Medical Image Analysis 7 (2003) 403–416www.elsevier.com/ locate/media

A generic framework for the parcellation of the cortical surface into¨gyri using geodesic Voronoı diagrams

a,c,d,e , a,d,e a,e a,e a,e* `A. Cachia , J.-F. Mangin , D. Riviere , D. Papadopoulos-Orfanos , F. Kherif ,c,e b´I. Bloch , J. Regis

a ´ ´Service Hospitalier Frederic Joliot, CEA, 91401 Orsay, Franceb ´ ´Service de Neurochirurgie Fonctionnelle et Stereotaxique, La Timone, Marseille, France

c ´Departement Traitement du Signal et des Images, CNRS U820, ENST, Paris, Franced ´ ´Imagerie Cerebrale en Psychiatrie, INSERM ERM0205, Orsay, France

e ´ ´Institut Federatif de Recherche 49 (Imagerie Neurofonctionelle), Paris, France

Abstract

In this paper we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri aredefined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto

¨the cortical surface. The parcellation stems from two nested Voronoı diagrams computed geodesically to the cortical surface. The firstdiagram provides the zones of influence of the sulci. The boundary between the two zones of influence of each selected pair of sulci

¨stands for a gyrus seed. A second diagram yields the gyrus parcellation. The distance underlying the Voronoı diagram allows the methodto interpolate the gyrus boundaries where the limiting sulci are interrupted. The method is illustrated with 12 different hemispheres. 2003 Elsevier B.V. All rights reserved.

Keywords: Brain; Cortex; MRI; Parcellation; Gyri; Sulci; Morphometry

1 . Introduction 2001; Good et al., 2001). While none of these tools can beconsidered perfect, simply because of the huge complexity

The recent advent of automatic methods dedicated to and variability of brain anatomy, it is assumed thatbrain morphometry has attracted much interest from the analyzing hundreds of brains overcomes the failuresneuroscience community (Ashburner and Friston, 2000; observed when analyzing a few.Fischl et al., 1999; Thompson et al., 2000; Chung et al.,2001; Toga and Thompson, 2002). These tools provide a 1 .1. Brain morphometrynew way of addressing issues related to the links betweenanatomy and function; they allow the study of the in- Most of the brain anatomy analysis methods applied onfluence of various parameters on the anatomicalsub- a large scale rely on acoordinate system, which may bestratum: sex, handedness (Davatzikos and Bryan, 2002; either three dimensional (like the well known TalairachMangin et al., 2003a), cognitive features (Maguire et al., grid; (Talairach and Tournoux, 1988)), or two dimensional,2000), genetic features (Thompson et al., 2001), pathology with spherical topology (for studies of cortical thickness,(Woermann et al., 1999; May et al., 1999), etc. Longi- see (Fischl et al., 1999; Mac Donald et al., 2000; Chung ettudinal studies of brain maturation or the ageing process al., 2003)). With each method, the coordinate system ishave also received increasing attention (Paus et al., 1999, provided to each brain via a specific warping process. This

process, calledspatial normalization, matches as far aspossible the different brains under study with a template

*Corresponding author. Tel.:133-1-6986-7813; fax:133-1-6986-endowed with the coordinate system. Morphometry is then7786.performed on a point-by-point statistical basis, either onE-mail addresses: cachia@shfj.cea.fr(A. Cachia),mangin@shfj.cea.fr

(J.-F. Mangin),http: / /anatomist.info. data-related features (Ashburner and Friston, 2000) for

1361-8415/03/$ – see front matter 2003 Elsevier B.V. All rights reserved.doi:10.1016/S1361-8415(03)00031-8

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studying relative or absolute tissue amount, or on de- quality local coordinate system for complex structures likeformation-related features (Chung et al., 2001) for compar- cortex sulco-gyral patterns.ing the spatial organization of the tissue (translation,rotation, shrinking, etc.).

This coordinate-based spatial normalization paradigm 1 .2. Segmentation of cortical gyrihas had a tremendous impact on morphometry strategiesbecause of its versatility. A number of different normaliza- Image analysis methods dedicated to the cortex almosttion algorithms, however, are used throughout the world, always focus on cortical folds (i.e. the sulci), because theyeach potentially leading to different results (Hellier et al., can be defined simply using geometric properties (depth,

12003). For instance, the widely distributed SPM software curvature, medial axes, etc.). The usual neuroscience(Friston et al., 1995; Ashburner and Friston, 2000) allows approach to cortical surface segregation, however, is gyrusthe user to choose the template or the number of basis based. A gyrus is usually considered to be amodule of thefunctions used to model the warping. This observation cortex endowed with dense axonal connections throughoutmeans that what is called ‘‘spatial normalization’’ is far local white matter (Van Essen, 1997). Unfortunately,from clear, which can be explained by the fact that nobody cortical gyri are relatively difficult to define from a purereally knows what the gold standard is in terms of brain geometrical point of view, even if they are assumed to bewarping. Furthermore, nobody knows to what extent the delimited by two parallel sulci.matching of two different brains with acontinuous de- Two directions of algorithmic research aim at providingformation makes sense from an anatomo-functional point automatic methods to perform structure-based mor-of view. phometry. The first approach stems directly from the

