Cachia JF Mangin D Ri vi ere Kherif N Boddaert A Andrade D apadopoulosOrf anos JB Poline I Bloch M Zilbo vicius Sonigo Brunelle and J egis Abstr act In this paper we pr opose new epr esentation of the cortical surface that may be used to study the c ID: 26629 Download Pdf

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Cachia JF Mangin D Ri vi ere Kherif N Boddaert A Andrade D apadopoulosOrf anos JB Poline I Bloch M Zilbo vicius Sonigo Brunelle and J egis Abstr act In this paper we pr opose new epr esentation of the cortical surface that may be used to study the c

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 Primal Sk etch of the Corte Mean Curv ature: Morphogenesis Based Approach to Study the ariability of the olding atterns A. Cachia, J.-F Mangin, D. Ri vi ere, Kherif, N. Boddaert, A. Andrade, D. apadopoulos-Orf anos, J-B. Poline, I. Bloch, M. Zilbo vicius, Sonigo, Brunelle, and J. egis Abstr act In this paper we pr opose new epr esentation of the cortical surface that may be used to study the cortex olding pr ocess and to eco er some putati stable anatomical landmarks called sulcal oots usually ur ried in the depth of adult brains.

This epr esentation is primal sk etch deri ed fr om scale space computed or the mean cur atur of the corti- cal surface. This scale-space stems fr om diffusion equation geodesic to the cortical surface. The primal sk etch is made up of objects deﬁned fr om mean cur atur minima and saddle points. The esulting sk etch aims ﬁrst at highlighting signiﬁcant elementary cortical olds, second at epr esenting the old mer ging pr ocess during brain gr wth. The ele ance of the frame- ork is illustrated by the study of central sulcus sulcal oots fr om antenatal to adult age. Some

esults ar pr oposed or ten differ ent brains. Some pr e- liminary esults ar also pr vided or superior temporal sulcus. ywor ds Sulcogenesis, spatial normalization, mor phometry ariabil- ity HE adv ent of methods dedicated to the automatic analysis of lar ge databases of MRI images of brain anatomy has raised lar ge interest in the neuroscience community [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. These tools, in- deed, pro vide ne ays of addressing issues related to the com- parison of brain populations. The allo the study of the inﬂu- ence of arious parameters (se x,

dominant hemisphere, cogni- ti features, genetic features, pathology etc...) on the anatomi- cal substr atum [13], [14], [15], [16], [17]. Longitudinal stud- ies of brain maturation or ageing process ha also recei ed increasing attention [18], [19], [20]. The comple xity and the inter -indi vidual ariability of the cortical folding patterns, ho w- er is still challenging issue for these tools. Indeed, nobody really kno ws ho to match the cortical folds across brains, and at which xtent such matching is rele ant from neuroscience point of vie A. Spatial normalization Most of the brain anatomy

analysis methods rely on the con- cept of spatial normalization, which consists in arping all the brains to ards template endo wed with 3D (v olumet- rical) or 2D (spherical) coordinate system. This referential A. Cachia, J.-F Mangin, D. Ri vi ere, A. Andrade, Kherif, D. apadopoulos- Orf anos, M. Zilbo vicius, and J-B. Poline are with the Service Hospitalier Fr ed eric Joliot, CEA, 91401 Orsay France I. Bloch and A. Cachia are with the epartement raitement du Signal et des Images, CNRS U820, ENST aris, France N. Boddaert, Sonigo and Brunelle are with the Service de Radiolo- gie ediatrique,

Neck er Hospital, aris, France M. Zilbo vicius, N. Boddaert, A.Cachia, J-F Mangin and J.-B. Poline are in with the Brain Imaging in Psychi- atry team, INSERM ERM205 Orsay France J. egis is with the Service de Neurochirur gie onctionelle et Stereotaxique, La imone Hospital, Marseille, France. All authors apart J. egis are with the Institut ed eratif de Recherche 49, aris, France then underlies further statistical studies. This coordinate based spatial normalization paradigm has made tremendous impact on morphometry strate gies because of its ersatility num- ber of dif ferent normalization

algorithms, ho we er are used throughout the orld, each one potentially leading to dif fer ent results. or instance, the widely distrib uted SPM softw are (http://www .ﬁl.ion.ucl.ac.uk/spm/ [4], [7]) allo ws the user to choose the template or the number of basis functions used to model the arping. This observ ation means that what is called spatial normalization is ar from being clear which is xplained by the act that nobody really kno ws what may be the gold stan- dard in terms of brain arping. Furthermore, nobody kno ws today to which xtent matching tw dif ferent brains with con-

tinuous deformation mak es sense from an anatomical point of vie ith re gards to the problems induced by the ariability of the cortical folding patterns, recent hypotheses claim that some answers could stem from better understanding of the brain gro wth processes [21], [22]. This paper proposes ne rep- resentation of the cortical surf ace that aims at highlighting some of the fold mer ging ents that occur during the cortical surf ace folding process. B. Gyr ogenesis and sulcal ariability Cortical folding, the gross anatomical landmarks of the corti- cal surf ace, xhibit arious forms in dif

ferent adult brains [23], which pre ents from using them as straightforw ard and accu- rate referential. The origin and meaning of this ariability are still lar gely unclear and discussed [24], [22]. Ne ertheless, undle of con er ging ar guments, stemming from ontogenesis, phylogenesis, and architectonic considerations, ha led to hy- pothesize that simple and stable or ganization of the folding, related to the fetal stage, may underlie the apparently ariable and intricate sulcal patterns of adult cortices [25], [21]. Indeed, the ﬁrst cortical folds that appear on the fetus corte x,

called sulcal oots seem to be especially stable (in number position and orientation) across indi viduals. During the follo wing gyral xpansion, ho we er these sulcal roots become urried into the depth of the corte after ha ving mer ged with each other to uild lar ger folds. Because variable mer ging vents can occur dif- ferent sulcal patterns can be observ ed at adult age (see Fig. 1). The more usual patterns ha led to the standard sulcus nomen- clature [23], ut some brains are ery dif ﬁcult to read according to this nomenclature, either because the main sulci are split into pieces or

orst because the sulcal root mer ge ents ha cre- ated unusual sulci.

