Inferring brain variability from di�eomorphic deformations of

currents: an integrative approach

Stanley Durrleman a,b Xavier Pennec a Alain Trouvé b Paul Thompson c Nicholas Ayache a

aAsclepios team project, INRIA Sophia Antipolis Méditerranée, 2004 route des Lucioles, 06902 Sophia Antipolis cedex, FrancebCentre de Mathématique et Leurs Applications, ENS Cachan, 61 avenue du président Wilson, 94235 Cachan cedex, France

cLaboratory of NeuroImaging, Dept of Neurology, UCLA School of Medicine, 225E Neuroscience Research Building, Los

Angeles, CA, USA

Abstract

In the context of computational anatomy, one aims at understanding and modelling the anatomy of the brain andits variations across a population. This geometrical variability is often measured from precisely de�ned anatomicallandmarks such as sulcal lines or meshes of brain structures. This requires (1) to compare geometrical objects withoutintroducing too many non realistic priors and (2) to retrieve the variability of the whole brain from the variability ofthe landmarks.We propose, in this paper, to infer a statistical brain model from the consistent integration of variability of sulcal

lines. The similarity between two sets of lines is measured by a distance on currents that does not assume any typeof point correspondences and it is not sensitive to the sampling of lines. This shape similarity measure is used in adi�eomorphic registrations which retrieves a single deformation of the whole 3D space. This di�eomorphism integratesthe variability of all lines in a as spatially consistent manner as possible.Based on repeated pairwise registrations on a large database, we learn how the mean anatomy varies in a population

by computing statistics on di�eomorphisms. Whereas usual methods lead to descriptive measures of variability, suchas variability maps or statistical tests, our model is generative: we can simulate new observations according to thelearned probability law on deformations. In practice, this variability captured by the model is synthesized in theprincipal modes of deformations. As a deformation is dense, we can also apply it to other anatomical structures de�nedin the template space. This is illustrated the action of the principal modes of deformations to a mean cortical surface.Eventually, our current-based di�eomorphic registration (CDR) approach is carefully compared to a pointwise line

correspondences (PLC) method. Variability measures are computed with both methods on the same dataset of sulcallines. The results suggest that we retrieve more variability with CDR than with PLC, especially in the direction of thelines. Other di�erences also appear which highlight the di�erent methodological assumptions each method is based on.

Key words: computational anatomy, brain variability, sulcal lines, shape statistics, non-linear registration, currents, largedeformations, di�eomorphisms

1. Introduction

From the ever growing databases of medical im-ages, there is considerable interest in extracting themost relevant information to characterize normalanatomical variability within a group of subjects aswell as between di�erent groups, to detect anatom-

ical abnormalities, to classify new images accord-ing to their pathologies, and for understanding dis-ease progression. However, modeling the individualanatomy and its normal variability across a popu-lation is di�cult as there are commonly no phys-ical models for comparing di�erent subjects, andanatomical shapes are complex and require large

Preprint submitted to Elsevier June 17, 2008

number of degrees of freedom to model adequately.Moreover, anatomical landmarks such as curves orsurfaces embedded in R3 as well as deformations ofthe 3D space do not belong to usual vector spaces.De�ning statistical models is therefore di�cult andspeci�c tools have to be developed to accuratelymeasure anatomical variations. If anatomic varia-tion were better understood, tools encoding thesevariations could have a signi�cant impact in neuro-science to minimize the in�uence of the anatomicalvariability in functional group analysis, and in clin-ical medical image analysis to better drive the per-sonalization of genericmodels of the anatomy (calledalso template, atlas or prototype in the literature).Instead of analyzing the anatomical variability di-

rectly in the 3D intensity space, it is often prefer-able to extract precisely de�ned anatomical land-marks such as sulcal lines (Thompson et al., 1996a;Mangin et al., 2004), cortical surface models (Fis-chl et al., 2001; Tosun and Prince, 2005), or mod-els of some sub-cortical structures (Hazlett et al.,2005; Vaillant et al., 2007). The data to be analyzedare thus curves, surfaces or volumes represented bystructured or unstructured point sets. The �rst at-tempts to compare such shapes was based on de�n-ing correspondences between points (Zhang, 1994;Chui and Rangarajan, 2003). However, the samplingof two di�erent geometric subjects can be so dif-ferent that such a correspondence assumption in-troduces a bias that often hides the �real� under-lying geometrical variability. To overcome this dif-�culty, some authors proposed to measure varia-tions of some features extracted a priori such aslength, area, volume, complexity, principal modesof variation of the cloud of sampled points, etc. SeePaus et al. (2001) for instance. Although these ap-proaches, that derive scalar measures from structuremodels, are relatively easy to set up from a computa-tional point of view, they fail to capture �ne geomet-rical variations between subjects like for instance atwisting of the extremity of a sulcus, which cannotbe readily described by a set of a priori selectedfeatures. Also, data analysis often proceed by com-puting dense displacement �elds that encode vari-ations in shape and volume among individuals, of-ten based on deformable registration of shapes. Thedeformation that maps one shape onto another hasbeen proved to be useful for measuring signi�cantanatomical variability among di�erent subjects (Fil-lard et al., 2007a; Vaillant et al., 2007; Ashburnerand al, 1998; Durrleman et al., 2007). Due to thepresence of noise and of sampling e�ects, it may be

advantageous to allow a trade o� between the reg-ularity of the deformation and the precision of thematching, instead of exact matching. This raises theneed to develop a consistent deformation frameworkand a shape similarity measure that does not relyon point correspondences nor on features selected a

priori.In this perspective, one interesting framework

consists of modeling geometrical primitives as cur-rents (Vaillant and Glaunès, 2005; Vaillant et al.,2007; Durrleman et al., 2007). The idea is to char-acterize shapes via vector �elds, which are used toprobe them. For instance, a surface is character-ized by the �ux of any vector �eld through it, aline by the path integral of any vector �eld alongit. Conversely, associating a �ux to any vector �eldspeci�es an object which is more general than a sur-face or a curve and which is called a current. Thisway of embedding shapes in a Hilbert space allowsone to de�ne algebraic operations such as additionor averaging, and to measure distance between geo-metrical primitives via an inner product that doesnot assume a speci�c type of point correspondence.Discrete and continuous objects are handled in thesame setting, o�ering a way to measure the sam-pling quality and to guarantee numerical stability.This framework has been used to compute andvisualize mean lines and surfaces, and to performprincipal component analysis on datasets of suchprimitives, suggesting the e�ciency and generalityof the approach (Durrleman et al., 2008).However this similarity measure is too weak to