An alternative approach consists of following the classi- iconic spatial normalization scheme: a manual segmenta-cal way of performing morphometry, namely defining tion of the template is warped towards any new brain inanatomical structures by a segmentation method and order to obtain an automatic segmentation (Collins et al.,deriving shape descriptors (surface, volume, moments, 1995). While this approach gives good results for stableetc.) andinter-structure descriptors (relative spatial posi- brain areas such as the deep nuclei, it is more questionabletion, connexity, etc.) that will be compared across brains. for the cortex (Tzourio-Mazoyer et al., 2002) because theThis alternative is sufficiently attractive to be applied warping algorithms are disturbed by the high inter-in-manually (Sowell et al., 2002; Kim et al., 2000), although dividual variability of the folding patterns (Ono et al.,

`tedious work has to be performed, which prevents large- 1990; Riviere et al., 2002). Therefore, a concurrentscale studies. The motivation behind this structure-based strategy for the cortex consists of linking blind geometricmorphometry is that some neuroscience results highly parcellations with pattern recognition methods (Mangin etrelated to brain architectural organization may be either al., 1995b; Le Goualher et al., 1999; Lohmann and von

`lost during the non-perfect iconic spatial normalization or Cramon, 2000; Riviere et al., 2002; Cachia et al., 2003;are inaccessible via a coordinate-based approach (Mangin Rettman et al., 2002) in order to achieve a better definitionet al., 2003b,c). of the sulco-gyral shapes to be compared across brains.

Finally, it should be noted that some morphometry Many other dedicated segmentation schemes have beenapproaches arehybrid becauselocal coordinate systems defined for various applications (Duncan and Ayache,are used to compute some shape descriptors. For instance,2000).using a surfacic coordinate system dedicated to simply No data-driven parcellation algorithm dedicated to gyralconnected objects allows the computation of simple aver- shapes has been proposed in the literature, because of theage distances between objects and opens the door to lack of gyrus geometrical definition. Therefore, a gyrusspherical harmonics-based measures (Gerig et al., 2001). segmentation method has to rely on a preprocessing,The detection of cortical folds using two-dimensional yielding the detection of the sulci delimiting the gyri. Aregular meshes allows similar shape studies (Le Goualher first possible approach would be to add someconstraintset al., 2000; Davatzikos and Bryan, 2002). While some of into warping algorithms in order to impose a corre-the shape descriptors used in this approach are still spondence between these sulci (Thompson and Toga,disturbed by inter-individual variability in the definition of 1996; Collins et al., 1998; Cachier et al., 2001). Thusthe local coordinate system, the fact that a dedicated warping of the template gyral segmentation would providesystem is used for each brain structure overcomes some of a reliable result. An alternative, proposed in this paper,the problems related to the standard normalization consists of devising amorphological definition of the gyriparadigm. Global warping algorithms imply a trade-off from the limiting sulci.between the constraints imposed by the different shapes Thus, the gyrus segmentation method proposed in thisembedded in the brain, which usually leads to a low- paper relies on a two-stage strategy. First, the main cortical

sulci are automatically extracted and identified using acontextual pattern recognition method that may be viewed

1http: / /www.fil.ion.ucl.ac.uk/spm/. as a structural alternative to the brain warping approach

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`(Riviere et al., 2002). Second, the dual gyri are defined aspatches of the cortical surface yielded by the computation

¨of two nested Voronoı diagrams, the initial seeds of whichare inferred from the identified sulcus bottom lines. Thecomputed gyri correspond to a set of pairs of sulciprovided by the user. These sulci can be chosen from thelist of sulci defined by the neuroanatomist who hasperformed the manual identification used to train the

`pattern recognition system (Riviere et al., 2002). Themethod is generic because the user can select any pair ofsulci, and because an alternative identification of the sulcican be provided either manually or automatically for alarge-scale study if a learning database is constituted.Some user-friendly tools can be downloaded fromhttp: //anatomist.infoto perform these manipulations.

The next section gives an overview of the main steps ofthe method: detection and labeling of the sulcus bottomlines (Section 2.1), transition from their 3D representationsin the raw data to their 2D projections on the corticalsurface (Section 2.2) and gyral parcellation of the corticalsurface (Section 2.3). For a better understanding, thetechnical aspects are detailed separately (Sections 3 and 4).Finally, the method is applied to 12 different brainhemispheres and the results are discussed (Section 5).