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 Fig. 1. Up: map of the ﬁr st fetal cortical folds (sulcal oots) of the tempor al lobe [25]. The folds ar numer oted accor ding to their date of appear ance dur ing the fetal sulco enesis. Dif fer ent mer ging vents between these sulcal oots can lead to very dif fer ent folding patterns at adult Left: The superior tem- por al sulcus is supposed to stem fr om the mer of four dif fer ent sulcal oots. Right: The usual mer ging vents leading to the superior tempor

al sulcus have not occur ed. One of the simplest xamples of sulcus, which will be used to illustrate the method described in this paper is the central sul- cus (see Fig. 2). This lar ge sulcus is ery interesting landmark on the cortical surf ace because it is the limit between motor and somesthesic areas. Hence, the study of its shape has been the subject of numerous studies [26], [27], [28], [29], [17]. The central sulcus is supposed to be made up of tw sulcal roots that mer ge in almost all cases. In ery rare cases, ho we er this mer ge does not occur and the central sulcus is split in the

middle by gyrus [30], [31]. Ne ertheless, in most of the non interrupted cases, this gyrus is still visible on the alls of the sulcus [32], [33], [34]. This gyrus initially separating the tw sulcal roots, indeed, has just been uried into the depth of the corte x. Hence, stable simple fetal pattern seems to mak the link between all the possible conﬁgurations. This paper aims at de elopping method inferring this primal pattern from local information about the curv ature of sulcal alls and fundi, which gi es clues about the localization of the sulcal roots. Surf ace curv ature, indeed,

embeds more information than the geodesic depth that has been pre viously proposed to se gment sulci into smaller units [35], [36]. In the follo wing, we propose ne representation of the cor tical surf ace that may be used to study the corte folding process and to reco er putati stable anatomical landmarks, the sulcal roots, usually uried in the depth of adult brains. This represen- tation is primal sk etch [37], [38] deri ed from scale space [39], [40] computed for the mean curv ature of the cortical sur ace. This scale-space stems from dif fusion process geodesic to the cortical surf ace.

The primal sk etch is made up of objects central sulcus white matter white matter for Central Sulcus stage early Two sulcal roots Raw data 36 week foetus 18 months Adult Burried gyrus Standard adult Examples of central sulcus interruptions Fig. 2. Evolution of the centr al sulcus shape during br ain gr owth. Up: Antena- tal ima es allow the econstruction of the fetus corte surface on whic shallow dimples corr esponding to ne gative mean curvatur ar eas ar highligted in blue At that sta the centr al sulcus is made up of two sulcal oots. Middle left: 18 months after birth, the gyrus separ ating

the two sulcal oots is still visible on white matter surface Middle right: At adult sta only slight deformations of the centr al sulcus walls give clues on the pr esence of urried gyrus. Do wn: In some ar br ains described in liter atur the two sulcal oots of the centr al sulcus have not mer ed [30], [31] deﬁned from mean curv ature minima and saddle points, lik in pre vious approaches [41], [42]. The resulting sk etch aims ﬁrst at highlighting signiﬁcant elementary folds, second at representing the fold mer ging process during brain gro wth. The long term aim of the

method consists in reco ering auto- matically map of the sulcal roots from an adult brain. This map ould pro vide an appealing set of landmarks to match dif- ferent adult folding patterns. The sulcal roots could then be used to add some reliable constraints into standard arping al- gorithms [43], [44], [45], [46]. Such map could also be used to study brain de elopment from antenatal MR images. Statis- tics on sulcal root chronology of appearance may indeed be- come precious tool for early detection of de elopment prob- lems. Some interesting cogniti or clinical information could also be

embedded into the ariable mer ging ents occuring be- tween the sulcal roots. Finally since the sulcal roots are de- ﬁned as indi visible cortical folds, the could be used to er come the problem induced by sulcus interruption in algorithmic approaches relying on one dimensional lines [47] or tw dimen- sional meshes [48], [49].

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 The ﬁrst stage of the method consists in xtracting smooth mesh representing the cortical surf ace of each hemisphere from T1-weighted MR image. This mesh is endo wed with the actual spherical

topology of this surf ace, which allo ws the implemen- tation of geodesic dif fusion and inﬂation operations. A. Spherical triangulation of cortical hemipher es ﬁrst sequence of treatments pro vides binary mask of each hemisphere corte with spherical homotop This se- quence, which includes bias correction [50], brain mask se g- mentation [51], hemisphere mask se gmentation [52], and de- tection of the gre y/white interf ace [53], is freely ailable on http://anatomist.info. standard acet tracking algorithm is used to compute ﬁrst spherical mesh made up of acets from the