capture the broad range of possible di�erences be-tween shapes: it is bene�cial to couple it with adeformation framework. The large deformation dif-feomorphic metric mapping (LDDMM) framework(Trouvé, 1998; Grenander and Miller, 1998; Dupuiset al., 1998; Miller et al., 2002, 2006) is ideal forthis task as shown in Vaillant and Glaunès (2005)and Glaunès and Joshi (2006) although it might bepossible to adapt other di�eomorphism registrationsmethod proposed for images (e.g. Ashburner andFriston (2003); Avants et al. (2006)). The deforma-tion that matches a pair of shapes is sought withina group of regular di�eomorphisms in order to op-timize a trade-o� between the regularity of the de-formation and matching accuracy, as measured bythe dissimilarity measure on currents (Vaillant andGlaunès, 2005; Durrleman et al., 2007). As a result,the registration decomposes the di�erences betweentwo shapes into (1) a deformation that captures a�global� misalignment and (2) a residual term (rep-

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resenting the di�erence between the registered shapeand the target shape) that contains possible non-di�eomorphic variations as well as physical and nu-merical noise. In the approach followed here, we per-form our statistical analysis of shape on the defor-mation term only. Our results on a dataset of sul-cal lines show that this method can be used to de-tect and characterize anatomical variability withina group of subjects.Moreover, this di�eomorphic framework enables

to register multiple objects in a spatially consistentway. Indeed, a dataset of anatomical landmarks of-ten consists of a set of several shapes (e.g. a set of sul-cal lines or set of meshes representing di�erent sub-cortical structures for instance (Mangin et al., 2004;Duchesnay et al., 2007; Gorczowski et al., 2007). Ifone set of manifolds, such as a distributed set of sul-cal landmark curves, is embedded in another man-ifold which also varies, such as the cortical surface,one often aims to measure the variability not onlyof the embedded landmarks but also of the wholeunderlying brain surface or 3D brain volume. Theframework based on currents enables precisely to de-�ne a distance between multiple objects sets even ifthey are not labeled or if all subjects have not thesame number of objects. (In these cases the distancewill just be less precise than for labeled objects).The di�eomorphic framework in turn �nds a singledeformation of the underlying image domain thatintegrates the variability of all objects in as consis-tent manner as possible. By contrast, several meth-ods such as in Fillard et al. (2007a) analyze the vari-ability of each shape individually; there is a needfor an extrinsic extrapolation scheme to retrieve avariability in the space between the objects. Theapproach proposed in Hellier and Barillot (2003);Cachier et al. (2001) makes a model of deforma-tion that has constraints on sulci, cortex, and wholebrain, all within a single optimization framework.Earlier work like Thompson and Toga (1996) justused the matching of low order manifolds �rst, andused these as hard constraints or boundary condi-tions on subsequent mappings one dimension higher.Altough there are many other registration frame-works in the literature, we focus in this paper on thecurrent-based di�eomorphic registrations (CDR) tobuild brain variability models.This paper aims to present and discuss such a

framework, based on di�eomorphic registration ofcurrents, in the case of curves. We apply the methodon a dataset of labeled sulcal lines to infer the vari-ability of the brain surface within a population. In

Section 2, we explain the framework for registrationof sulcal lines. How this di�ers from the pointwiseline correspondence (PLC) approach, proposed inFillard et al. (2007a), is discussed in depth from amethodological point of view. In Section 3, we per-form a statistical analysis of the underlying brainsurface based on these registrations. This allows ustomeasure and visualize how the brain surface variesin a population. A comparison with the results ob-tained in Fillard et al. (2007a) on the same databaseillustrates some of the di�erent methodological as-sumptions each method is based on.

2. Registering Sets of 3D Curves

Registering a set of 3D curves L0 onto anotherset of 3D curves L1 can be formulated as the taskof looking for the most regular deformation φ thattransports all curves of L0 and best matches thecurves of L1. We follow here the approach intro-duced in Glaunès (2005): the unknown deformationis sought in a subgroup of di�eomorphisms and itsregularity is measured based its distance to the iden-tity (i.e. no deformation), the similarity measure iscomputed by embedding the curves into a space ofcurrents. As it is common practice in deformable im-age registration, we �nd the registrations by mini-mizing a cost function that balances the regularityof the deformation against the matching �delity.

2.1. Non-parametric Representation of Curves as

Currents

The space of currents is a vector space that maybe equipped with a norm that measures geometri-cal similarity between curves. In this space, curvescould be discrete or continuous and may consist ofseveral di�erent parts. All these objects are han-dled in the same setting and inherit many interest-ing mathematical properties: linear operations, dis-tance, convergence, etc. Moreover, this de�nition ofdistance between curves does not make any assump-tion about point correspondences, even implicitly.This framework di�ers therefore from usual meth-ods such as that in Joshi and Miller (2000) wherelandmark matchings are performed or those in ChuiandRangarajan (2003); Granger and Pennec (2002);Cachier et al. (2001) where curves are considered asunstructured point sets and di�erent kind of �fuzzy�correspondences are assumed. We recall here how tobuild a space of currents and how to compute explic-

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itly a similarity measure on curves. For more detailson the theory we refer the reader to Vaillant andGlaunès (2005); Glaunès (2005); Durrleman et al.(2008) and references therein.In the framework of currents, curves are seen via

the way they integrate vector �elds. Any continuouscurves or any �nite set of polygonal lines (denotedhere generally L) can be characterized by the pathintegral of any vector �eld ω along it:

∀ω ∈ W, L : ω −→∫

L

< ω(l), τ(l) >R3 dl (1)

where τ(l) is the unit tangent vector (de�ned almosteverywhere) of L at point l and W is a test spaceof smooth vector �elds (See Fig.1). More generally,a current L is de�ned as a continuous linear map-ping from the test space W to R. As a set of map-pings, the space of current (denotedW ∗) is a vectorspace: (L1+L2)(ω) = L1(ω)+L2(ω) and (λL)(ω) =λ(L(ω)). For curves, this means that the path inte-gral along two curves is the sum of the path integralalong each curve: the addition corresponds there-fore to the union of the two curves. Scaling a curvemeans scaling the path integral along the curve.Suppose now, that we can provide the test

space W with a norm (‖.‖W ) that measures theregularity of the vector �elds in W . We can de-�ne then a norm of a current L as the supre-mum path integral of any regular vector �eld (i.e.