¨Fig. 1. Definition of a gyrus from two parallel sulci using the Voronoı2 . Overview of the methoddiagram principle. (A) Two schematic parallel sulci. (B) Definition of the

¨Voronoı diagram of the sulcal lines, i.e. parcellation of the domain in theThe main problem complicating the morphological zones of influence of the sulci. (C) The boundary between the two zones

definition of the gyri is the interruption of the delimiting of influence provides the gyrus seed. (D) The gyrus delimited by the twosulci, because these interruptions are highly variable. Theparallel sulci can be obtained as the zone of influence of the previous

boundary seed. The initial two sulcal lines must be removed from theidea proposed in this paper overcomes this difficulty using¨domain to prevent the front propagation underlying the Voronoı diagram¨the Voronoı diagram principle (Aurenhammer and Klein,

construction crossing them (a gyrus should end at the bottom of the2000). Such a diagram is a parcellation of space from a set limiting sulci). The remaining gyrus boundaries will be induced byof seeds. Each parcel is thezone of influence of one of the competition with the other gyri.seeds, namely the domain of space closer to this seed thanto any other seed. If a set of lines approximately located atthe level of the crowns of the gyri of interest can beprovided as gyrus seeds, the whole gyral parcellation can

sented by a set of voxels obtained from a skeletonbe defined from a geodesic distance computed conditional-

segmentation (Fig. 2). For each sulcus, discrete topologyly to the cortical surface. Each gyrus will be the zone of

properties (Malandain et al., 1993; Mangin et al., 1995a)influence of its own seed (Fig. 1). In order to impose the

enable us to obtain the subset of voxels corresponding tosulcus bottoms as parts of the boundaries between these

the bottom lines (main part and branches, seeFig. 3),zones of influence, they are removed from the cortical

called sulcal bottom lines. Another outcome of this pre-surface to avoid the distances to be propagated across these

processing stage consists of two smooth meshes of thelines. Hence the resulting diagram is inferred from an

cortex hemispheres endowed with a spherical topologyiterative dilation of the gyrus seeds that are stopped either

(Mangin et al., 1995a, 1996), each mesh corresponding toat the level of the sulcus bottoms, or when two zones of

the interface between the gray and white matter. It is ontoinfluence contact each other.

this representation of the cortical surface that the sulcalbottom lines will be projected (cf. Section 3) to define the

2 .1. Sulcus segmentation and identification limits between the dual gyri. Therefore, to obtain access tothe method described in this paper, the user has to provide

The first stage of the method, which has been described a list of pairs of sulcus names; each pair will usually`in (Riviere et al., 2002),automatically provides the correspond to two parallel sulci, possibly interrupted,

identification of the main sulci, each sulcus being repre- defining a gyrus.

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Fig. 2. Top: Result of sulcus extraction and identification (left: the white matter mesh used as a spherical model of the cortex). The different colors`correspond to the various names used by our neuroanatomist to train the recognition system (Riviere et al., 2002). Bottom (left) Only the sulci used by the

user as a boundary for the gyral parcellation are represented. Several small sulci, coded with the same color, have been gathered into one to create a longboundary; others, yielding useless gyral subdivisions, have been discarded. (right) The corresponding gyral parcellation. Hence, according to the userrequirements, several different parcellations can be obtained using different choices of gyrus boundaries.

2 .2. Projection of the sulcal bottom lines onto the Another important constraint is the localization of thecortical surface projection that should correspond to the deepest part of the

fold on the cortical surface (seeFig. 5(A)). After compar-The morphological definition of the gyri given above ing several possible approaches (cf. Section 3.1), we chose

(seeFig. 1) assumes a processing geodesic to the cortex first to project each bottom point using the tangent plane tospherical topology. The initial sulcal bottom lines, how- the sulcus. Then the projection of each sulcus bottomever, are not embedded in the spherical mesh representing connected component is regularized (cf. Section 3.2) inthe cortex. Therefore, they have first to be projected into order to impose a global consistency: the whole set ofthis 2D space. bottom points is projected using the affine transformation

To make the projected bottom lines behave like walls which minimizes the squared distance to the initial project-for geodesic distance propagation, their topology has to be ed points (Fig. 3).preserved as far as possible during the projection. Preserv- Finally, morphological closing operations, processeding the topology during the projection process is not geodesically to the triangulated surface (cf. Section 4.2),straightforward because of sampling differences between guarantee as far as possible that the projection of eachthe initial 3D volume, in which the sulcus skeleton has connected component is a simply connected object for thebeen defined, and the triangulation of the cortical surface. mesh topology.The mesh sampling, moreover, is not regular, because itresults from an adaptive decimation algorithm whichadapts the size of the triangles to the local curvature. 2 .3. Gyral parcellation

The initial topology of the bottom line cannot bepredicted because the sulcus can be interrupted, namely Once the sulci have been projected, the remainingconsist of several smaller folds and branches, the bottom processing is embedded in the spherical topology of thelines of which do not touch each other. Hence, the cortical surface. The two stages of computation rely on the

¨projection is performed connected component by con- Voronoı diagram notion. These diagrams of zones ofnected component. influence are computed via iterative dilation of the seeds