corte mask [54]. Then, the center of each acet is con- nected to the center of the neighboring acets in order to yield spherical mesh of triangles. This algorithm, which preserv es the initial topology relies on look-up table of conﬁgurations lik in the standard marching cube algorithm. Finally decimation including smoothing is performed to discard stair artef acts re- lated to the underlying discretization. The decimation algorithm is inspired by the algorithm used in the Vtk package [55]. The embedded smoothing operation iterati ely mo es the nodes to- ards their neighborhood gra vity

center which may be related to some usual surf ace olution processes [58]. This mesh construction includes some smoothing operations that may remo some interesting anatomical information. Ne v- ertheless, this smoothing process is required to get initial accept- able surf ace representations and reliable mean curv ature estima- tions. In the future, surf ace olution approaches could pro vide alternati scale-space computation that might be less restric- ti e. B. Mean cur atur estimation Dif ferent approaches can be used to describe and study ﬁne details of the cortical surf ace folding

patterns. Depth maxima ha been used to detect concept similar to sulcal roots in [35]. In this paper mean curv ature is proposed as richer descriptor (than the depth) of the arious features that can be observ ed along sulcus bottoms and alls, which is illustrated in ﬁgure 8: fold bottoms appear as local minima of while gyrus cro wns appear as local maxima. Hence, urried gyri sep- arating tw sulcal oots appear as areas of positi curv ature along the sulcus alls. Other curv ature related features, such as oenderink curv ature metric (the L2 norm of the principal curv atures, or the

logarithm thereof) or the maximum principal curv ature, may be interesting for our purpose and should be in- estigated in the future. It should be noted that an isophote mean curv ature related measure (Lvv) has also been proposed to dis- tinguish sulci from gyri [48]. While this approach could be used to get mean curv ature estimation related to our cortical surf ace representation (which may be considered as one gi en isophote), its main interest is in the deﬁnition of gyrus and sulcus sk eletons as surf aces of singularities [74]. In this paper mean curv ature is directly estimated

from the mesh thanks to its relati smoothness. used an approxima- Fig. 3. Appr oximation of the mean curvatur fr om an irr gular mesh [60]. and denote espectively the triangle angles and ar eas; corr espond to the dihedr al angles between the normals the edg lengths ar noted tion proposed in [60], [61] that tak es into account some local properties of the mesh, as triangle angles and areas di- hedral angles between normals and edge lengths (see Fig. 3). This method may be considered as less rob ust than the usual quadratic patch based approaches, ut as chosen in

this paper for its lo wer computational urden. It should be noted that it is too soon to claim that one kind of curv ature based map is more adapted to our purpose than another one. C. etus and baby brains The pre vious chain of processing is used when the brain has reached high le el of myelinisation of axons, namely for more than tw years old brains. At this stage, indeed, standard T1- weighted images based on in ersion reco ery sequences gi good contrast between gray and white matter leading to an ac- curate deﬁnition of the cortical inner surf ace. Since the most interesting part of

the folding process is occuring during antena- tal stage, ho we er we ha initiated research program aiming at performing longitudinal studies from antenatal MR images obtained from clinical studies. In the case of antenatal and small baby brains, ho we er the axon myelinisation is still in progress, which means that T1- weighted images sho more or less contrast between gray and white matters according to the brain areas. Therefore, we ha adapted the pre vious chain of processing to T2-weighted im- ages, which pro vide better contrast (see Fig. 2) [62]. Unfor tunatelly T2-weighted images

usually ha lar ger slice thick- ness, especially with antenatal imaging, where acquisitions ha to be ery ast because of the fetus frequent motions and for his mother comfort. Since the fetus brains are ery small, this is leading to partial olume problems in the deﬁnition of the cor tical inner surf ace. Hence, we ha chosen to study the corte outer surf ace, which is located between the cerebro spinal ﬂuid and the brain tissues. It should be noted, ho we er that the struc- tural study of the folding process can be done from the corte outer surf ace as from the inner surf ace, as

ar as the surf ace rep- resentation is reliable. The current se gmentation toolbox is semi-automatic and is still ar to yield perfect results. An yw ay designing perfect surf ace detection method is challenge because of the frequent artef acts induced by fetus motions and the arious contrast mod- iﬁcations occuring during the myelinisation. Ne ertheless, for fe fetus brains, the detected surf aces are suf ﬁciently clean to visualize small dimples bound to become cortical folds as con- nected components of ne gati curv ature (see Fig. 2). These images allo us to question the

ﬁrst sulcal root maps, which had been inferred from arious descriptions of the literature (see

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 Fig. 1) [25]. Unfortunately the arious weaknesses of the cur rent acquisition process lead to an wful percentage of success: one acquisition er one hundred can pro vide meaningful cor tical surf ace representation. Hence, we are currently orking on aster acquisition schemes using ne MR sequences [56] and multi-coil approaches [57], because the main problem to be solv ed is motion between slices. In the future, longitudinal

acquisitions at se eral time steps could allo to follo the fold- ing process subject by subject, pro vided that such studies are ethically acceptable. The curv ature map of the cortical surf ace contains much ge- ometrical information that may be related to the anatomical elements that ha to be detected (sulcal roots, urried gyri). These elements, ho we er correspond to dif ferent le els of scale (see Fig. 9). Moreo er scale based point of vie is re- quired to distinguish anatomical elements from noise features bound to appear in curv ature approximations, due to se gmenta- tion/triangulation