‖ω‖W ≤ 1) along L: ‖L‖2W∗ = Sup‖ω‖W≤1 |L(ω)|.

The distance between two curves (‖L− L′‖2W∗ =

sup‖ω‖W≤1 |L(ω)− L′(ω)|) is therefore obtained forthe vector �eld that best separate the two lines, inthe sense that the di�erence between the path inte-grals along both curves is the largest possible. Thisdistance between curves is geometric: it does not

depend on how curves are parametrized and does not

assume any point-correspondences between curves.

For computational purposes, we suppose, fromnow onward, thatW is a reproducible kernel Hilbertspace (r.k.h.s.) with kernel KW (see Aronszajn(1950); Saitoh (1988) for details): vector �elds in Ware convolutions between any square-integrable vec-tor �elds and the kernel. This framework is generaland includes for instance radial basis functions. Inthis setting, the vector space of currents is a densespan of the set of all delta Dirac currents δα

x , whichis de�ned by: δα

x (ω) = 〈ω(x), α〉R3 for any ω ∈ W .A Dirac current may be seen as an oriented segmentα entirely concentrated at one point x. Although acurve has an in�nite set of tangents, polygonal linesmay be approximated in the space of currents by a

Figure 1. Measure of dissimilarity between lines modeled ascurrents: given two lines L and L′ (in red and blue) we com-pute the di�erence between the path integral of a vector �eldω (here drawn in black) along both lines. The maximum dif-ference obtained when ω varies among all possible regularvector �elds (i.e. ‖ω‖W ≤ 1) is a measure of the geometricaldissimilarity between the two lines. In this way, we de�ne adistance between shapes without assuming point correspon-dences. The more we allow the test vector �elds ω to havehigh spatial frequencies, the more �nely we measure geomet-rical di�erences. In this application, manual delineation ofthe sulci is typically accurate to within a 1-2 mm Hausdor�distance to a gold standard developed from multiple raters,so the matching of features at a slightly coarser scale thanthis is empirically reasonable.

�nite sum∑

k δτkck

where ck is the center of the kth

segment and τk the tangent of the line at ck.In this setting, it has been shown (Vaillant and

Glaunès, 2005; Glaunès, 2005) that the norm onW ∗ comes from an inner product 〈., .〉W∗ . Onbasis elements, this inner product is

⟨δαx , δβ

y

⟩W∗ =

αtKW (x, y)β. The inner product between two

polygonal lines L =∑n

i=1 δτici

and L′ =∑m

j=1 δτ ′

j

c′j

(where n is not necessarily equal to m) is thereforegiven by:

〈L,L′〉W∗ =n∑

i=1

m∑j=1

(τi)tKW (ci, c′j)τ

′j (2)

This enables to compute explicitly the distance be-tween two curves:

d2(L,L′) = ‖L′ − L‖2W∗ = ‖L‖2

W∗+‖L′‖2W∗−2 〈L,L′〉W∗

(3)In our applications, we choose KW to be isotropicand Gaussian: for all points x, y ∈ R3, KW (x, y) =exp(−‖x− y‖2

/λ2W )Id.

We observe that the distance between two curves(Eq.3), induced by the Hilbertian inner productEq.2, measures geometrical di�erences both in poseand shape (See Fig.1). If the points on one curve areat a distance much larger than λW from the points

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on the other curve, then curves are considered as or-thogonal (〈L,L′〉W∗ ∼ 0) and their distance is largewhatever their respective shapes are. By contrast,if two parts of the curves are located within an areaof size λW , local alignment of the tangent vectorsis taken into account by the inner product withinthe sums in Eq.2, thus measuring shape variations.Furthermore within this area, variations at a scalemuch smaller than λW are not taken into accountthanks to the smoothing e�ect of the kernel. Suchvariations are considered as noise. Finally, this dis-tance captures �rst misalignment and then shapedissimilarities until a noise level quanti�ed by λW isreached. Used as a data �delity term, this distanceintegrates a denoising process, to some extent, intothe modeling, preventing systematically over�ttedregistrations.

2.2. Di�eomorphic Registration

We use here the large deformation frameworkfounded in the paradigm of Grenander's groupaction approach for modeling objects (see Grenan-der (1994); Miller et al. (2006); Glaunès and Joshi(2006); Marsland and Twining (2004)). This frame-work enables to �nd a globally consistent deforma-tion of the underlying space that best matches thesets of lines. This di�ers from Fillard et al. (2007a)where each line is registered individually withoutassuming spatial consistency of the displacement�eld between lines.We build our deformations as di�eomorphisms φv

1,solutions at time t = 1 of the �ow equation:

∂φt(x)∂t

= vt(φt(x)) (4)

with initial condition φ0 = idR3 (i.e., φ0(x) = x:no deformation). The time-varying vector �eld v =(vt)t∈[0,1] is the speed �eld in the Lagrangian coordi-nates. We suppose, from now onwards, that at everytime t, vt belongs to a r.k.h.s. V with kernel KV . Wedenote ‖.‖V the norm on this space that measurethe spatial regularity of the vector �eld. To measurethe regularity of the �nal di�eomorphism, we inte-grate the regularity of this speed �eld along time(Grenander and Miller, 1998; Miller et al., 2002):

v ∈ L2([0, 1], V ): ‖v‖2L2([0,1],V ) =

∫ 1

0‖vt‖2

V dt.Our registration problem is to map a set of n la-

beled sulcal lines L0 = ∪ni=1L0,i to another labeled

set L1 = ∪ni=1L1,i. We must �nd therefore a time-

varying vector �elds (vt)t∈[0,1] that minimizes thefollowing cost function J :

J (v) = γ ‖v‖2L2([0,1],V ) +

n∑i=1

‖φv1.L0,i − L1,i‖2

W∗

(5)where γ is a trade-o� between the regularity of thedeformation and the �delity to data.

