o metrics and euler-lagrange quations of …alain trouve´ universit´e paris 13, institut galil ee,...

19 Jun 2002 9:23 AR AR164-16.tex AR164-16.sgm LaTeX2e(2002/01/18)P1: IBC10.1146/annurev.bioeng.4.092101.125733

Annu. Rev. Biomed. Eng. 2002. 4:375–405doi: 10.1146/annurev.bioeng.4.092101.125733

Copyright c© 2002 by Annual Reviews. All rights reserved

ON THE METRICS AND EULER-LAGRANGE

EQUATIONS OF COMPUTATIONAL ANATOMY

Michael I. MillerCenter for Imaging Science, The Johns Hopkins University, Baltimore, Maryland 21218;e-mail: [email protected]

Alain TrouveUniversite Paris 13, Institut Galilee, 93430 Villetaneuse, France;e-mail: [email protected]

Laurent YounesCMLA, ENS-Cachan, 94235 Cachan Cedex, France; e-mail: [email protected]

Key Words morphometrics, shape, image understanding, deformable templates,medical imaging

■ Abstract This paper reviews literature, current concepts and approaches in com-putational anatomy (CA). The model of CA is a Grenander deformable template, anorbit generated from a template under groups of diffeomorphisms. The metric spaceof all anatomical images is constructed from the geodesic connecting one anatomicalstructure to another in the orbit. The variational problems specifying these metricsare reviewed along with their associated Euler-Lagrange equations. The Euler equa-tions of motion derived by Arnold for the geodesics in the group of divergence-freevolume-preserving diffeomorphisms of incompressible fluids are generalized for thelarger group of diffeomorphisms used in CA with nonconstant Jacobians. Metrics thataccommodate photometric variation are described extending the anatomical model toincorporate the construction of neoplasm. Metrics on landmarked shapes are reviewedas well as Joshi’s diffeomorphism metrics, Bookstein’s thin-plate spline approximate-metrics, and Kendall’s affine invariant metrics. We conclude by showing recent ex-perimental results from the Toga & Thompson group in growth, the Van Essen groupin macaque and human cortex mapping, and the Csernansky group in hippocampusmapping for neuropsychiatric studies in aging and schizophrenia.

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376RECENT PROGRESS IN CA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377THE METRIC SPACE OF ANATOMICAL IMAGES. . . . . . . . . . . . . . . . . . . . . . . . . 378

Geodesics Connecting Geometric Transformationvia Diffeomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Inducing the Metric Space on Anatomical Images. . . . . . . . . . . . . . . . . . . . . . . . . . 381

1523-9829/02/0815-0375$14.00 375

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376 MILLER ¥ TROUVE ¥ YOUNES

Expanding the Metric Space to IncorporatePhotometric Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

EULER-LAGRANGE EQUATIONS FOR INEXACTIMAGE MATCHING AND GROWTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

COMPUTATIONAL IMAGE MATCHING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Beg’s Geometric Transformations via Inexact Matching. . . . . . . . . . . . . . . . . . . . . 383The Psychophysics of the Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Space-Time Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Photometric Matching with Tumors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

FINITE DIMENSIONAL SHAPE SPACES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386Diffeomorphic Landmark Matching in EuclideanSpace and the Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Mapping the Cerebral Cortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391Landmark Matching Via Other Metric Distances. . . . . . . . . . . . . . . . . . . . . . . . . . . 391

PROBABILISTIC MEASURES OF VARIATIONAND STATISTICAL INFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

INTRODUCTION

Revolutionary advances in the development of digital imaging modalities com-bined with advances in digital computation are enabling researchers to make ma-jor advances in the precise study of the awesome biological variability of humananatomy. This is emerging as the exciting new field of computational anatomy(CA) (1, 2). CA, as first defined in Reference (2), has three principal aspects:(a) automated construction of anatomical manifolds, points, curves, surfaces, andsubvolumes; (b) comparison of these manifolds; and (c) the statistical codificationof the variability of anatomy via probability measures allowing for inference andhypothesis testing of disease states. This review will focus on aspects (b) and (c).Although the study of structural variability of such manifolds can certainly betraced back to the beginnings of modern science, in his influential treatise “OnGrowth and Form” in 1917, D’Arcy Thompson had the clearest vision of what layahead, namely:

In a very large part of morphology, our essential task lies in the comparison ofrelated forms rather than in the precise definition of each; and thedeformationof a complicated figure may be a phenomenon easy of comprehension, thoughthe figure itself may have to be left unanalyzed and undefined. This processof comparison, of recognizing in one form a definite permutation ordeforma-tion of another, apart altogether from a precise and adequate understandingof the original ‘type’ or standard of comparison, lies within the immediateprovince of mathematics and finds its solution in the elementary use of a certainmethod of the mathematician. This method is the Method of Coordinates, onwhich is based the Theory of Transformations.


METRICS OF COMPUTATIONAL ANATOMY 377

The study of shape and structure has certainly progressed a long way; Thomp-son’s vision is precisely the mathematical structure we term anatomy in CA (1, 2),a Grenander deformable template (3) in which the space of anatomical imagery isan orbit under groups of transformations. There are two types of transformations,the first of the geometric or shape type studied through mappings of the coordi-nate systems of the anatomies. Variation in the image space is accomodated byintroducing groups of diffeomorphic transformations carrying individual elementsfrom one to another. The diffeomorphic transformations (invertible differentiablemappings with differentiable inverse) fill out the anatomy. The second transforma-tion type is of the photometric values accomodating the appearance or creation ofnew structures. Equally exciting is that in recent years, metric space structures havebeen associated with the orbits of anatomical imagery. The metric is calculatedvia equations of motion describing the geodesic connection between the elements.Thus the original vision of Grenander’s metric pattern theory is in large part beingcarried out in CA: Metric distances between patterned shapes and structures aremeasured via distances between the mappings.

In this paper, we first describe some of the recent work in the mapping litera-ture of computational neuro-anatomy. We then review the basic model of CA asa Grenander deformable template, an orbit generated under groups of diffeomor-phisms. Next, we define metrics between anatomies associated with shortest lengthpaths connecting one anatomical mapping to another in the orbit, measuring bothgeometric and photometric variation. The variational problems specifying thesemetrics are reviewed along with the associated Euler-Lagrange equations. Interest-ingly, the Euler-Lagrange equations of motion for the geodesics characterizing themetric generalize the Euler equations of Arnold (4) from flows through the groupof volume-preserving diffeomorphisms to those appropriate for the more generalspace of non-volume-preserving diffeomorphisms appropriate for the study ofgeneral shapes. In this context, metrics on landmarked shapes are reviewed. Weconclude by showing results on the study of growth and development, human andmacaque cortex, and computational neuropsychiatry in aging and schizophrenia.

RECENT PROGRESS IN CA

The area of mathematical codification of biological and anatomical structure hasbeen exploding over the past several decades. Digital electronic databases are cur-rently available (5), especially for colocalization of volume datasets such as thoseencountered with PET/SPECT, CT, and MRI (6–14). Suitable atlases supportingneuromorphometric analyses (15) are now becoming available with the advent oflarge volumetric image data sets with large numbers of voxel samples. Complemen-tary to the atlas development work, there has been great progress in the applicationof D’Arcy Thompson’s transformations in cataloguing and in studying shapes. Theearliest landmark and volume mapping work of biological coordinates, pioneeredin the early 1980s by Bookstein (16) and Bajcsy (7), continues today in manygroups including the Ayache-Gourdon-Thirion group (17–22), the Bajcsy-Gee



