups ([31,35–37,38–40]). These techniquesare based on highly nonlinear transformations and, there-fore, have high flexibility in adapting the shape of one brainto that of another, enabling very localized structural char-acteristics to be quantified through the resulting deformationfield. One of their limitations, however, is that matchingimage intensities does not necessarily match correspondinganatomical features.

In this paper we present a morphometric method usinggeometry-based shape transformations. Geometry-basedshape transformations have also been used by other groupsin this field [41–45]. Our shape transformation is based onpoint correspondences defined on distinct features, such asthe boundary of a brain structure, which are elastically inter-polated in areas, which present no distinct features. We havepreviously reported an earlier version of this approach [27],which was later applied to a study of sex-differences in thecorpus callosum [46]. In this paper we concentrate on a newmethodology fordetermining point correspondencesalongthe boundary of a structure, and on thevalidation of thismethod using simulated images. Since our method tries tomatch regions with similar geometric structures, its assump-tion is that geometry has a biological substrate. We believethat this is a reasonable assumption, which is backed to alarge extent by microstructural and functional imagingstudies. However, the exact relationship between structureand function in the brain is still largely unknown. Moreover,it is likely that it depends on the particular structure underconsideration, especially in the cortex. The validity of

C. Davatzikos / Image and Vision Computing 19 (2001) 63–7464

Fig. 1. The problems of volumetric measurements based on a-priori partitioning of a structure. (a) The partitioning of the corpus callosum here depends largelyon the callosal curvature, which makes the volumetric analysis unreliable. (b) Since the region of interest is not known in advance, it might happen tofall intotwo or more partitions, or to constitute only a small fraction of a partition, thereby not affecting volumetric measurements significantly, and remainingundetected.

interpolation models, such as the one used herein, will even-tually be tested using functional activation experiments.

2. Methods

In order to simplify the description of our method and tomake the presentation of the results easier and more clear,we will mostly restrict our discussion to a two-dimensional(2D) model. The same framework, however, can be used for3D curvature-based matching.

2.1. Regional shape analysis

Consider, for a moment, that we are interested in measur-ing the length of an object. Typically, three steps areinvolved. First, we need to define a measurement unit.Second, we need to have a way of comparing the lengthof a given object with our measurement unit; stretching ameasure over the extent of the object accomplishes exactlythis. Third, having expressed the lengths of two objects, ormore generally two groups of objects, in terms of the samemeasurement unit, we must be able to compare them; this iswhat standard arithmetic, or more generally statistical meth-ods, do. The same principles apply to the problem ofcomparing anatomical shapes using shape transformations.Here, the unit is a template shape. The means of quantita-tively representing a shape is ashape transformation, whichmaps the template to an individual shape; a shape transfor-mation quantifies the characteristics of a shape with respectto the template (the unit). Shape comparisons can beperformed by comparing the corresponding transforma-tions, provided that the same template is used for all shapes.An illustrative example of a shape transformation quantify-ing the differences between two shapes is shown in Fig. 2.Specifically, the shape of Fig. 2b was derived from that ofFig. 2a by applying the forces shown in Fig. 2a. The shapeof Fig. 2a was then used as the template, and a shape trans-formation that mapped it to that of Fig. 2b was found. Theresulting expansion/contraction of the shape of Fig. 2a wasthen calculated from the shape transformation, and isdisplayed in Fig. 2c as a gray-scale image, in which darkreflects contraction and bright reflects expansion. Theregions of high expansion and contraction are displayed inFig. 2d and e, respectively. This demonstrates that certainproperties of a shape transformation can quantify shapedifferences.

For certain kinds of shape comparisons, it can be shown[27] that, in principle, the choice of the template does notaffect the comparison between two shapes or two groups. Inpractice, however, the choice of the template affects theshape transformation, and therefore to some extent it affectsthe resulting comparisons. This issue was addressed in moredetail in Ref. [27].