φ.L represents the geometrical transportation ofthe curve L by the deformation φ. This formulationis compatible with our framework based on currents.The path integral of ω along a deformed curve φ(L)equals the path integral along L of the pulled-backvector �eld: φ ? ω(x) = (dxφ)tω(φ(x)). This is achange of variables formula within Eq.1, from whichwe deduce a general action of di�eomorphism on anycurrents: φ.L(ω) = L(φ ? ω). In particular, on basis

element, this gives: φ.δαx = δ

dxφ(α)φ(x) : an in�nitesimal

segment α at point x is transported into φ(x) anddeformed by the Jacobian matrix: dxφ. Combinedwith Eq.2 and 3, this makes computable the �delityto data term in Eq.5, once a deformation is given.To minimize the cost function in Eq.5, we take

advantage of a dimensionality reduction property.Although the vector �elds vt are dense, it has beenshown (for instance in Miller et al. (2002); Vaillantand Glaunès (2005)) that, in case of discrete curves,

L0 =∑N

i=1 δτici, the minimum of Eq.5 is achieved for

a vector �eld vt which interpolates the trajectoriesof the points of L0:

∀x ∈ R3, vt(x) =N∑

i=1

KV (x, ci(t))αi(t) (6)

where the momenta (αi(t)) is a set of N vectors (in3D) for each time t and ci(t) = φt(ci) are the tra-jectories of the points of L0. Based on Eq.4 evalu-ated at x = ci, these trajectories depend only onthe momenta αi(t). This means that the minimiz-ing dense vector �eld vt is entirely determined by aset of 3 ∗ N parameters for each time t. Once thetime interval [0, 1] is discretized, the cost functionEq.5 depends on a �nite set a parameters and maybe therefore minimized by a standard gradient de-scent scheme. All computational details of this gra-dient descent can be found in Vaillant and Glaunès(2005); Glaunès (2005).For KV we choose an isotropic and Gaussian ker-

nel with standard deviation λV . This parameter con-trols the regularity of the speed vector �eld vt ateach time t and hence the regularity of the �nal dif-feomorphism. λV de�nes roughly the scale of the dif-feomorphism's spatial consistency (called also rigid-ity). This is then the scale at which the underlyingdeformation tries to integrate the geometrical infor-

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mation. If λV is much smaller than the distance be-tween lines, the �nal deformation can vary dramat-ically in space, each piece of lines are then matchedalmost independently and the deformation is negli-gible outside the data. On the contrary, the greaterλV , the more consistently the deformation tries toexplain the variation of each lines, jointly with lessand less precise matching.

2.3. Registration Results

As part of a collaborative project involving theAsclepios-LONI associated team Brain-Atlas, weused a dataset consisting of cortical sulcal land-marks (72 per brain) delineated in a large numberof subjects scanned with 3D MRI (age: 51.8 +/- 6.2years). In order to compare our measures of variabil-ity with the ones of Fillard et al. (2007a), we usedthe same set of 72 mean lines that the authors of Fil-lard et al. (2007a) computed from the same dataset.For 34 subjects in the database, we register this setof mean lines onto every individual subject's set ofsulcal lines. The registrations were computed by J.Glaunès' algorithm detailed in Glaunès (2005). Thisalgorithm depends on the 3 parameters: λV , λW andγ. To understand the impact of these parametersand the speci�city of this current-based di�eomor-phic registrations (CDR) with respect to a pointwiseline correspondence (PLC) method (Fillard et al.,2007a), we ran the registration algorithm for severaldi�erent sets of parameters.Figure 3-a,b and c show for di�erent parameter

values the registrations in the superior temporalarea of the cortex in the right hemisphere (viewfrom inside), the brain faces to the left of this �g-ure, and region surrounding the Sylvian �ssure, onthe lateral surface of the right hemisphere, is mag-ni�ed. From 3-a to 3-b, λW is doubled: greater vari-ations are considered as noise and the matching isless precise (area 1). However, when λW is too small,lines with few sampled points are not matched cor-rectly (area 2). Small curves have small weight inthe data �delity term and matching them is notworth the cost of the deformation: the algorithm islocally in a minimum. Greater λW makes distanceloss larger due to the curve's motion to its target.The local minimum issue is avoided. In both cases (aand b), the deformation kernel's size is very small:λV = 5mm whereas the diameter of the brain isabout 120mm. The speed vector �eld is highly ir-regular and each curve is matched almost indepen-

Figure 2. Registration of the mean lines set (in blue) towardsone subject's lines set (in red). A unique deformation trans-ports all the mean blue lines to the registered green lines. Thespatial consistency constraint as well as the smoothing e�ectof the norm of currents prevents over�tted registrations fromoccurring. The residuals (i.e., the di�erence between the reg-istered green lines and target's red lines) contains physicaland numerical noise as well as possible non-di�eomorphicvariations. They are considered here as noise: the statisticson brain shape will rely only on the di�eomorphism. A movieof this deformation can be seen at �rst author webpage:www-sop.inria.fr/asclepios/personnel/Stanley.Durrleman.

dently. This is particularly obvious in area 3, whichis close to the supramarginal and angular gyri ofthe temporo-parietal cortex, where the speed vector�eld varies dramatically between two close pointsthat belong to two di�erent curves. There is almostno deformation between the curves. By contrast, in�gure 3-c, λV = 25mm and the deformation tends toexplain the sulcal lines variability with a consistentdeformation of the underlying space. This makes thespeed vector �eldmore regular, as in area 3.WhereasλW is the same as in �gure 3-a, small curves are nowmatched since they are �pushed� by the large curvesin the surroundings. To match the larger curves, thespace must deform consistently in this area with thee�ect of moving small curves to their target. Thisglobal constraint also leads to larger residual match-ing errors than in �gure 3-a (area 1). Such residualerrors contain both geometrical noise on lines (quan-ti�ed by λW ) and some variability that cannot beexplained consistently with other curves in a neigh-borhood of size λV , which is regarded as noise in themodel. Besides λV and λW , γ re�nes the compro-mise between the regularity of the deformation andthe precision of the matching.Figure 3-d shows, in the same anatomical region,

how a pointwise line correspondence method (PLC)

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a-λV =5mm,λW =5mm,γ=0.01 b-λV =5mm,λW =10mm,γ=0.01

c-λV =25mm,λW =5mm,γ=0.01 d-PLC method

Figure 3. Registration of the same subject for three di�erent sets of parameters (a,b,c) and with a pointwise line correspondencesapproach (d). In these �gures, the superior temporal area of the cortex is magni�ed (arrow 1 points the extremity of theSylvian Fissure). The parameters in�uence the precision of the matching (like in area 1), the regularity of the deformation�eld (area 3) and the way the deformation integrates geometrical information (area 2).