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Fig. 3. A sketch of the method mapped on an inflated version of the cortical surface for clarity. Most of the remaining surface curvature is related to thegyral parcellation targeted by the algorithm. Each sulcus bottom line connected component is first projected (point-to-point (A.1), and globally using anaffine transformation (A.2)). Then the projection is closed using geodesic mathematical morphology, and skeletonized in order to obtain an actual

¨continuous line (A.3). A first Voronoı diagram is computed for the seeds corresponding to these projected lines using a geodesic distance (B). This diagramprovides a sulcal-based parcellation of the surface. The seeds that will stand for the gyri are boundaries of this first diagram related to the pairs of sulciinitially defined by the user (C). Finally, a second diagram is computed for these gyral seeds after removal of the sulcal seeds from the mesh in order toprevent the geodesic distance crossing a sulcus bottom (D).

according to the geodesic Euclidean distance (cf. Section cortical surface (seeFig. 3(D)). The sulcal lines are first4.1). removed from the mesh in order to prevent the dilation

¨The first step leads to the definition of a gyral seed from from crossing these targeted boundaries. Then, a Voronoıeach pair of sulci given by the user. This definition relies diagram is computed for the gyral seeds yielded by the

¨on the computation of the Voronoı diagram of the labeled previous step. Finally, this intermediary diagram is used assulcal lines (seeFig. 3(B)). seed for the computation of the last diagram computed for

The goal of this diagram is the detection of the the whole mesh, in order to fill the sulcal lines.boundaries between the zones of influence of the pair ofsulci given by the user (seeFig. 3(C)). Such a boundarywill further represent the seed of the corresponding gyrus. 3 . Projection of the sulcal bottom linesThe set of boundaries of the diagram is sometimes called aSKeleton by Influence Zone (SKIZ) (Lantuejoul and The more technical part of the parcellation method is theBeucher, 1981). The boundaries are the nodes with at least transition from the voxel world towards the corticaltwo different labels in their direct neighborhood. Hence, spherical world. The parcellation algorithm assumes thatthe boundaries of interest are sets of nodes with exactly each connected component of a sulcus bottom line can betwo labels in their neighborhood corresponding to one of considered as an impassable barrier. Thus, each continuousthe user-specified sulcus pairs (Fig. 4). This definition of line of bottom voxels has to be transformed into athe gyral seeds leads to a localization equidistant to the continuous line of mesh nodes. In order to achieve thistargeted sulcal boundaries (seeFig. 1). This localization result, each connected component of the sulcal bottom lineseems to be the best one, considering that the gaps between is first globally projected using an affine transformation,the sulcal boundaries will be filled via competition be- which prevents the creation of large gaps between thetween neighboring seeds. With this choice, the virtual projected points. The projected set is then post-processedboundaries will be equidistant to the midline of the two using geodesic morphological techniques in order to obtaincompeting gyri. a simply connected line.

The second step leads to the gyral parcellation of the Because of weaknesses in the discrete topology charac-

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Fig. 4. Selection of the gyrus seeds. Schematic example for frontal lobe parcellation. (A) Definition of the sulcal line (C.S., Pre.C.S., etc.) of the sulci¨involved in a gyral pair (Pre.C.G., F1, F2, F3). (B) Definition of the Voronoı diagram of the sulcus influence zone. (C) Selection of the gyrus seeds: only

¨the boundaries corresponding to a pair of sulci delimiting a gyrus are taken as gyrus seeds. (D) Voronoı diagram of the selected gyrus seeds. The sulcal¨lines are removed from the domain before the seed dilation process. The distance underlying the Voronoı diagram allows the method to extrapolate the

gyrus limits where the sulci do not provide a boundary.

`terization scheme used to define the bottom lines (Riviere An important point is the choice of the geometricet al., 2002), a pre-processing step has been added before properties used to drive the projection process. Threetheir projection. Their continuity is sometimes broken different approaches have been compared to finally choosebecause some of thebottom points are labeled asjunction a method using the plane tangent to the sulcus surface atpoints (i.e. points connecting a sulcus to a branch or each bottom point. In the following, we describe first aanother sulcus). Therefore, a first volumetric closing comparison of these three point-to-point projection meth-process related to the 26-connectivities is applied to the ods. Then, we introduce the regularized global projection,sulcal bottom lines before projection. This closing is made which is an adaptation of the well-known ICP algorithmup of a standard dilation followed by a homotopic erosion (Besl and McKay, 1992; Feldmar and Ayache, 1996).which preserves the topology (Malandain et al., 1993; Finally, we describe the geodesic post-processing.Mangin et al., 1995a). For the sake of clarity, the following explanations

Fig. 5. Projection using local mesh information (curvature or geodesic depth). In this example, the sulcal bottom line is projected onto the corticalsurfacealong the line of maximal geodesic distance to the gyrus crowns.