Let V t andV s be two domains in which the template andan individual shape are defined, respectively. A shapetransformation is a vector field,U(·), from V t to V s,which maps corresponding regions to each other. In theapproach described in this paper, the geometry of theshapes that are being matched plays an important role indetermining the shape transformation. Specifically, amongthe infinite number of ways that one can map a templateshape to a target shape, the one that best matches thegeometric structures of the boundaries of the shapes ischosen. The degree of matching is determined via criteriadescribed later (see Eq. (3)). The interior of a shape, in theabsence of additional internal boundaries, is warped elas-tically, in a way that the boundaries of the two shapes arebrought into coincidence. We note that the elastic interpo-lation in regions which display no distinct features is some-what arbitrary. However, it is necessary, since oftenmeasurements on the boundaries of structures are causedby processes in the interior of the structures. For example,a complex pattern of brain atrophy will induce a complexpattern of boundary deformation. The purpose of the inter-polation scheme used herein is to infer such patterns oftissue atrophy from observed shape measurements onboundaries of structures. Clearly, as the imaging capabil-ities improve and provide higher anatomical detail, morefeatures can be used in determining the shape transforma-tion. Moreover, biomechanical models of tissue deforma-tion [47] might also increase the accuracy of the kind ofanalysis presented in this paper.

2.2. Boundary parameterization

In order to obtain a description of the geometric structureof the boundary of a shape, we apply an active contouralgorithm initially developed for extracting ribbon-likeobjects [48]. The active contour is initially placed close tothe boundary of a structure of interest, and subsequentlydeforms toward it, attracted bya boundary massfunction,

C. Davatzikos / Image and Vision Computing 19 (2001) 63–74 65

Fig. 2. Demonstration of the use of shape transformations in comparing two synthetic images. The shape in (b) was created from the shape in (a) by applyingthe forces shown in (a). The shape transformation that maps (a) to (b) was then determined by the method described in the paper. The amount of expansion,calculated from the resulting shape transformation, was used to determine regions of differences between the two shapes, and is displayed as an imagein (c),where dark reflects contraction and bright reflects expansion. (d) The region of more than 15% expansion shown in white. (e) The regions of more than 15%contraction shown in black.

m(x), which is defined as follows:

m�x� � i7�I �x� p g�x��iwhereI(·) is the image intensity andg(·) is a low-pass filter.

The external force acting on each point on the activecontour is then defined as

f �x� � c�x�2 x; �1�wherec(x) the center of the boundary massm(x) included ina circular neighborhood,N(x,r (x)), centered onx (see Fig.3). The radiusr (x) of the neighborhood is adaptive, as itsdependency onx implies. Specifically, it is given by

r�x� � min r . 0f jZN�X;r�

m�X� dX $ Mg;

whereM is a threshold. The neighborhood adaptivity makesconvergence faster, since relatively larger neighborhoodsare used to quickly pull toward the boundary the activecontour points that are far from the boundary. When theactive contour, however, is in the vicinity of the boundary,the small neighborhood size guarantees that the activecontour will balance on the boundary, the points of whichcoincide with the center of mass around them and thereforeexperience zero external force.

2.3. Curvature-based boundary mapping

The second step in our procedure is the most critical one;

it is the one that determines point correspondences alongtwo boundaries to be matched. Specifically, consider atemplate anatomical shape,T, bounded byBt, and thecorresponding structure,S, in an individual anatomy,bounded byBs. From a purely geometric point of view,there are infinitely many ways of mappingBt to Bs.However, here we seek the map that best matches the under-lying anatomies, according to certain criteria. We nowdefine this map and these criteria.

Let Bt(s), s[ [0,1], be the parameterization of the bound-aryBt obtained by the active contour algorithm described inSection 2.2, and letBs(s) be the parameterization ofBs. Letk t(s) andk s(s) be the corresponding curvature functions,estimated as described in Appendix A. We seek a repara-meterization,r(s), s[ [0,1], of B 0 t�s� which brings thesetwo curvature functions in best agreement. It is importantto note that we do not want a parameterization that matchesthevaluesof the curvatures but rather one that matches theirpatternof variation. The reason for this is illustrated in Fig.4. Curve 2 here could be the curvature of the template,denoted byk2(s), and Curve 1 the curvature of the indivi-dual’s structure, denoted byk1(s). A reparameterization,r(s), of Curve 2 attempting to maximize the similaritybetween k1(s) and k2(r(s)) would cause an erroneousstretching along Curve 2 around its peaks. For example,two points along Curve 2 with parametric coordinatess1

and s2 are shown in Fig. 4, together with their locationafter a reparameterization attempting to matchk1(s) andk2(r(s)). We note thatk1�s1� � k2�r�s1�� K which impliesthat the goal of matching the curvatures of Curve 1 with thatof the reparameterized Curve 2 is achieved here. However,the point correspondences determined this way are wrong;the solutionr�s� � s; i.e. the one leaving the parameteriza-tions unaffected, is the one providing the best point corre-spondences in this example. This has motivated us to look ata criterion, which is independent from the actual values ofthe curvatures, but depends on their pattern of variation.