set up in Fillard et al. (2007a) handles the same data.Lines are registered individually and point corre-spondences are assumed between source and targetlines. Extremal points are supposed to be matched.In between points are matched via a closest neighborprocedure after B-spline smoothing and resampling.As no correlations between curves are assumed andno correspondence �eld computed outside the data,this matching can be seen as an approximation of ourregistrations when λV tends to zero. Since the cor-respondences between points do not take noise into

account, it is also the limit as λW tends to zero butavoiding the local minimum issue. The di�erent waythat PLC handles lines matching have importantconsequences. For instance, the tangential variationof two curves is mainly captured at the extremities ofthe curves, and minimized elsewhere. With the cur-rent approach, these variations are captured moregeometrically all along the curves. Moreover, thePLC approach does not take noise into account (inthe sense that noisy point correspondence �eld com-puted from the manual traces is regarded as true),

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and does not model any deformation of the spacebetween the curves. As we will see in the next sec-tion, PLC approach needs afterwards to handle twoadditional processing, denoising and extrapolationof the variability measures, to compute brain shapestatistics. This method consists therefore of 3 dis-tinct processing steps: matching, denoising, extrap-olation, each with its own assumptions and parame-ters. By contrast, the approach proposed here basedon currents' di�eomorphic registrations (CDR) inte-grates denoising, matching and extrapolation withina single consistent framework. Matching based oncurrents avoids the need to de�ne a principle for en-forcing speci�c point correspondences. Denoising isperformed jointly with the matching while minimiz-ing the cost function. Extrapolation is performedon the basis of a deformation of the underlying bio-logical material. The whole framework is explicitlycontrolled by 3 parameters λV , λW and γ that ef-fect a compromise between the di�erent processingsteps. This method, however, discounts variabilitythat is not compatible with the modeling. Residualmatching errors may hide non-di�eomorphic varia-tions between subjects although one would like totake them into account. Setting the 3 parametersis di�cult since they are not independent and de-termine jointly the �nal residual matching errors.After extensive experiments, we choose the typicalcoherence scale of di�eomorphisms λV = 25mm,the typical noise scale on lines λW = 5mm and thetradeo� γ = 0.01 by visually inspecting the results.Changing these parameters would smoothly a�ectthe typical correlation size of the following variabil-ity maps. These values highlight the speci�city ofour framework, that will explain, in turn, the di�er-ent variability maps retrieved by our CDR methodand the PLC approach on the same dataset as inFillard et al. (2007a).

3. Statistics on Deformations

3.1. Tangent-space Representation of

Di�eomorphisms

To compute statistics on deformations, we takeadvantage of an additional property of the minimiz-ing di�eomorphisms. It has been shown (in Milleret al. (2006) for instance) that the di�eomorphismretrieved by the minimization of Eq.5 is geodesic:among all time-varying vector-�elds that enables togo from Id to φ1, the minimizer of Eq5 has the small-

est norm in L2([0, 1], V ). As a consequence, the mo-menta (αi(t))i solve the Euler-Lagrange equations(Miller et al., 2002, 2006): they are entirely deter-mined by their initial values: αi(0). This is the usualtangent-space representation as highlighted in Vail-lant et al. (2004) or in Pennec et al. (2006) for �nitedimensional manifolds. This representation enablesto generate randomly deformations of L0: given anyset ofN vectors α0

i , we can solve the Euler-Lagrangeequations (partial di�erential equations) to give themomenta at every time: αi(t). We deduce then fromEq.6 evaluated at every x = ci, the speed of thetrajectories of the points ci of L0: vt(ci). Integrat-ing the �ow equation Eq.4 enables then to computethe whole trajectories φt(ci). The generated defor-mation does not act only on the line L0 but it isa di�eomorphism of the whole 3D space. Based onthe interpolation property Eq.6, we can compute thespeed (and then the trajectory by Eq.4) of any pointx of the space, thus computing the entire di�eomor-phism. The purpose of our forthcoming statisticalestimations is to learn a law on the momenta α0

i ,so that di�eomorphisms simulated according to thislaw model the variability within the studied popu-lation.From the previous 34 registrations of the mean

lines to each subject's lines, we store 34 sets of initialmomenta: (αs

i ) for i = 1 . . . N and s = 1 . . . 34whereN is the total number of points ci within the set ofmean lines: L0 =

∑Ni=1 δτi

ci. We de�ne an inner prod-

uct (resp. a norm) on this space of momenta as theinner product (resp. norm) of its associated dense

vector �eld v0(x) =∑N

i=1 KV (x, ci)αi(0) based onEq.6. Since V is a r.k.h.s. with kernel KV , the innerproduct between two sets of 3N momenta (from theregistration of two di�erent subjects) (αp

i )i and (αqi )i

is equals to∑N

i,j=1(αpi )

tKV (ci, cj)αqj . In the sequel,

we denote this inner product between two vectors inR3N : 〈αp, αq〉V ∗ . Our statistics on di�eomorphismsare then reduced to statistics in R3N provided bythis inner product.

3.2. Mean of Deformations

Since the mean lines we used as a templatewere obtained in Fillard et al. (2007a), they arenot consistent with our registration framework.The deformations are not centered around theidentity (i.e. no deformation), so the vectors inR3N do not have zero-mean. To measure theinduced bias, we compute the mean of the ini-

8

tial momenta at each sample: αi = 134

∑34s=1 αs

i .

The norm of this bias is given by ‖α‖2V ∗ =∑N

i,j=1 αiKV (ci, cj)αj . Numerically we �nd in

our experiments: ‖α‖V ∗ /√

134

∑34s=1 ‖αs − α‖2

V ∗ =0.39. This means that the bias is less than 0.4 timesthe standard deviation, far below the usual 3σthreshold to decide a statistical signi�cance. 1

We now substract the mean �eld to each subject's�eld so that the analyzed data are centered for thefollowing computations of second order statistics.