artif acts, or biased estimations of the discrete curv ature (see Fig. 8). The scale-space paradigm has been de- eloped to deal with such problems where all the scales may be of interest. A. Geodesic diffusion of the Cur atur map alternati approaches can be used to create amily of curv ature maps olving to ards smoothness, either dif fusion process geodesic to the cortical surf ace [63], [64] or geometric olution of the surf ace itself [58]. Surf ace olution according to function of curv ature has been widely used in image analy- sis to describe 3D shapes. The standard implementation of this

kind of olution using the le el set frame ork, ho we er is not adapted to our goal. This frame ork, indeed, allo ws topological modiﬁcations of the track ed isosurf ace, and pro vides no con- stant parameterization that may be used to simply track objects across scales. mesh based implementation, in return, may be used for the olution process, ut ould not necessarily be pro viding the causality property required to deal with the scale space [39], [40]. At each iteration, indeed, some curv ature esti- mation errors ould be made. It should be noted, ho we er that such an implementation is

used to inﬂate the cortical surf ace for visualization purpose [59]. In the follo wing, the scale space of the curv ature map is com- puted from the heat equation [40] geodesically to the cortical surf ace. This is an arbitrary choice made to xperiment with the anatomical structural ideas mentioned abo e. fe xper iments using inﬂated ersions of the initial surf ace to compute the geodesic dif fusion ha sho wn fe consequences on the sul- cal structures of interest in the scale space. Some more studies ha to be done, ho we er to get better idea of the inﬂuence of the

intrinsic curv ature of the corte on the dif fusion process [65]. In the future, the beha vior of alternati anisotropic dif- fusion schemes [66] could also be considered. or instance, further ork could consist in looking for the dif fusion scheme maximizing the similariry across subjects of the structural rep- resentations inferred from the indi vidual scale spaces. The heat equation, which corresponds to parabolic artial Dif ferential Equation (PDE), lends itself with relati ease to being adapted for the speciﬁc case of an irre gular 2D lattice em- bedded in 3D space [63], [64]. B.

Numerical implementation The numerical implementation of the heat equation is carried out as an iterati process of the form: ! #"$% !& ! !& $% ')( +, !& (1) for each node and each temporal iteration step $% where is the Laplacian estimate at the node of the ﬁeld of alues (i.e. the map of the curv ature). The implementation of partial dif ferential equations on ir re gular lattices can lead to comple numerical problems; the causality property usually required by the scale-space frame- ork may be lost because of discrete phenomena. This point is be yong the scope of

this paper and ould require further study B.1 Spatial parameterization The act that the cortical lattice is embedded into olume raises question concerning the proper axis system upon which to base the estimation of the local partial deri ati es. conﬂict xists between the need to emplo spatial coordinates in the Euclidean space to obtain an ambiguity-free description of the position of each node, and the wish to perform strictly surf ace- based smoothing. The deﬁnition of coordinate system intrinsic to the surf ace ould allo for strictly 2D-based smoothing. This implies

parameterization, i.e. the deﬁnition of mapping function such that /.0!12435&768-9;:<!=>& and being the ne coordinates that allo to refer to ery point in the surf ace without ambiguity The dif fusion equation is then solv ed along and in strictly 2-dimensional ashion. possible parameterization ould consist in the application of ﬂattening procedure to the cortical lattice representation [59]. This ay the ﬂattened corte ould be contained in plane, and the parameterized coordinates /:?,=& ould corre- spond simply to the ;.2@,1>@A& coordinates of the plane in

question. Ho we er cortical ﬂattening leads to signiﬁcant amount of met- ric distortion (10-20 in erage, locally attaining much higher alues [59], [6]) and requires the introduction of cuts because of the corte spherical topology Using the mapping to sphere ould be an alternati solution [8], ut spherical coordinates include tw poles leading to some other dif ﬁculties. B.2 Local planar parameterization In vie of this, the adopted parameterization as simple local transformation that maps each surf ace element (a node and its ﬁrst neighbours) into plane, while eeping

unchanged both the edge distances and the angular proportions between the edges. This neighborhood parameterization [69], [70] amounts to locally ﬂatten the surf ace element, and oids the se ere areal distortion that ould result from an xplicit global ﬂattening. An indi vidual mapping function -7 is thus deﬁned for each node independently of the surrounding surf ace elements. An arbitrarily-oriented orthogonal referential is deﬁned for each surf ace element, centered at its central node, and all the required

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IEEE TRANSA CTIONS ON MEDICAL IMA GING

2002 estimations, notably local Laplacian, are based on this ne co- ordinate system. This local-based approach is made possible since, for each iteration step of the numerical solution of the dif fusion equation, the estimations in olv only the dif ferences between the alues associated with each node and its nearest neighbors. B.3 Estimation of the Laplacian operator The adopted (ﬁnite dif ference) approach [71], [72] assumes that the ﬁeld, implicitly described by the function +/:?,=& is no sampled with an irre gular planar lattice composed of in- terconnected nodes. aylor

series xpansion around gi en point :DC =EC has the form: F "HGJI "HKE I GL L KL LJ "NKOG PC "QR$TS & (2) where PC +/:2CJ!=ECU& ;: != V WYX V Z and V WYX V [ de- note the partial deri ati of +/:?,=& at point ;:0CJ,=ECU& :DC =EC and '8\ "K Writing 2) for surf ace element consisting of central lattice node, located at ;:DC!=ECU& and its neighbours ;: ,= &0B '^] J_A_`_ leads to acb 'ed f g where G2k Kk l m G2kpKk _`_A_q_A_A_ _A_`_ _A_`_ Gsr t u '^d wk 4 J_A_`_A, gyx and the deri ati es at are '8z PC EI PC OI PC OI PC OI PC =?{ In this ashion,