Let bt(s) andbs(s) be two trinary functions obtained fromk t(s) andk s(s) as follows:

bt�s� �1; kt�s�2 mt . ast;

21; kt�s�2 mt , 2ast;

0; 2ast # kt�s�2 mt # as t;

8>><>>: �2�

and similarly forbs(s). Here,m t ands t are the mean andstandard deviation ofk t(s) within the unit interval, and simi-larly for m s ands s; we typically seta � 1: According to thisdefinition, the functionsbt(s) andbs(s) are 1 and21 at thesignificant positive and negative peaks ofk t(s) and k s(s),respectively, and zero elsewhere. Therefore, they do notdepend on the actual values of the curvatures but only ontheir pattern of variation. However, we note that this patternis affected to some extent by the thresholdas t. In Section 3(see Fig. 8), we will show the robustness of our method withrespect toa .

We now define the reparameterizationr(s) of Bt(s) to be


Fig. 3. The external force acting on the active contour pulls it toward thecenter of the nearby boundary mass, which is determined through a localedge detector. The neighborhood size used for the center of cortical masscalculation is determined adaptively. It is larger for points likeA, which arefurther away from the boundary than points likeB.

Fig. 4. An illustration of an undesirable reparameterization of Curve 2, sothat its curvature matches that of Curve 1�k2�r�s1�� k1�s1� � K�: Thepattern, not the actual value, of the curvatures should be matched instead.

the function that minimizes the following functional:

F�r�s�� Z1

0�bt�r�s��2 bs�s��2 ds1 L

Z1

0�r 0�s��2 ds; �3�

subject to the cyclic boundary condition

r�0� � r�1�:The first term inF(r(s)) effectively attempts to match thecurvature patterns ofBt(·) andBs(·); the second term is asmoothing term. The corresponding Euler equation, whichprovides a necessary condition, is the following:

b0t�r�s��bt�r�s��2 bs�s��2 Lr 00�s� � 0; �4�whereb0t�r�s�� is the derivative ofbt(·) with respect to itsargument, evaluated atr(s).

Eq. (4) is discretized using finite differences, and solvedusing successive over relaxation (SOR) [49]. Since this is alocal descent method, a good initial estimate ofr(s) must beobtained; we do this by finding the circular shift that mini-mizes the functionalFI(r(s)) defined as follows:

FI�r�s�� Z1

0�bt�r�s��2 bs�s��2 ds:

The initial estimate,ro(s), of r(s) is then defined as follows:

ro�s� � �s1 a� mod1;

where

a� argminA

{F I �r�s��ur�s� � �s1 A� mod1}:

This typically provides a good starting point for the SORminimization, assuming thatT andS are reasonably simi-lar. However, if corresponding curvature peaks do not over-lap at all after this initialization, then registration of theunderlying contour segments is not guaranteed. We notethat even if such segments do not overlap initially, theymight be brought into overlap during the contour’s repara-meterization due to forces exerted from nearby segments ofhigh curvature. Some issues regarding the implementationof the procedure described in this section are given inAppendix A.

The computational complexity of the algorithm is verylow, at least in 2D. In our formulation we typically subsam-ple the contour in order to determine the reparameterizationdeformation, and therefore computational time depends onthe subsampling rate. However, typically it takes a fewseconds on a silicon graphics workstation.

The problem of curve reparameterization was put byTagare [50] in the more general framework of bi-morphismsdefined on a torus. We note that the energy functionalF(·)of our formulation effectively discriminates betweenequivalence classes of curves, which is a restricted case ofthat in Ref. [50], under certain assumptions. However, thismight be a desirable restriction, since curves whose curva-tures have different values but the same pattern (such as thetwo curves shown in Fig. 4) are considered equivalent for

the purposes of our application. We note, however, that forother applications that might not be the case.