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consider only one connected component of a sulcus bottom crowns. The geodesic depth of all these nodes is thenline, which is called a ‘‘ sulcal line’’. All the processing is null.applied independently to each connected component. 4. Compute the geodesic distance to these crowns, using

the front propagation described below.For the two last approaches, which embed the constraint3 .1. Point-to-point matching

on the localization of the projected nodes as a trade-offbetween the Euclidean distance and a local mesh property,The algorithms described in this section projects each

Curv / Depthv t the selection of the weighta influences thevoxel M of the sulcal line onto a nodeM of thei i

accuracy of the projection. In practice, this weight istriangulation representing the cortical surface. Three differ-chosen empirically, according to the raw data resolutionent approaches have been tested, each corresponding to a

tand the cortical mesh properties, so that the nodeMdifferent a priori constraint on the localization of theCurv / Deptht minimizing the distanced lies in the bottom ofprojected line. In each case,M is defined as the meshi vv the fold, close to the voxelM .node minimizing a distance toM . ii

3 .1.1. High curvature location3 .1.3. Geometrical approachIn this approach, the projected sulcus bottom points are

In this approach, the sulcal lines are obtained as theconsidered to correspond to high curvature locations on theintersection of the continuation of the sulcal surface withspherical mesh. A high curvature is assumed to be anthe cortical mesh. The sulcal surface is assumed toindicator of the exact location of the cortical fold bottom.represent a notion closed to the medial axis of the fold,To be more precise, since the spherical mesh is oriented,which should split the cortical surface at the right placethe targeted nodes are endowed with a low negative

v(Fig. 6). More precisely, for each pointM of the sulcalcurvature. Thisa priori constraint is embedded in the iv line, the tangential planeP to the sulcal surface isdefinition of the distance between a bottom pointM and ii jestimated from itsm 26 neighborsN included in the foldany mesh nodeM: i i →

skeleton. It is easy to show that the normal vectornCurv v v Curvd (M ,M)5 d (M ,M)1a Curv(M), (1) defining the best fitting tangent plane, in the sense ofi E i

minimizing the sum of the squared distance of themCurv v iwhere a is a positive weighting constant,d (M ,M) 1 miE i points hN , . . . ,N j to the plane, corresponds to thei idenotes the 3D Euclidean distance and Curv(M) is a m jismallest eigenvalue of the matrixV 5o (M 2j51 icurvature-related characterization of the mesh at nodeM. v j v TM )(M 2M ) (Chung et al., 2001). The projected pointi i iThis discrete curvature is heuristically defined by t v vM of M is the closest node toM , according to the 3Di i i%%& %&

Curv(M)5 sign(MB ?n )d (M,B), (2)M E

whereB is the mesh neighborhood barycenter (B 5o M ,i%&

with M the neighbors ofM), and n the normal of thei M

mesh at nodeM. This heuristic is used for the smoothnessof the resulting curvature map.

3 .1.2. Maximal depthIn this approach, the projected sulcus bottom points are

assumed to be located as deeply as possible. The deepestmesh areas are considered to correspond to the exactlocation of the cortical fold. As in the previous approach,this a priori constraint is included in the definition of the

vdistance between the bottom pointM and any mesh nodei

M:Depth v v Depthd (M ,M)5 d (M ,M)1a Depth(M), (3)i E i

where Depth(M) corresponds to the geodesic depth at nodeM. The geodesic depth computation follows the followingsteps (a similar approach may be found in (Rettman et al.,2002)):1. Apply a 3D morphological closing to the white matter

binary mask.2. Apply a 3D erosion of 5 mm to the closed mask. Fig. 6. Projection of the sulcal bottom line using the intersection of the

plane tangent to the sulcal surface with the cortical surface mesh.3. Define all the mesh nodes outside this mask as gyrus

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Euclidean distance, included in a stripe of 1 mm defined on sulcus bottom. Each sulcal line point is randomlyboth sides of the tangent planeP . projected onto one of these two targets (Fig. 7), and thei

final result after regularization of the projection is not3 .1.4. Comparison of the three methods predictable.

Thanks to the following stages of the robust projection • The definition of the geodesic depth is not a straight-process, the three point-to-point projection methods intro- forward task since it requires the definition of aduced above generally give acceptable results. Specific reference level (i.e. the null depth), which is not alwaysconfigurations, however, have led us to choose the method clear from an anatomical point of view, especially forbased on the tangent plane as the most robust method. This buried areas like the insula. When one gyrus is toocan be understood from the fact that the assumption that buried to be included in the reference level, the line ofthe best location of the fold projection corresponds to high maximal depth is ill-defined.curvature or maximal depth of the mesh is not always A comparison of the whole projection process applied toverified: the central sulcus with the three point-to-point projection• During the gyrogenesis process, some gyri are buried in methods is given inFig. 8. In the following results, the

the depth of the cortex and may locally fold the bottom tangent plane method will always be used.´and the walls of some sulci (Regis et al., 1995; Yousry

et al., 1997; White et al., 1997; Boling et al., 1999). 3 .2. Robust projectionSuch locations turn out to correspond to a minimalgeodesic depth relative to the rest of the sulcus bottom In this section, the point-to-point projection performed(see the case of the central sulcus inFig. 8). In fact, the by the algorithm described in the previous section isbottom line of a sulcus on the mesh usually presents regularized, mainly to minimize the width of the gaps thatsome depth variations. This is problematic because the may exist in the middle of the set of projected points.