estimates of the tw o-dimensional Laplacian ;:<!=>& V Z V [ are obtained at each node of the lat- tice. This approach amounts to solving, for each node, linear system in olving the relati positions of the node and its neigh- bours and the corresponding ﬁeld alue dif ferences. It is still ﬁnite dif ferences method, in the sense that the partial deri ati es are estimated by dif ferences between ﬁeld alues in neighbour ing points, ut its range of application is xtended to an ar bitrarily irre gular grid. The resolution of the system, for each node, is done in

least-squares ashion a}| (3) This system of equations is solv ed once for eac node at the be ginning of the dif fusion process. Practically the estimation of the Laplacian operator at each mesh node entails the multiplication of the pseudo-in erse esti- mation of the matrix by the ector containing the dif fer ential data :2C!: :2CJJ_A_A_ :2r :DC ), with denoting the number of neighbors [71] R PC PC ` ‚;:2 (4) with ,where denotes the element on the rd line and th column of the pseudoin erse matrix. primal sk etch is constructed from the curv ature map using the algorithm

proposed by Lindeber in [42]. This primal sk etch is xpected to xhausti ely describe the structure of the scale space of the curv ature map, and therefore to pinpoint its rele ant embedded objects. These objects are alle ys of the curv ature landscape xisting during range of scales. In the follo wing, we present brieﬂy the main steps of the primal sk etch construction. A. The gr ey-le el blobs At each le el of scale (i.e. dif fusion time ), some 2D objects called gr y-le vel blobs (GLB) are xtracted from the smoothed curv ature map. Each GLB is bassin, which may represent cortical fold

at this le el of scale. One GLB is deﬁned for each local minimum. The GLB spatial xtent (a set of lattice nodes) is deﬁned from ater rise lik algorithm. The ater pouring from each minimum ﬁlls bassin, which altitude is deﬁned by the lo west surrounding saddle point (see Fig. 4). In practice, each minimum is gi en dif ferent label that marks gro wing area. The gro wing is performed altitude by altitude, follo wing the ater rise idea. An area be gins to gro when it is reached by the ater When tw gro wing areas with dif ferent labels get in touch, their gro wing is

stopped. The surrounding areas, ho w- er are mark ed by background label and the background fol- lo ws the ater rise. When gro wing area gets in touch with the background area, its gro wing is stopped too. When the highest altitude has been reached, the lattice is made up of GLB supports and background area. In ersing the rise of ater could allo the deﬁnition of some GLBs for each local maximum, which could also be of interest to study the gyral patterns. Local minimum Saddle point I(X,Y) support Background point Background Fig. 4. Deﬁnition of gr y-le vel blobs fr om local

minima and saddle points (2D case) B. The scale-space blobs Each GLB is deﬁned by tw xtremal points, local mini- mum and saddle point, whose beha viors in the scale-space are well kno wn from theoretical point of vie The GLBs appear

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 or disappear according to possible ents called bifur cations 1/ cr eation (a GLB appears) 2/ annihilation (a GLB disappears) 3/ mer (tw GLBs mer ge into one) 4/ split (one GLB splits into tw GLBs) (see ﬁg. 5). In scale-space satisfying the causality property the cr eation relation is

theoretically possible only at the ﬁrst le el of scale. Between tw of these ents, it is pos- sible, with spatial erlap criterion, to track GLB from one scale to slightly coarser or ﬁner one. The chains of GLBs link- ing tw bifurcations (see Fig. 6) deﬁne multiscale objects called scale space blobs (SSB), which are xpected to correspond to some anatomical structures embedded in the curv ature map. s+ds a) b) c) d) Fig. 5. The basic matc hing elations between GLBs thr ough the scales (fr om left to right): a) plain link b) annihilation c) mer d) cr eation e) split. The GLBs

ar connected in SSB with plain links In scale-space satisﬁying the causality pr operty the creation elation is theor etically possible only at the ﬁr st le vel of scale Fig. 6. Deﬁnition of the Scale-Space Blobs fr om the Gr y-Le vel Blobs. GLBs be- longing to the same hain can be seen as dif fer ent instances of same SSB. The or ganization of the primal sk etc in the space and thr ough the scale yields an xplicit structur al multi-scale description of the sulcal topo gr aphy(fr om Coulon et al.[67]). a) b) c) Fig. 7. Iter ative construction of the scale space Example of

cases equiring adaptive sampling C. Adapti sampling of the scale space Due to the sampling of the scale parameter situations dif fer ent from the described in Fig. may occur (see Fig. 7). In this case, an adapti sampling pro vides an intermediate scale le el at which this problematic situation can be ercome. Ev ents in the linear scale space ha been sho wn to arise loga- rithmically the computed intermediate scale between and eriﬁes then: ; !& '…„ †‡ ˆ nŠ‰;‹ „ †‡`ˆ i.e. 'Œ Ž limit in the number of successi

reﬁnements must be ﬁx ed, and it may happen that this number is not suf ﬁcient to solv particular problematic situation. Ho we er reaching the limit in the number of successi reﬁnements is not of great importance since the missed SSBs ha an xtremely short lifetime and can then be considered as not signiﬁcant [42]. A. The central sulcus primal sk etch The Figure 10 pro vides glimpse on the primal sk etch fo- cused on the central sulcus of an adult brain. The structure of this sub-sk etch is consistent with our initial aims. First, the three highest