Related is also the work in Ref. [51], which presents aheuristic approach to the problem of determining pointcorrespondences between two curves. In that approach, asimilarity in the angles between consecutive points of corre-spondence is favored by an energy function. This is equiva-lent to matching the curvatures of the two curves. As wediscussed earlier (see Fig. 4), directly matching the curva-tures might be problematic.

Finally, a different approach to the problem of determin-ing point correspondences was presented in Ref. [52]. Thatapproach is novel in that it does not attempt to determinepoint correspondences independently from the shape match-ing step. Instead, the point correspondences are determinedin a way that the determinant of the covariance matrix of thewhole sample set is minimized, which effectively maxi-mally “packs” shape information into the first few eigen-vectors of the covariance matrix. Although this is aninteresting idea from a shape representation perspective, itmight result in correspondences that are not necessarilyanatomically meaningful. More importantly, it might failto reveal shape changes occurring with diseases, such as alocalized loss of tissue, since a representation of those viashape transformations often requires an abrupt local defor-mation of the template, which would not be favored by theenergy function minimized in Ref. [52].

We note, here, that emerging imaging methods, such asdiffusion tensor imaging, will eventually enable us to deter-mine point correspondences not only along the boundariesof structures, but also in their interior [53].

2.4. Shape deformation

The procedure described above determines a map,M,from Bt [ V t to Bs [ V s, as follows:

M : V t ] Bt�r�s�� ! Bs�s� [ Vs; s [ �0;1�; �5�essentially defining a large number of point correspon-dences fromV t to V s. These point correspondences couldbe utilized by a landmark-based morphometric technique,such as the one developed by Bookstein [54,19,18].Although there would be certain advantages in thisapproach, a shortcoming of it is that it determines morpho-logical differences on the boundary of a structure. Clearly,morphological effects, such as for example tissue loss due tonormal aging or a disease, can occur anywhere in the inter-ior of a structure.

In order to address this issue, we have followed anelastic transformation method, instead. We believe thatelastic transformations are good models for anatomicalshape transformations because they tend to preserve therelative positions of anatomical structures, while they areflexible enough to allow for considerable inter-individualvariability.

We define this transformation,U(·), to be the one obeying


the linear elastic deformation equations resulting from anexternal force field,F(·), that vanishes only when the mapM in Eq. (5) is satisfied:

lDU�u�1 �l 1 m�7 Div U�u�1 F�u� � 0; �6�whereu [ V t andF(u) is defined as follows:

F�u� �Bs�s�2 u; u [ Bt

0; otherwise:

(�7�

Here,

u � Bt�r�s��; if u [ Bt;

and therefore

s� �r21+B21t ��u�:

Hence, we can writeF(u) more precisely as

F�u� � Bs�r21+B21t �u��2 u; if u [ Bt;

0; otherwise:

(

The equations in (6) are discretized using finite differences,and are solved iteratively using successive over-relaxation.

The resulting transformationU(·) quantifies the shapeproperties ofS with respect to the templateT, whichserves as a metric unit. Therefore, two individual structurescan be compared by comparing the corresponding transfor-mations point-wise. Several quantities can be extracted fromU(·). In this paper we focus our attention on one particularscalar field, which is representative of many morphologicaleffects of various factors, including gender, age, anddiseases. This field is defined as the determinant of the

Jacobian ofU(·):

V�u� � det{7U�u�} : �8�This scalar field quantifies regional volumetric differencesbetween two brains, and will therefore be referred to asregional volumetric difference function (RVDF). IfV1(u)and V2(u) are the RVDF’s corresponding to a structure intwo different brains, and ifV1(u) . V2(u), then this impliesthat this structure is more bulbous in the first brain than inthe second brain, in the vicinity ofu. Similarly, if V1(u) andV2(u) represent measurements of a structure of the samebrain at two different times, thenV1(u) . V2(u) implies aloss of tissue between these two times. Since many braindiseases are associated with regional tissue loss (for exam-ple Alzheimer’s and Parkinson’s diseases), the RVDF’s areof special interest in many studies.