3.3. A Gaussian Model on Deformations

To compute the covariance structure of the set ofdeformations, we perform a Principal ComponentAnalysis (PCA) on the set of vectors αs ∈ R3N foreach subject s. For this purpose, we build the 34×34symmetric matrix whose coe�cients are 〈αp, αq〉V ∗ .If V 1 ∈ R34 is the �rst eigenvector of this matrix,the �rst mode of initial momenta is given by: mi =∑34

s=1 V 1s αs

i (the normalization factor has been set

to 1 so that ‖m‖2V ∗ = λ1, i.e. the eigenvalue corre-

sponding to V 1). Given this �rst mode of initial mo-menta m, we follow the procedure explained in sec-tion 3.1, to generate the di�eomorphism determinedby m. We call this deformation, the �rst mode of de-formation. It illustrates, to the �rst order, how themean anatomy varies within the population. Sincethe di�eomorphism is dense, we can apply it not onlyon the mean lines points but also on a mean corti-cal surface to which the mean lines are close (Fig.4-middle). This deformation shows the variabilitybetween 0 and σ (Fig. 4-right). Repeating the pro-cedure for −α give the �rst mode of deformation be-tween 0 and −σ (Fig. 4-left). Complete movie of the�rst deformation mode is available at �rst author'swebpage 2 . This illustrates the generative propertyof the modelling: the lines and surfaces build by thisprocedure do not belong to the original database.These results show that we learn here a complete

statistical model of the whole brain surface defor-mation constrained by the sulcal lines. This di�ersfrom other methods that measure only the variabil-

1 Performing a real statistical test would imply the estima-tion of the number of degrees of freedom (since the initialmomenta are not independent) as well as the curvature ofthe space (Bhattacharya and Patrangenaru, 2003; Oller andCorcuera, 1995; Pennec, 1999). This is particularly di�cultdue to the in�nite dimension of the space.2 www-sop.inria.fr/asclepios/personnel/Stanley.Durrleman/

ity of the sulcal lines. This is possible due to in-tegrative power of the proposed approach: the dif-feomorphisms integrate the information of all sul-cal constraints to �nd the most acceptable defor-mation of the brain volume. As for the modes ofdeformation, this enables to generate new observa-tions (new brain surfaces) according to the learnedprobability law on deformations. This determine atleast visually how these new observations comparewith the original data and therefore understand thevariability that the model captured. For this rea-son, such models are called generative models: wenot only learn how to factorize an observation into adeformation and residual noise, but also how to re-construct it. Such models o�er an approach to clas-sify new subjects according to characteristics suchas gender, handedness, pathologies, etc., and iden-tify systematic di�erences in anatomy that correlatewith these features. Given a previously unseen indi-vidual anatomy, we can decompose it into a globaldeformation driven by its sulcal lines position anda residual noise. Our statistical model tells us howprobable such a deformation may be within a givenpopulation. Other methods try to retrieve similarcorrelations but with descriptive statistics such asstatistical test for instance (Narr et al., 2007; Hamil-ton et al., 2007; Luders et al., 2004). In the PLCapproach (Fillard et al., 2007a) no deformation iscomputed outside the lines. In this framework, thelines variability is computed from the displacement�eld at each mean lines samples positions. Then anextrinsic extrapolation scheme enables to retrieve avariability measure on the whole brain surface. Inthis aspect, the link between the observations andthe mean surface is broken by the succession of dis-tinct processing: the statistical model is learned onthe lines only and the extrapolation scheme does notdirectly infer a probability model on the brain sur-face. PLC approach deduces from the observationsvariability maps on the brain surface (see next sec-tion) but the way to reconstruct surfaces from thesemaps is missing. There are other methods that ex-trapolate sulcal line deformations to a full corticalsurface, based on covariant partial di�erential equa-tions that are invariant to the surface parametriza-tion (Thompson et al., 2002), based on harmonicmappings that minimize a surface-to-surface defor-mation energy (Shi et al., 2007; Wang et al., 2005).Some of these methods even extend the surface de-formation to the full volume, using interpolation(Thompson and Toga, 1996). In each of these cases,the di�erential operator governing the mapping may

9

−σ 0 +σ

Figure 4. First mode of deformation obtained by a PCA on the initial vector speed �elds. Original mean brain surface (Center)and its deformation at −σ (Left) and +σ (Right). Colors measure the displacement of each point along the deformation process(in mm).

be regarded, after suitable normalization, as the ex-ponent of a Bayesian prior on the space of allowabledeformations, so in a sense there is an assumed prob-ability law that captures the variability and spatialcovariance of the mappings in between the explicitlyde�ned landmarks, even when a partial di�erentialequation or variational method is used to interpo-late the mappings.

3.4. Comparison of Variability Maps

To compare our CDR-based variability measureswith those computed with a PLC approach on thesame database we create variability maps similar tothose in Fillard et al. (2007a): in absence of gener-ative models, PLC approach performs such descrip-tive statistics. At each point x of the mean surface,we computed the covariance matrix of the 34 initialspeed vectors v0(x), computed with Eq.6 for each setof initial momenta. These 3×3 matrices (called alsotensors) show how locally one point is varying amongthe studied population, as proposed in Thompsonet al. (1996b, 1998). We notice that this variancecontains less information than the former principalcomponent analysis. Here each point are consideredindependently whereas the PCA takes into accountthe correlations of all points' motion together. More-over, due to the di�eomorphic approach, the initialvector �eld is dense and no extrinsic extrapolationscheme is required to compute the covariance matri-ces at each point of the mean surface. By contrast,PLC approach computes these 3 × 3 matrices fromthe correspondence �elds at the mean lines samples.These tensors are then extrapolated to the wholebrain surface using a log-Euclidean framework (Pen-nec et al., 2006; Arsigny et al., 2006) without any

guarantee that the obtained variability measures arecompatible with an underlying deformation of thebrain surface. This gives an aggregated measure atthe population level that is not based on individ-ual deformation mappings. The two approaches arebased on radically di�erent assumptions and thefollowing variability maps show how these di�erentmodels in�uence the results. Di�erences stem fromthe di�erent way lines are matched, noise is removedand variability is extrapolated to the brain surface.

3.4.1. Regularity of the VariabilityThe �gure 5 shows the covariance matrices built

from the initial vector speed at the mean lines points(ci) in our Current-based Di�eomorphic Registra-tions (CDR) (Fig.5-a) and from the correspondence�eld in the PLC approach (Fig.5-b). We notice thatthe point matching method leads to irregular tensor�elds at extremities of the lines and between lines,whereas the global regularity constraint of the dif-feomorphism in CDR imposes the retrieved variabil-ity to be spatially smoother. CDR thus discountsany variability contained in the residual matchingerrors, which is considered here as noise. In PLC,the tensor �eld is denoised separately by removing�unreliable� large tensors at the end of lines beforethe extrapolation step.