Depthdistanced biases the projection towards depth local These gaps can be created either by an erroneous estima-maxima and creates some gaps in the projection. tion of the tangent plane, which can occur, for instance,Furthermore, the extremities of the sulcus are attracted because of the presence of branches along the sulcustowards the depth, which shortens the projected line. In skeleton, or because of the presence of large triangles

curvsome cases, these buried gyri also disturbd because along the bottom of a flat sulcus, which disturbs thethey present a positive surface curvature. computation of the intersection with the tangent planes.

• Most of the time, the aspect of the folds on the white A simple way to overcome the consequences of thematter surface is well described as V-shaped surfaces, problems mentioned above consists of reducing the num-namely surfaces whose points with a locally minimal ber of degrees of freedom used to project the initial set ofcurvature form a single bottom line corresponding to bottom points of the sulcal line. The point-to-point projec-the projection target. Unfortunately, some U-shaped tion includes no constraint of consistency for the point set.sulci, whose bottom on the white matter surface is flat, In fact, a low-dimensional transformation should be suffi-can be observed. In such cases, the target line is split cient to project the sulcal line on the mesh. Therefore, ininto two spurious lines located on both sides of the this section, we propose to use the affine transformation

Fig. 7. Curvature information (left) and splitting of the projected sulcal line when the bottom of a fold is flat.

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Fig. 8. Top: Projection, closing and thinning of the central sulcus. From left to right: projection using curvature, geodesic depth and a tangent plane. Frombottom to top: first projection using point-to-point matching, affine transformation, closing and thinning (for the sake of clarity, the projectionsof the sulcalline, buried in the depth of the cortex, are mapped on an inflated version of the cortical surface). Bottom: From left to right: curvature and geodesic depthmap used to drive the projection; the presence of a buried gyrus locally deforms the wall and the ridge of the central sulcus.

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which minimizes the sum of the squared distances between matter mesh is considered as a graph, whose edge lengthsthe bottom points and their point-to-point projection. In are simply their Euclidean length. The lengthd(n) isorder to obtain a robust estimation, an outlier detection initialized by zero for the setS and an infinite valuemethod is included in a two-stage process. It should be anywhere else. Nodes are then processed in a specificnoted that this method does not need to be very sophisti- order, determined by the length of the path toS. Lengthscated (Huber, 1981), because some additional morphologi- and order are calculated recursively during the searchcal post-processing is still required to fill the gaps appear- process: ifn9 is a neighbor ofn, thend(n9) is upgraded bying in the oversampled areas of the mesh. A detailed d(n)1 d(n,n9) when it leads to a length decrease. Thisexplanation of the least square estimation of the affine algorithm can be understood as the propagation of a thicktransformation is proposed in Appendix A. The following front, made up of the points located in a short range ofis the method of outlier rejection: distance to the set. The front is initialized with the setS.

init1. A first estimation of the affine transformationA is The front is then iteratively updated by removing thevmade after discarding the pointsM , whose projection closest point toS after having propagated the length to itsi

tM is too far away. This first outlier rejection results neighbors, and adding them to the front if they had not yeti

from a single threshold on the Euclidean distance (in been reached.initpractice 10 mm). Then, a new projectionM is This algorithm only provides a coarse approximation ofi

vcomputed for eachM as the closest mesh node to the the exact Euclidean geodesic distance. Hence, it cannot beiinit vtransformed pointA M . It is assumed that the first used to develop a Mathematical Morphology, becausei

initestimation ofA is a good approximation of the final dilations cannot be composed correctly. The result, how-transformation. Hence, it will be used to obtain a better ever, is sufficient to perform simple quasi-isotropic dila-outlier detection. tions, by applying a threshold to geodesic distance maps.

final2. A second estimation of the affine transformationA In practice, the front propagation is stopped as soon as theis made after discarding a fixed percentagep of the threshold is reached, for the sake of efficiency. The front

vpoints M , which present the greatest Euclidean dis- propagation corresponding to the dilation of the initialit inittance betweenM and M (in practicep510%). The projected set is stopped regularly in order to check thei i

goal is to discard points whose point-to-point projection topology of the dilated set (practically, the checking isis not consistent with the majority of the bottom points. performed each 0.5 mm).

final ¨This final transformationA is then used to perform The Voronoı diagrams are computed using the samevthe final projection of the whole set ofM , the thick front propagation algorithm, with the additionali

finalprojected nodesM corresponding to the closest update of a label imagel(n). A different label is initiallyifinal vmesh node to the transformed pointA M . given to each seed. The set of seeds is the setS fromi

which is computed the distance propagation. Each updateof d(n9) is followed by an update ofl(n9) by l(n), which