3. Experiments

3.1. Curvature-based matching

In the first experiment we tested the procedure for curva-ture-based boundary parameterization, on two magneticresonance brain images acquired from the midsagittalplane. The structure of interest was the corpus callosum.Fig. 5 (top) shows the initial placement of the active contourfor the template and for a typical subject. Fig. 5 (bottom)shows the resulting parameterizations after convergence ofthe active contour. We note that the active contour must beinitialized fairly close to the boundary of interest, if adjacentedges are present.

The curvatures of the resulting contours were then esti-mated as described in Appendix A, and were used for thereparameterization of the active contour of the template inFig. 5. The regions that corresponded to significant peaks orvalleys of the curvature are shown thicker in Fig. 6. Thisshows that the regions used as corresponding regions thatdrive the elastic boundary parameterization correspondwell. Fig. 7 shows the trinary functions of the templateand the target before (a) and after (b) the elastic reparame-terization. A good match is apparent in Fig. 7b, by examin-ing the pattern of curvature variation.

In order to display the resulting point correspondences inthis example, we selected 8 points whose parametric coor-dinates were equally spaced in the unit interval, which aredisplayed in Fig. 8; here, the first point is shown as a cross,in order for the reader to identify individual point correspon-dences. For comparison purposes, in Fig. 8 we show thepoints with the same parametric coordinates for both thetemplate and the target images. The top row in Fig. 8shows the resulting correspondences fora � 1 and thebottom row fora � 2: As Fig. 8 shows, our elastic repar-ameterization procedure seems to create reasonably goodpoint correspondences, regardless of the value ofa , theparameter involved in the definition of the trinary functions


Fig. 5. The initial (top) and final (bottom) configurations of the activecontour, applied to the template, and to a midsagittal MR brain image.

Fig. 6. The regions whose curvatures was above or below the correspondingthresholds are shown thicker for the template (left) and the target (right).

bt(·) andbs(·). The overall accuracy of the method will betested quantitatively in Section 3.3 on simulated data, inwhich the actual deformation is known.

3.2. Synthetic deformations

In our second experiment we tested the accuracy withwhich our methodology can localize subtle differencesbetween two shapes. In particular, we simulated a hypothe-tical “tissue loss” in the corpus callosum shown in Fig. 9a,by applying a uniform shrinkage of 50% within the threecircular regions shown in Fig. 9b. We accomplished this bysolving the Navier–Stokes equations, assuming an initialstrain tensor equal to 0.5I within the three circular regions.At equilibrium, the three circular regions shrunk to abouthalf of their initial areas. The resulting shape is shown inFig. 9c.

We then used the shape of Fig. 9a as the template anddeformed it to the shape of Fig. 9c, as described in Section 2.The resulting RVDF is shown in Fig. 10a as a gray scaleimage; the darker the image is, the higher the contraction ofthe template is. For better visualization of the RVDF, wethresholded it at three different thresholds: 15, 20, and 25%,

as shown in Fig. 10b–d, respectively. In general, the esti-mated shape differences are in agreement with the locationsof the three circular regions of atrophy in Fig. 9b. We note,however, that the regional volumetric differences appear tobe much more diffused than the original circular regions ofFig. 9b. This is due to the fact that the shape transformationrelies entirely on boundary information, because of theabsence of any features in the interior of the structure; there-fore, shape differences deeper in the interior of the structurecannot be accurately resolved in the absence of internalfeatures. For similar reasons, the RVDF has much lowervalue than 50% in the three areas of synthetic contraction.Accordingly, the RVDF’s are only indicative of the approx-imate location of a regional volumetric difference, and arebound to underestimate the actual degree of the underlyingtissue loss.

We then examined the dependency of our results on theselection of the elasticity parameters. In particular we usedvalues forl andm ten times higher and ten times lower thantheir default values used above, which werel � 2 × 10210

;

m � 10210: The resulting regions of contraction were

virtually identical for all parameter values, as shown inFig. 11.


Fig. 7. The superposition of the trinary functions before (a) and after (b) reparameterization.

Fig. 8. Point correspondences resulting after elastic reparameterization of the template for two difference values ofa : 1 (left) and 2 (right).