3.4.2. Variability in the Direction of the Lines

One drawback of PLC's method as underlined inFillard et al. (2007a) is the fact that it systemati-cally under-estimates the variability in the directionof the lines. This variability is indeed essentially cap-tured at the extremities of the lines and minimizedin between. As a pragmatic solution, in the PLC ap-proach, the large extremal tensors were removed be-

10

Whole Brain Detail

a - Current-based Di�eomorphic Registrations (CDR)

Whole Brain Detail

b - Pointwise Line Correspondences (PLC)

Figure 5. At each sampling point, ellipsoids represent the square root of the empirical covariance matrix of the initial speedvectors (left hand side) or displacement �eld (right hand side). With PLC method, extremal points are supposed to be matched:this induces a high variability at the extremities of lines (area 1, right). This is avoided by the current approach (area 1,left). With PLC, each line is registered individually: the variance can vary dramatically where lines cross (area 2, right). Thissituation can occur where a sulcus has a branch, in �Y�-shape con�guration, and the junction may not be considered by thePLC approach. The global regularity constraint of CDR leads to smoother results (area 2, left).

fore extrapolating the variability, and the �nal mea-sures minimize the variability in the direction of thelines. This aperture problem is particularly visibleon the top of the brain as shown in �gure 6. By con-trast, the models based on currents (CDR) manageto represent a larger part of this variability. This ef-fect is of particular importance since this tangentialvariation is one of the major variation trends withinthe population as shown in Fig.4. Anatomically, anylateral splaying of the central and pre-central sulci(at the top of the brain) is usually a sign atrophy,consistent with widening of the interhemispheric �s-sure. If this variation is discounted, for example bydiscarding tensors at the extreme points of sulci,any future registration approach that uses the ten-sor �elds to model variation will underestimate thetrue anatomic variation in these areas. Otherwise,the variability which is orthogonal to the directionof the lines is in good agreement for the most part.

3.4.3. Distinction between correlated and

anticorrelated motions

In our CDR framework the tensors at every pointsof the mean surface are computed from the initialspeed vectors at these points that are interpolationsof the initial speed vectors at the mean lines sam-ples (see Eq.6): the interpolation is performed beforecomputing the variability measures. By contrast,in the PLC method the covariance are computedon the mean samples and then extrapolated to thebrain surface. As shown �gure 9 this di�erence the-oretically enables CDR to distinguish between areaswhere points are deviating from the mean anatomyin a correlated or anti-correlated manner. This is apossible explanation of the di�erent variability mapsretrieved in area 4 of �gure 8.

11

Variability map Covariance matrices

a - Current-based Di�eomorphic Registrations (CDR)

Variability map Covariance matrices

b - Pointwise Line Correspondences (PLC)

Figure 6. In the variability maps, a variability in the direction of the lines is retrieved in area 3 (extremities of central sulci) byCDR and not by PLC. The covariance matrices in this region show that the variability is mainly longitudinal. Since, in PLCmethod, large tensors at the endpoints of the sulcal lines are removed before the extrapolation, the variability in the directionof the lines is missed and the total variability is unreasonably small.

a - CDR b - PLC

Figure 7. The registration method in�uences the way tangential variability is taken into account. With point correspondencesthe tangential variability is mainly captured at the endpoints of the lines and minimized in between. The approach based oncurrent (CDR) retrieves a tangential component of variability all along the lines.

4. Discussion and Conclusion

In this paper we present a methodological frame-work to build global brain shape statistics by mea-suring and consistently integrating the variabilityof anatomical constraints such as sulcal lines. This

framework is based on two methodological tools:lines are modeled as currents, and multiple objectsets are matched by a single di�eomorphic defor-mation. By modeling lines as currents, we are ableto measure geometrical dissimilarity between curveswithout assuming point correspondences between

12

Left Hemisphere Right Hemisphere

a - Current-based Di�eomorphic Registrations (CDR)

Left Hemisphere Right Hemisphere

b - Pointwise Line Correspondences (PLC)

Figure 8. Area 4 is surrounded by 4 major sulci: the Sylvian �ssure (a), postcentral sulcus (b), intraparietal sulcus (c) andsuperior temporal sulcus (d). In the left hemisphere the �rst two vary, with respect to the sample mean, mostly in a decorrelatedmanner with respect to the last two sulci whereas their respective motions are much more correlated on the right hemisphere.The CDR approach tries to combine the motion of all lines and therefore leads to a small variability in area 4 (the perisylviancortex) in the left hemisphere and to a large one in the right hemisphere. In these perisylvian areas, the variability is likelyto di�er by hemisphere as the right hemisphere perisylvian sulci are torqued forward and at a higher angle of elevation thantheir counterparts in the left hemisphere (Thompson et al., 1998). With PLC method this asymmetry in the magnitude ofanatomical variability is not retrieved directly.

a - CDR b - PLC

Figure 9. Extrapolation schemes in the simple case of anti-correlated vectors. Right: In PLC framework the tensors arecomputed at the sample points and then extrapolated in the middle point: the tensor in the middle is similar to the two others.Left: In CDR approach one �rst extrapolates the vector �eld and then computes at each point the covariance matrix. Sincethe vectors are anti-correlated, the �eld is close to zero at the center and the variability measured at this point is negligible.

objects. Discrete and continuous lines are handled inthe same setting, thus guaranteeing numerical sta-bility and nice convergence properties. This distanceis also robust to noise, preventing small perturba-tions of lines from hiding true underlying geomet-rical di�erences. On the other hand, the di�eomor-phic framework consistently integrates the geomet-rical variations of a set of currents into a single de-

formation.We avoid modeling the variability of eachobjects independently. On the contrary, we try toexplain the variability of the sulcal constraints by aglobal deformation of the underlying image domain.By inferring aGaussianmodel on such deformations,we can de�ne a generative statistical model thatroots the variability measures into a rigorous modelfor individuals. Principal trends of variations within

13

the database can be highlighted by looking at thedeformation of a mean brain surface. Such statisticalmodels o�er an approach for classifying new obser-vations according to their pathologies, gender,etc.The synthesis of the geometrical variations into prin-cipal modes of variations may also make it easierto identify spatially correlated anatomical patternsand may lead to new scienti�c �ndings.This framework however raises several questions.