4 . Morphological processing corresponds to the underlying seed dilation. When all thepoints of the mesh have been reached, the label image

¨The last stage of the projection method aims at closing corresponds to the Voronoı diagram.the potential gaps in the projected set of points. The goal is

¨to create a barrier for the computation of the final Voronoı 4 .2. Simple points and homotopic skeletonizationdiagram. This is achieved through a dilation of theprojected set on the mesh of the cortical surface. The size The dilated projected point set, which is now simplyof the dilation is the minimal size yielding a simply connected, is skeletonized in order to obtain anactual lineconnected object, which is controlled from the computa- (i.e. a chain of node points). The skeletonization istion of the number of connected components and the Euler homotopic so as to preserve the topology obtained duringnumber. Finally, a homotopic skeletonization preserving the previous step. Its implementation on the mesh relies onthe topology is applied to the dilated set in order to obtain the definition of simple points in two-dimensional domainsa thin line, namely a chain of nodes connected by some (Serra, 1982): a point of a binary image is simple, namelytriangle edges. it can change color without topological change, if its

immediate neighborhood contains exactly one white con-¨4 .1. Geodesic dilations and Voronoı diagrams nected component and one black connected component.

This characterization is used to develop the followingAll the geodesic distances used in the method described skeletonization algorithm:

in this paper stem from the thick front propagation idea (1) Put all the nodes of the object connected to outsideproposed in(Verwer et al., 1989).This method is based on into the front.a special case of theA*-algorithm (Hart et al., 1968), (2) Mark all the non-simple points of the front aswhich provides the lengthd(n) of the shortest path in a ‘‘immortals’’.graph from a set of nodesS to any other noden. The white (3) Repeat until the front is empty:

A. Cachia et al. / Medical Image Analysis 7 (2003) 403–416 413

(a) repeat until the front is empty: to long neighboring parallel sulci, in order to obtain as far(i) remove the next point of the front; as possible the usual parcellation described in anatomical(ii) compute its topological characterization if it atlases. The results proposed inFig. 9 share striking

is not ‘immortal’; similarities across the six brains. More work has to be done(iii) if it is a simple point, remove it from the on sulcus pair selection, however, to obtain a more

object and update the topological characteri- intuitive parcellation.zation of its neighbors; otherwise, mark it as These results are also qualitatively comparable to atlas‘immortal’; descriptions, except for the occipital lobe. This lobe does

(b) put all the nodes connected to outside into the not include gyri corresponding to our morphologicalfront. definition. Comparing these results with manually defined

gyri has not yet been carried out, first because it is tediousand difficult work which has to be done on a large

5 . Results and discussion database to be significant, and second because the parcella-tions yielded by our method are not intended to correspond

The method has been applied to the two hemispheres ofexactly with the standard method. The standard corticalsix different brains extracted from a database of brains parcellation was proposed by anatomists 100 years ago,whose sulci had been identified manually by a neuro- when the functional segregation of the brain into functionalanatomist in order to train the pattern recognition system. areas was not known. Thus, this standard point of view isThe list of sulcus pairs selected by the user corresponded not necessarily the only meaningful one to compare

Fig. 9. A typical result obtained for the two hemispheres of six different brains. Each color corresponds to a different gyrus. The boundaries betweengyriappear in white. For the external face of the brain, the set of pairs of sulci selected for this experiment aims at defining the three horizontal frontal gyri andthe polar frontal face, the three horizontal temporal gyri, the pre- and postcentral vertical gyri corresponding to motor (cyan) and somesthesic (green) areas,and the two parietal lobules. Some unsatisfactory attempts were made to parcellate the occipital lobe. The lowest images provide a superimposition in theTalairach proportional system of the mesh of the six brains. Other kinds of parcellations can be obtained if the user selects a different list of sulcuspairs.

414 A. Cachia et al. / Medical Image Analysis 7 (2003) 403–416

populations. The goal of the method described in this paper While the first application of automatic cortical parcella-is to provide an automatic and reliable parcellation tech- tion methods relies on morphometric studies, splitting thenique, which can be applied on large databases, using the cortex into standard pieces could also be used as asame morphological definition for all the brains. Therefore, referential system to compare functional results, using aour efforts are aimed at searching for parcellations which volume-of-interest-based statistical method (Evans et al.,can be consistently inferred from the sulcus identification, 1991). Another interest in the cortical parcellations intowhatever the inter-individual variability of the sulco–gyral gyral patches stems from the recent development of MRpatterns. diffusion imaging for fiber tracking (Poupon et al., 2001).