3.3. Population comparisons

The methodology described herein can be used for deter-mining differences between two populations (e.g. normalsvs diseased). A major obstacle in such studies is inter-subject morphological variability. Depending on howlarge the difference between two populations is relative tothe within-population inter-subject differences, it might bevery difficult to detect an underlying morphological differ-ence between the two populations. In the experimentpresented in this section, we tested the performance of ourmethod on two small groups, one of which differed system-atically from the other, as explained below.

We randomly selected 18 midsagittal images from maleright-handed normal subjects of about the same age, whichcontained the cospus callosum. We then randomly dividedthese images into two groups. We simulated a systematicatrophy (shrinkage) between these two groups in the poster-ior part of the structure. In particular, we outlined the regionof atrophy in each subject of one of the two groups and weapplied a uniform contraction by 50% in that region. Fig. 12shows two representative examples of the original imagesand the resulting images with the contracted corpus callo-sum.

We then applied our method for determining point corre-spondences between these 18 subjects and a new subject,serving as the template. The resulting point correspondencescan be viewed in Fig. 13, in which 8 representative pointswere selected along the structure’s boundary, the first one ofwhich is represented by a cross. We then calculated theRVDF’s for both groups, and applied point-wiset-tests inorder to identify regions of significant difference. Theregion in which the RVDF of the group on the top of Fig.12 was significantly�p . 0:001� higher than the RVDF of

the group on the bottom of Fig. 12 is shown in Fig. 14. Theregion of significant difference was correctly detected. Wenote a couple of false positives (small clusters of whitepixels), which are probably due to the relatively smallsample size we have. Simply because of their small size,these clusters are generally ignored.

Finally, we tested the robustness of the detection of thisregion of simulated atrophy, by applying our algorithm withdifferent values of rigidity in the elastic transformation.Each time, ap-value image was found by the algorithm,such as the one in Fig. 14. We then outlined the true regionof simulated atrophy, and we calculated the average of thep-value within that region, for each rigidity parameter value.Fig. 15 shows a plot of the results. We see that for a rangeover 7 orders of magnitude, the region of atrophy isdetected.

In our validation experiments we tested the method onsimulated images, primarily because in those images weknow the ground truth. Although ideally we would like tomeasure the accuracy and sensitivity of this method on realdata, this is practically not possible, because we don’t knowthe actual degree of atrophy or other anatomical abnormal-ity. However, our previous study reported in Ref. [46],which found significant sex differences in the corpus callo-sum and significant correlations with cognitive perfor-mance, provides additional evidence for the highsensitivity of this method in detecting shape differences.

4. Conclusions

We presented a methodology for anatomical shape analy-sis, focusing on the brain, using shape transformations. Ashape transformation that adapts an anatomical template toan individual anatomical structure is a means for quantify-ing the shape properties of the structure, using the templateas ashape unit. The shape transformation described in thispaper was based on geometric properties of the boundary ofa structure, which were used to drive an elastic deformationtransformation.

The main assumption in our approach has been that thegeometry of neuroanatomical shapes and their underlyingfunctional organization are directly linked together. This isclearly a hypothesis. However, the remarkable consistencyof the brain structure across individuals, as well as existingstudies on white matter connectivity and on functional acti-vation of the brain suggest that anatomical shape is linked toa large extent to the underlying functional organization.


Fig. 9. (a) A template shape, (b) three circular regions in which a uniform shrinkage of 50% was applied, and (c) the resulting shape.

Fig. 10. (a) The RVDF calculated by the algorithm shown as a gray scaleimage; the darker a region is, the higher the contraction of the template was.Thresholded RVDF for thresholds equal to (b) 15%, (c) 20%, and (d) 25%.

This is the case primarily for subcortical structures and forthe primary cortical regions. For higher level corticalregions this hypothesis could potentially fail; therefore,our methodology should be restricted to those regions forwhich a structure-function correlation exists.

In our approach we used an elastic transformation, forreasons that were explained in Section 2.4. Several othertransformations have been proposed in the literature, includ-ing thin plate splines, fluid, and others. Comparative studies,aimed at determining to what extent the model of the shapetransformation affects the results of the shape analysis,would be very interesting, and would help interpret theresults of neuroanatomical studies.