scale-space blobs are link ed by an ent which seems to correspond to the mer ge of the central sulcus sulcal roots de- scribed by neuroanatomists [21]. Second, the spatial localiza- tion of the tw sulcal roots related blobs are separated by ur ried gyrus, re ealed by slight deformation of the central sul- cus all, as described by the model. Moreo er the tw sulcal roots ha longer life time throughout scales than noise related blobs. ﬁne analysis of the lo wer part of the sk etch (the superior sulcal root), ho we er sho ws that some instabilities may stem from spurious split ents induced

by the elongated shape of the sulcus related blobs. These splits, ho we er are not necessar ily spurious and may be related to what we call the castle all ef fect. Let us imagine landscape with castle all shape, namely all with tw thick to wers as xtremities. If the alti- tude of the all is higher than the to wer altitudes at the be gin- ning of the smoothing process (a weird castle indeed), the whole ediﬁce is represented by only one blob Because of their thick- ness, ho we er the to wer altitudes decrease at lo wer rate than the all altitude during smoothing. Hence, after while, the

ini- tial blob is split into tw to wer related blobs. In should be noted that this kind of ents respects the causality property which pre ents only the creation of ne le el sets. When such ents, ho we er occur because of noise induced by some weaknesses of the numerical scheme relati to the irre gularity of the mesh, the primal sk etch may need some postprocessing. An interesting alternati e, based on Finite Element Method (FEM), oiding local planar parameterization and in ersion of ill conditioned matrices, can be found in [64], and is under in estigation. It may ercome some of these

problems. B. Stability elati to cur atur based featur Se eral choices made to perform the studies described in this paper are arbitrary (mean curv ature and heat equation). hy- pothesize, ho we er that the structural anatomical information we are interested in is stable across arious curv ature based

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 features. In order to illustrate this idea, the central sulcus pri- mal sk etch computed from our mean curv ature estimation has been compared to an equi alent sk etch computed from map of the distance between each node and its mesh

neighbor hood barycenter '‘ signed by the scalar product be- tween ’ and the normal at node (see Fig. 11). While some ariability could be observ ed among the objects xtracted at the lo west le els of scale, the highest part of the sk etches rely on isomorphic sets of bifurcation point trajectories deﬁning the same candidates for the central sulcus roots. C. Repr oducibility acr oss indi viduals The primal sk etch has been uilt from 10 dif ferent left hemi- spheres in order to check if the tw sulcal root model of the central sulcus could be highlighted as an in ariant of

the corte morphogenesis. The supports of the tw gre le el blobs sup- posed to correspond to the tw putati sulcal roots are mapped on an inﬂated ersion of the surf ace, for the sak of visualization (see Fig. 13). or each brain, the scale has been chosen slightly before the ent leading to the mer ge of the tw sulcal roots into the whole central sulcus. In each case, the tw gre le el blobs are located on both sides of the central sulcus all defor mations looking lik urried gyrus. It should be noted that the spatial xtent of the gre le el blobs is not supposed to ha an accurate localization

po wer Dedicated urried gyrus detection algorithms initialized by the localization of the gre le el blob minima could indeed pro vide better result for further studies. The goal of the primal sk etch is mainly structural: highlighting the fetal structure of the adult folding patterns. D. Discussion The pre vious results ha sho wn that some hidden informa- tion about the morphogenesis of the cortical folding patterns could be reco ered from the remaining curv ature of the surf ace at adult age. This information could help to ercome the prob- lems related to the inter -indi vidual ariability of

these patterns. Our results, ho we er are related to one of the simplest sulci, and more ork is required to analyze whether the current primal sk etch structure is suf ﬁcient to highlight stable or ganisation of the sulci made up of more sulcal roots. or instance, it has to be pro en that the primal sk etch scale space blob set includes all the sulcal roots that ha to be localized. The link between fetal sulcal roots and their putati primal sk etch analogue, indeed, may be more intricate than for the central sulcus case, because se eral mer ge ents are then in olv ed. This could call

for the de elopment of more sophisticated anisotropic scale spaces. While ﬁrst manual xploration is required to try to match the primal sk etch based representations with our current sulcal root maps (see Fig. 12), an automatic strate gy should be de- vised to get more reliable generic model. Fe approaches ha been proposed for such inference of high le el models un- derlying the brain anatomy Some ideas could stem from similar ork done from skull crest lines [73]. Another attracti direc- tion consists of Mark vian models for the comparison of primal sk etches de eloped to match acti

ation maps across indi viduals [67]. The underlying idea consists in labelling simultaneously lar ge number of sk etches, each label corresponding to an entity relati ely stable across indi viduals in terms of shape, localiza- tion and surrounding. Finally when such sulcal root based generic model will ha been inferred from set of brains, auto- matic labelling methods might be used to match it with an ne primal sk etch [35], [74]. It should be noted, besides, that the group analysis of geodesic scale-spaces of 2D maps painted on the cortical surf ace is generic tool, which may be of interest

in other conte xts such as in fMRI/PET studies with statistical maps [63], [67], or for the study of cortical thickness maps [68]. In this paper we ha sho wn that ne approaches to the un- derstanding of the ariability of brain structure can be de vised taking into account the ariability of the brain morphogenesis. Such approaches could ercome the dif ﬁculties of iconic tem- plate based methods to deal with structurally dif ferent patterns. The ne structural multi-scale based representation of the sul- cal folding patterns presented in this paper will be used to infer ﬁner grained