group (7, 8, 23–27), the Bookstein group (16, 28–33), the Dale-Fischl group (34–37), the Duncan-Staib group (38–44), the Evans group (45–53), the Friston group(48, 54, 55), our own Grenander-Miller-Trouv´e-Younes group (1, 2, 56–76, 76a),the Kikinis-Jolesz group (77–82), the Davatzikos-Prince group (83–92), the Sapirogroup (92a–92c), the Hurdal-Sumners group (93), the Mazziotta-Toga group (11,94–98), the van Essen group (66, 99–101), and the Wandell group (92c, 101a,101b), to name a few. Many of these approaches have studied higher dimensionaltransformations and mappings. Concurrently, efforts have proceeded along thelines of establishing the power of these approaches in lower and moderate dimen-sional settings in which the image structures being matched are more globallydefined, with the variability of global structures being the primary goal of success-ful registration. Complementary to the volume work, there has been great progresson anatomical mapping via subsets of landmarked points. For such approaches,predefined subsets of the anatomy provide registration information and become theprincipal features about which variational studies between the various coordinatesystems proceed.

Structured approaches via deformable templates and contour and surface mod-els studying the variations of substructures via boundaries and surfaces via vec-tor field transformations have emerged in many of these efforts. Detailed grossmacroscopic studies of cortical folding and localization of functional and anatom-ical boundaries are emerging in both macaque and human brains, with automatedmethods for generating the sulcus and gyral principle curves becoming available.Active shape methods have presented significant advances as well for studyinganatomical shapes, including the active contour and surface deformation workby Terzopoulos (102–105), Pentland & Sclaroff (106–108), and Cootes & Taylor(109–111).

Because of the sheer complexity of human neuroanatomy, in particular thehuman brain, the study of brain geometry has emerged as the study of the sub-manifolds of anatomical significance including its landmarks, curves, surfaces,and subvolumes all taken together forming the complete volume. The completemethodology combines mappings that carry all of the submanifolds of points,curves, surfaces, and subvolumes together. This is precisely why in our own groupthe fundamental transformation group that we have studied are spaces of diffeo-morphisms, as they carry these submanifolds consistently. The transformations areconstrained to be 1-1 and onto, and differentiable with differentiable inverse, sothat connected sets remain connected, submanifolds such as surfaces are mappedas surfaces, and the global relationships between structures are maintained.

THE METRIC SPACE OF ANATOMICAL IMAGES

Geodesics Connecting Geometric Transformationvia Diffeomorphisms

The mappings of the geometric or shape type are exclusively invertible, 1-1, ontocontinuous mappings with continuous inverses that are differentiable, henceforth



called diffeomorphisms. These transformations form a group, denoted byG withelementsg∈G, and the coordinates that they transform are subsets ofX. Generally,the low-dimensional transformations—rotation, scale, and skew—are studied intandem with the infinite dimensional local diffeomorphic transformations. As orig-inally proposed by Christensen et al. (59), one models the transformations as arisingfrom an evolution in time, or a flowg(t), t ∈ [0, 1] corresponding to the transportequations from continuum mechanics [see Dupuis (65) and Trouv´e (60, 76) fortheir technical development]. The forward and inverse maps are uniquely definedaccording tog−1(g(x, t), t)= x for all t ∈ [0, 1], x∈ X, implying the equations offlow are linked according to

∂

∂tg(x, t) = v(g(x, t), t),

∂

∂tg−1(y, t) = −Dg−1(y, t)v(y, t),

g(0) = g−1(0)= id, (1)

id the identity map, the Jacobian operator giving thed× d matrix forRd-valuedfunctions (Df )i j = ∂ fi

∂xjand thed× 1 row vectorDf = ( ∂ f

∂x1, . . . ,

∂ f∂xd

) for scalarvalued functions defined ond–dimensional domain. The flow equations are depic-ted in panel 1 of Figure 1.

Definition 3.1 Define the group of transformationsG to be diffeomorphismsg(1)∈G : x 7→ g(x, 1)∈ X, 1-1,onto, with continuous inverse and differentiablesolutions of Equation 1, withv(t), t ∈ [0, 1] sufficiently smooth and vanishing atthe boundary of X for each t to generate smooth solutions g(1),g−1(1) (60, 65, 76).