In our experiments we extracted a single scalar fieldfrom the shape transformation, namely the RVDF, whichquantifies regional volumetric differences betweensubjects. Since many brain diseases are believed to beassociated with regional tissue loss, this is one of themost important quantities to be extracted from a shapetransformation. However, several other interesting quan-tities can be extracted. In particular, the directions of theprincipal strains, derived from the strain tensor of thevector field U(·), can be measured, and could reflectdirectionally preferential shape differences between twostructures.

Current and future work in our laboratory focuses on the3D formulation of our methodology, and in particular of thecurvature-based matching procedure, as described inSection 2.1.

Acknowledgements

This work was supported in part by a Whitaker Founda-tion Research Grant, by the NIH grant 1R01-AG14971, bythe NIH contract NIH-AG-93-07, and by the NIH grant1R01AG13743-01 under the Human Brain Project.

Appendix A

Curvature Estimation.Let B1,…,Bn be then points of a(discrete) boundary curve resulting from the active contouralgorithm of Section 2.2. In order to obtain a robust estima-tion of the curvature of the boundary at each of these points,we find the least-squares fit of a fourth degree polynomialcurve for each coordinate (x andy) of Bi, by considering its2Mi neighboring points along the boundary. As the depen-dency ofMi on i implies, the number of neighbors used inthe least-squares estimation is determined adaptively.Specifically, lete(m) be the mean squared error of fit for agiven value,m, of Mi:

ex�m� � 12m1 1

X�i 1 m� bmodn

j��i 2 m� bmodn

�Bxi 2 �aj4 1 bj3 1 cj2 1 dj

1 f ��2;whereBx

i is the x-component ofBi (and analogously forthe y-component). The equation above has been basedon the fact that the contour can be locally represented


Fig. 11. The robustness of the detected regions of contraction with respect to the elasticity parametersl andm : (a) values 10 times lower than the defaultvalues, (b) the default values, and (c) 10 times higher values.

Fig. 12. Top: Two representative images of the corpus callosum, overlaid on which are shown the regions in which a uniform contraction was applied, tosimulate atrophy. Bottom: The resulting images. Nine such simulated images were used along with nine other images to test the capability of our algorithm todetect group differences.

by two polynomial functions ofs, one for thex- andone for they-component. Sincesj � j=n; the expressionabove follows by incorporating powers ofn into theparametersa,b,c and d. Then Mx

i is the maximummfor which the error of fit remains below a thresholdE:

Mxi � max{muex�m� # E} :

Typically, we set E � 1: The estimation window isdifferent for the x- and y-components, since those aretwo separate functions ofs. From each of the resultingpolynomial functions, derivatives, and therefore the


Fig. 13. Point correspondences determined via elastic boundary matching with the same template, for 18 subjects of the study.

Fig. 14. The region (white) in which the RVDF of the group with thesimulated atrophy was significantly lower than the controls, for thep�0:001 significance level. This region is indeed the region in which asystematic 50% contraction was applied in the data. Notice a few falsepositives, which are probably due to the relatively small sample size.

Fig. 15. Robustness in detecting the region of simulated atrophy, as afunction of the rigidity parameter of the elastic transformation. A plot ofthe average p-value of the difference between the RVDF’s of the twogroups, within the true region of atrophy, as a function of the rigidityparameter. The region is detected (reflected by the low p-vales) for awide range of values of the rigidity parameter, spanning several orders ofmagnitude.

Fig. 16. Non overlapping peaks (a) are not brought into alignment by a localoptimization procedure. Smoothing (b) reduces this undesirable effect.

curvature, are calculated for each contour point. Typi-cally, dozens of neighboring points are used in thecurvature estimation, and therefore the resulting systemis well-determined.

Smoothing of the curvature trinary functions.If twocorresponding peaks of the trinary functions,bt(s) andbs(s), of the template and the individual structure, respec-tively, do not overlap at all (see Fig. 16a), then the localoptimization procedure described in Section 2.3 will not beable to align them, but will converge to a local minimum. Inorder to avoid this, we apply a low-pass filtering to thefunctionsbt(s) andbs(s), which effectively widens the spatialextent of their positive and negative peaks (see Fig. 16b),thereby reducing the possibility of having two correspond-ing non-overlapping peaks.

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