than usual generic model of the human cortical sur ace [25], [21]. Such model ould greatly impro our current understanding of the corte ariability and help for ﬁnding sta- ble anatomical landmarks. These landmarks ould be usefull, for instance, emplo yed as geometrical deformation constraints in arping processes [43], [44], [45], [46], or for deﬁning re- liable parcellations of the cortical surf ace [75]. In the future, longitudinal time series of brain images will be of great help to alidate our approach. Hence, long-term studies of brain gro wth processes will be used both as

answers to neuroscience ques- tions and as inspiration for ne methodological de elopment. A. Stability of the numerical scheme The numerical resolution of the pre vious artial Dif ferential Equation (PDE) must fulﬁll con ver ence and stability criteria to gi satisf actory results. ithout loss of generality let us consider one-dimensional heat dif fusion-lik (parabolic) PDE: :</.0! !& '”“ :;.?! !& (5) is time, the linear coordinate, and constant.) Assum- ing that the temperature ﬁeld (in our case the mean curv ature ﬁeld) is sampled by re gularly-spaced grid

(with spatial sam- pling step ), ﬁnite dif ferences approach to approximate the partial deri ati es yields /. , (6) "+: .2& ;. ! (7) where is the temporal sampling step, and stands for :;. , i.e. the alue of the function at the node and at the instant Using this, approximate alues of the function (as opposed to the xact alues can be computed. Substituting for in Eqs. and 7, we ha "—D˜– "H– (8)

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 where 'š™œ› ›Y highly important concept in this conte xt is the truncation err

or (so called because truncation of aylor series is implied by approximations such as the ones abo described) [76], [77]. The truncation error is simply the remainder of Eq. when is replaced by the xact solution The numerical resolution of the PDE will be said consistent if the truncation error tends to zero as and tend to zero. In other ords, consistenc ensures that ﬁner meshes and smaller iteration time steps will al ays lead to more accurate approximations of the actual PDE solutions at each time step and at each node. While ob viously ery desirable feature, consistenc does not

guarantee that the resolution will be stable. or stability to be attained, the actual error must tend to zero as the truncation error tends to zero, otherwise the errors will tend to propagate rapidly and the ﬁnal results will be de oid of sense. The olution of in time is related to the truncation error in the follo wing ay [76]: —&!Ÿ "H—>Ÿ "H—>Ÿ (9) The important point is that the coef ﬁcients of the terms on the right add up to unity and that, pro vided that —7 the are all positi e. This ay the error at time step is bounded by -5.#

˜ 'f J_A_`_` in the follo wing ay: (10) This is suf ﬁcient condition for stability [76]. Thus, alid numerical implementation of this simple parabolic equa- tion ould require that '“ ›Y The present case dif fers from the xample abo in tw points: 1) the (2-dimensional) mesh is not re gular (nonuniform internodal separation); 2) the estimation of the partial deri a- ti es, although based on ﬁnite dif ferences, relies upon the least- squares resolution of linear system. These speciﬁcities must be accounted for in discussion of the appropriate stability

guide- lines. The PDE to solv at each node location has the form :</.0,1, !& 'f sJI :;.?!12! !& "eI :;.0,12! !& (11) The approximate result satisﬁes (12) where m42 stands for estimated partial deri ati of at node !1 and time point (similarly for ). The error can be written as &#"ž (13) Combining Eqs. 13 and leads to A?k ˜ ˜ &˜ A?k ˜ (14) where (resp. ˜ stand for the element contained in the 3rd (resp. 4th) line and th column of the pseudo-in erse matrix pertaining to the node located at .s5!1%& ,

and stands for the error at the th neighbour of the node located at .E,1% & at instant . suf ﬁcient condition for the error to be bounded is that the coef ﬁcients af fecting all of the add up to unity (which is clearly the case) and that the are all positi e. Denot- ing ˜ this translates as acb 4 (15) From practical point of vie these constraints apply to the elements of the matrix. High elements in those lines of the matrix that play role in the estimation of the second deri a- ti es may lead to the violation of the left-hand condition in Eq. 15, and

thus to instability While the analogy will not be pur sued further it is note orthy that this constraint can be seen as putting limits to the ariance of the implied least-squares estima- tion [78]: high pseudoin erse alues denote linear systems with unacceptably high condition numbers (nearly colinear lines in the corresponding matrix), and therefore high estimation ari- ance, liable to cause important error propagation in an iterati process. The goal, thus, is to minimize estimation ariance in order to oid ha ving to emplo ery small that could ultimately re- quire prohibiti ely high

computation times. ith this purpose, simple condition number minimising algorithm as imple- mented at the lattice creation stage. It relies on the act that the orientation of the referential used to deﬁne relati neigh- bours coordinates is arbitrary Minimization of the condition number is performed by successi ely rotating the referential of the nodes that ail to comply with user -pro vided threshold. In cases where tw nodes are ery close to each other this proce- dure may not be enough to guarantee reasonably lo alue for the critical In such cases, an interv ention at the le el of the

lattice conﬁguration may be required. or typical xamples of cortical lattices submitted to node decimation (resulting number of nodes: 20,000-25,000 ), the fulﬁllment of the left-hand condition in Eq. 15 required of 0.5-0.7. Ho we er practice re ealed that as high as led to stable systems, sho wing that the error can be bounded en in cases where the suf ﬁcient condition for stability is violated. The right-hand condition in Eq. 15 is apparently more dif ﬁ- cult to fulﬁll. Ho we er the se eral violations that were identi- ﬁed in the tested lattices were

not an obstacle to stable numer ical resolution, sho wing that the fulﬁllment of the ﬁrst condition is in itself enough to guarantee satisf actory bounding of the error The application of postprocessing procedures (e.g. decima- tion) to the lattice may lead to situations in which node neigh-