Our statistical modeling focuses on the deformationterm whereas there is obviously no ground truth re-garding anatomical homology between brains. Evenso, a di�eomorphism can capture a large part ofthe geometrical variability between shapes and setsof shapes. It is clear that some of the �true� un-derlying variability is not captured by the di�eo-morphism and remains unmodeled in the residualmatching errors. These residuals contain numericaland physical noise, possible non-di�eomorphic vari-ations (changes in topology or folding patterns forinstance) as well as variations that are not compat-ible with the variations of other objects in the sur-roundings. Our statistical model focuses here on thedeformation term only and our results indicate thatit can model a great part of the variability. However,a more complete statistical framework would takeinto account the matching residuals as well. A givenobservation would be therefore decomposed into adeformation and residuals and the statistical modelwould say how probable such a decomposition mightbe. Building such a statistical framework is beyondthe scope of the present paper, but it must be thetopic of further investigation.Some comment is also necessary regarding

whether the norm on currents is appropriate for thedata, as this model does not take explicitly curva-ture into account and every points on lines play thesame role. When anatomical curves are matchedusing a smooth registration �eld, Leow et al. (2005)have previously explored the case where the curvesare modeled as level sets of an implicit functions,and no explicit point correspondence is enforced,allowing the mapping to relax along sulcal lines.They investigated the matching of anatomic struc-tures by directly constructing their implicit levelset representations and the proposed matching costfunctionals were shown to be closely related to theHausdor� metric. With this type of mapping, brainstructural variability was reduced by 10% in mostregions and up to 40% in other regions; greatestreduction was observed in the temporal and frontallobes, while a lesser reduction was observed in ar-

eas with greatest anatomic variability. Arguably,this results in mappings that have less distortionwhile still matching homologous gyral anatomy indetail from one subject to another. Contrary tosome other norms that explicitly take into accountsurface curvature (Fischl et al., 2004), or di�er-ential invariants within a curve (such as torsion)derived from the Frenet frames of the curves beingmatched (Guéziec and Ayache, 1994), our norm inthis paper, and the one in the paper by Leow et al.(2005) do not consider that there are particularpoints of anatomical interest along the curves thatcan be identi�ed as reliable landmarks. With someminor exceptions (such as the genu of the centralsulcus (Vaillant and Davatzikos, 1999; Goualheret al., 2000), curvature, at least at the �ner scaleof indentations within sulci, is not a reliable guideto functional or structural homology in the humancortex, and using points of maximum curvature toguide correspondence may be problematic. Our ap-proach is somewhat more agnostic in regard to pointmatching. At least for the human cortex, detailsvisible on MRI along the length of a sulcus wouldnot be reliable features for anatomical matching,although this does not preclude their identi�cationin future, e.g. using other modalities such as DTI.The third issue concerns the choice of sulcal lines

as constraints to retrieve the global variability of thebrain. The question of the anatomical signi�canceof the sulcal lines is often raised in the literature(Toga and Thompson, 2007; Thompson and Toga,2003). The sulci that we use here as topological con-straints for cortical matching are essentially thosewhich have been chosen to have functional signi�-cance, occur consistently in large numbers of normalindividuals, and are not so variable in their incidenceand relation to other sulci that it would precludetheir reliable identi�cation in large numbers of sub-jects. Moreover, the sulcal lines are labeled and sup-posed to be in a single part. However, the frameworkbased on currents is not limited to such databases:lines could be discontinuous and consists of severalparts, which may indeed be more accommodating ofinterrupted sulci, which are known to occur (Man-gin et al., 2004; Duchesnay et al., 2007). Even if thelines are not labeled, a matching based on currentsis still possible: we then consider all the lines as asingle current. Preliminary experiments on the bot-tom lines of the sulci automatically extracted froma few subjects Rivière et al. (2002) raises howeverseveral problems. The geometry of the lines can beso complex that it is not possible to de�ne a global

14

orientation of the sulci. The variability is too high toretrieve sensible correspondences in the absence ofany priors on sulcal labels. The quality of the linesextraction (and possibly the quality of the labellingitself) does not enable to �nd reasonable matchingsbetween subjects. The geometry of the sulci dependsactually on the process of extraction (whatever it ismanual or automatic). In order to better constrainthe registrations, one needs to account for more in-formation than only the most probable lines. Onewould like for instance to use probability maps ofthe presence of the sulci in order to account for vari-ability of the extraction itself. Sulcal ribbons couldalso contain more reliable geometrical information.A similar framework (but that does note requireorientation of lines) has been used in Auzias et al.(2008) with the sulci of Rivière et al. (2002). Inthe future, alternative cortical landmarks, includingperhaps the endpoints of �ber paths inferred fromDTI, may also supplant or partially replace the re-liance on sulci as a guide to anatomical homology inthe human cortex (Cathier and Mangin, 2006).The comparison with a pointwise line correspon-

dence (PLC) method is also di�cult to interpret.The comparison remains here largely qualitative andat a methodological level. Even so, it is clear thateach result is biased by the assumptions on whichthe variability measures are based. Each approachreveals new features from the database such as theprincipal modes of deformation, or unexpected pat-terns of anatomical correlation at distant points(Fillard et al., 2007b). A fair comparison betweenboth methods would rely on objective statisticalperformance metrics, such as their respective pre-dictive power for instance. In future, we will designstudies that aim to predict extrinsic informationabout the subjects (e.g. sex, handedness, diseasesubtype, prognosis), from the information encodedin the cortical deformations. In a sense, the bestmodel of anatomical variability is one that allowsmost reliable inferences and predictions regardingpopulation. However, such a deep comparison isbeyond the scope of this paper. Our purpose washere to present a general framework to computestatistics of brain shapes, to highlight its strengthand limitations and �nally to show its feasibilityand relevance for addressing a range of statisticalproblems in the �eld of computational anatomy.

5. Acknowledgments

We would like to thank Joan Glaunès (Labora-toire de mathématiques appliquées, Université Paris5) for providing the code of the registration algo-rithm and Pierre Fillard (Team-project Asclepios,INRIA-Sophia Antipolis) for giving access to his re-sults as well as for our fruitful discussions. Imagesand movies were obtained thanks to the freewareMedINRIA 3 . This work was partially supportedthe European IP project Health-e-child (IST-2004-027749).

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