The huge folding variability highlighted in the figure The methods used to detect the fiber bundles linking twoillustrates the difficulties preventing a simple geometrical different cortical areas are still in their infancy, but thisdefinition of the gyri. Some frontal sulci that are often long new possibility may lead to the development of dedicatednon-interrupted furrows can be split into several pieces in mapping methods. One possibility is the inference of thesome brains. This phenomenon disturbs both the sulcus matrix of connectivity of the main cortical gyri. For eachrecognition and the gyrus definition. Nevertheless, our individual, using gyral patches as input and output maymethod can extrapolate the usual parcellation to these allow the sorting of the huge number of tracked bundles.intriguing complex configurations. Hence, any brain can be Then individual matrices of connectivity could be com-processed in a rather consistent automatic and reliable way, pared on a statistical basis. This approach could providewhich opens the door to large-scale comparisons between new research and diagnostic tools for the pathologiespathological and standard subjects. According to the user’s related to brain connectivity.interest, different sulcus pair lists may be provided to themethod in order to compare gyral areas and shapes fromvarious definitions. One of the weaknesses of our ap- 6 . Conclusionproach, however, is the need to define a set of gyricovering the whole cortical surface or at least surrounding The complexity and variability of the sulco–gyraleach gyrus of interest. Competition between neighboring patterns of the human cortex remain a challenging issue forgyri is mandatory in our morphological definition. the research field aiming at normalizing brain anatomy for

Future work will aim at extending the method to a more brain mapping purposes. As matters stand, it is difficult togeneral morphological definition of cortical patches. The claim that a perfect solution can be reached using acurrent method is dedicated to gyri delimited by two long warping strategy. This paper contributes to the stream ofparallel sulci. Current neuroanatomical research on the research aiming at analyzing the cortex patterns from acortical folding process, however, could provide a reliable structural point of view closer to the neuroanatomist’sdetection of smaller sulcal units called sulcal roots that approach. The parcellation method assumes that a reliablemay allow a finer parcellation of the cortical surface identification of the main sulci can be made first for any

´(Regis et al., 1995; Cachia et al., 2003). These units can be subject, which is far from the case with the current patternconsidered as points of the cortical surface where the recognition system. We believe, however, that the currentfolding begins. Thus, sets of such points could be used to state of this system is sufficient to obtain interestingdefine the limits of gyrus subunits using extensions of the morphometry results if a large database of brains isidea described inFig. 4. This extension means that the processed, which can now be done without any userdefinition of each cortical patch seed will be related to a interaction. This belief relies on the large number ofset of more than two sulcal lines, without a specific significant results obtained by voxel-based morphometry inrelationship, like parallelism, between them. A similar spite of poor spatial normalization.extension could be used to define lobes, which could allowus to deal with the occipital lobe, for instance (other lobescan be obtained by merging several gyri).

The direct morphometric application of our method is A ppendix A. Least square estimation of the affinethe comparison of the cortical patch areas between brains.transformationOther related measures like the volume of grey matter inthe gyrus can be obtained. The gyral cortical patches can

In this appendix, we give the technical details leading tobe backprojected to the 3D mask of the grey matter, to be

a least squares estimation of the affine transformation¨used as 3D seeds. Then, a volumetric geodesic Voronoı 1 2 nmatching two set of paired 3D pointshX ,X , . . . ,X j (the

diagram can be computed in the cortex grey matter in 1 2 nset of bottom voxels) andhY ,Y , . . .Y j (the set of nodesorder to split it into gyri. The ratio between gyrus grey

provided by the firstpoint-to-point projection stage):matter volume and gyrus area would also provide an

i i iaverage cortical thickness measure. Finally, more sophisti-A ?X 5 L ?X 1 T 5Y (i 5 1, . . . ,n), (A.1)cated descriptors of the gyrus shape could be computed

i i i i T i i i i Tusing, for instance, 3D invariant moments (Poupon et al., where X 5 [X ,X ,X ] , Y 5 [Y ,Y ,Y ] , L is a linearx y z x y z

1998). transformation (i.e. a 333 real matrix) andT a translation

A. Cachia et al. / Medical Image Analysis 7 (2003) 403–416 415

C hung, M.K., Worsley, K.J., Paus, T., Cherif, C., Collins, D.L., Giedd,vector. Using condensed block matrix notation for thei i i i T J.N., Rapoport, J.L., Evans, A.C., 2001. A unified statistical approach˜affine transformation, namelyX 5 [X ,X ,X ,1] (n 3 4),x y z to deformation-based morphometry. NeuroImage 14 (3), 595–606.i i i i T ˜Y 5 [Y ,Y ,Y ] (n 31), A 5 [LuT ] (334), Eq. (A.1) canx y z C hung, M., Worsley, K., Robbins, S., Paus, T., Taylor, J., Giedd, J.,

be written as Rapoport, J., Evans, A.C., 2003. Deformation-based surface mor-phometry applied to gray matter deformation. NeuroImage 18 (2),1 2 n 1 2 n˜ ˜ ˜ ˜[A uX uX u . . . uX ] 5 [Y uY u . . . uY ] (i 5 1, . . . ,n). (A.2) 198–213.

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