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 bours are fe wer than the number of ariables estimated locally This number will be in the case of the 2nd order aylor x- pansion necessary to estimate the Laplacian, or if isotrop is assumed and the m,! V term is considered to

be zero, as in the current implementation. deﬁcit of neighbours will lead to in- determinac of the corresponding linear system, which means that the solution will not be unique. One of the options to pick one among the inﬁnity of solutions thus obtained is to select the solution with the smallest norm. This is the condition implied by the Moore-Penrose pseudoin erse [79] based resolution that we adopted. This as sho wn to be sensible option in practice: pro vided that the nodes with less than neighbours constitute small %) proportion of the total number of nodes, the ﬁnal

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apadopoulos-Orf anos, I. Bloch, and J. egis, Gyral parcellation of the cortical surf ace using geodesic oronoi diagrams, in MICCAI 02, ok yo LNCS-2488, Springer erlag, pp. 427434, 2002. [76] K.W Morton and D.F Mayers, Numerical Solution of artial Dif fer ential Equations Cambridge Uni ersity Press, 1994. [77] Gran ville Se well, The Numerical Solution of Or dinary and artial Dif fer ential Equations Academic Press, 1988. [78] Ronald Christensen, Plane Answer to Comple Questions The Theory of Linear Models Springer 1996. [79] James R. Schott, Matrix Analysis for Statistics John ile Sons,

1997.

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 11 Fig. 8. Mean curvatur of the corte inner surface (adult br ain), mapped on itself (left) and on an inﬂated ver sion [59] (right). Red (ne gative) ar eas corr e- spond to sulci, while blue (positive) ar eas corr espond to gyri. Fig. 9. Some isophotes of cortical surface mean curvatur at dif fer ent scales, mapped on itself (up) and on an inﬂated ver sion (down). Centr al sulcus includes two curvatur minima at middle scale and ﬁnally only one minima at highest scale The middle scale minima will corr

espond to two blobs in the ﬁnal primal sk etc h. The saddle point whic separ ates these two blobs is located at the le vel of the uried gyrus elated clues. Hence these blobs may corr espond to centr al sulcus sulcal oots. Noise? Noise? Noise? Central sulcus sulcal root Inferior sulcal root Superior Noise? Scale Fig. 10. Sub-sk etc of the primal sk etc focused on the centr al sulcus ar ea. Eac gr le vel blob is epr esented by its contour The contour is mo ved towar ds the outside br ain by an homothetic factor elated to the lo garithm of the scale Eac scale-space blob has its own color

Red points corr espond to the curvatur minima fr om whic the gr le vel blob gr owth be gins. Purple points between two blobs and gr een points between one blob and the bac kgr ound, corr espond to points stopping blob gr owth. The position and the or ganisation of the SSBs located in the centr al sulcus zone matc with the sulcal oot based model of this sulcus. buried gyrus buried gyrus buried gyrus buried gyrus buried gyrus Signed distance to neighbors barycenter Mean curvature SR1 SR2 SR1 SR2 Central Sulcus Central Sulcus Fig. 11. Up: two dif fer ent curvatur based maps: left, the mean

curvatur estimation used thr oughout the paper; right: the signed distance to the barycen- ter of the neighbor in the surface mesh. Middle: The gr le vel blobs at one le vel of scale located befor the mer of the centr al sulcus oots. Do wn: The structur of the upper parts of the centr al sulcus primal sk etc hes ar isomorphic, while the bifur cations occur at slightly dif fer ent times. SR1 SR2 SR3 SR4 SR1 SR2 SR3 SR4 SR1 SR3 SR4 SR2 Fig. 12. Left: The part of the primal sk etc corr esponding to the left supe- rior tempor al sulcus for thr ee dif fer ent br ains. The putative four sulcal oots

of our model have been labelled in these gr aphs (see ig 1). The structur of the sub-sk etc hes is slightly dif fer ent acr oss individuals, whic may eﬂect some dif fer ences occuring during the folding pr ocess. It should be noted that the sub-sk etc hes ar not necessarily connected tr ees, and that the superior tempor al sulcus is not necessarily epr esented by scale space blob Right: The corti- cal surface of the thr ee corr esponding br ains. The positive mean curvatur has been mapped with ed color map in or der to highlight putative gyri uried in the sulcus walls. The localizations

of the curvatur minima corr esponding to the putative sulcal oots have been superimposed on the 3D endering (r ed and gr een ectangles).

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IEEE TRANSA CTIONS ON MEDICAL IMA GING 2002 12 Adults Children s=80.64(1.91) s=56.95(1.76) s=47.85(1.68) s=114,20(1.91) s=47.85(1.68) s=67.77(1.68) s=40.21(1.60) s=56.95(1.76) 1.5 years old 6 years old s=80.64(1.91) s=40.21(1.60) Fig. 13. ariability of the centr al sulcus folding pattern among eight adult (up) and two hild br ains (down). or eac subject, mesh of the cortec inner surface mapped with its mean curvatur highlights the deep uried

sulcal shape Pinpointed blobs, mapped on slightly inﬂated ver sion of the surface ar supposed to corr espond to the two putative centr al sulcus sulcal oots (inferior and superior); the indicated scale (with its lo garithm in br ac ets) corr espond to their scale of apparition.

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