One principal aspect of CA is to define the metric distance between anatomiesthrough the mappings between them. We follow the approach taken in Refer-ences (60, 68, 71, 76); embed the diffeomorphisms in a Riemannian space anddefine distance between them via the geodesic length of the flowsg(·): [0, 1]→ G

Figure 1 Panel 1 (left) shows the Lagrangian description ofthe flow; panel 2 (right) shows the variation of the group flowelementg(·) by η(·).



which connects them. The geodesic length is defined through the square rootof the energy of transformation of the path,E(g)= ∫ 1

0 ‖v(t)‖2Ldt, v(t) = ∂g/∂twith ‖·‖L a proper norm on the vector fields onX (i.e., a Sobolev space withL a differential operator). Investigators have used linear differential operatorsLoperating on the vector fields to enforce smoothness on the maps and to definethe finite norm, generally differentiating only in space and constructed from theLaplacian and its powers.L is ad× d matrix of differential operatorsL= (Li j ) onRd valued vector fields of the form (Lv) j =

∑di=1 Li j vi , j = 1, . . . ,d inducing

a finite norm constraint at each timet. Denote the norm-squared energy densityaccording to

Ev(t) =∫

XEv(x, t) dx = ‖v(t)‖2L = 〈Lv(·, t), Lv(·, t)〉 <∞, t ∈ [0, 1], (2)

so that the solutions of Equation 1 are well defined. The energy density has been de-fined through powers of the Laplacian for the classic thin-plate splines (16, 29, 30)and the Cauchy-Navier operator for 3-dimensional elasticity (25, 56, 59, 97). Dif-ferential operators with sufficient derivatives and proper boundary conditionsinsure the existence of solutions of the transport equation in the space of dif-feomorphic flows (60, 65, 76).

Now the metric between transformationsg0, g1∈G is defined via the length ofthe shortest pathg(t), t ∈ [0, 1] with the boundary conditionsg(0)= g0, g(1)= g1.A crucial property of the metricρG : G × G → R+ is that it is invariant toG, sothatρG(g0, g1) = ρG(g · g0, g · g1) for g ∈G, whereg′ · g= g◦ g′.

Theorem 3.1The functionρG(·, ·) : G × G → R+ between elements g0, g1∈Gdefined as

ρG(g0, g1)2 = infg(·): ∂

∂t g−1(t)=−Dg−1(t)v(t),g(0)=g0,g(1)=g1

∫ 1

0Ev(t) dt, (3)

is a left-invariant metric distance onG. The geodesics satisfy the Euler-Lagrangeequations(proof in Appendix):

∂

∂t∇vEv(·, t)+ (Dv(·, t))t∇vEv(·, t)+ (D∇vEv(·, t))v(·, t)+ div v(·, t)∇vEv(·, t) = 0, (4)

where∇ is the gradient operator delivering a vector, ∇vEv(t)= 2L†Lv(t) with theadjoint defined as〈Lf, g〉= 〈 f, L†g〉, div v=6 i

∂v∂xi

the divergence operator.

Equation 4 was first derived by Mumford (112) who also showed that it reduces toBurger’s equation in one-dimension forL= id,∇vEv = 2v. Mumford used a varia-tional argument for calculating Equation 4 perturbing the geodesic byη(t), t∈ [0, 1]leaving the endpoints unchanged (see panel 2 of Figure 1 and appendix). Equation 4generalizes Arnold’s derivation for incompressible flow [Equation 1 of (4)]. Theleft-invariance follows from that fact that translation by another group element



leaves the distance unchanged becauseg(·) satisfies∂g(·,t)∂t = v(g(·, t), t), g(0) =

id, g(1)= g1 implyingg(0)◦ h= h, g(1)= g1◦ h, with an identical velocity field sothat ∂g(h(x),t)

∂t = v(g(h(x), t), t), g(0)= h, g(1)=g1(h). The distancesρG(id, g) =ρG(h, g◦h) are equal for allh ∈G. As pointed out by Ratnanather and Baigent (per-sonal communication), the left-invariance of the metric allows the Euler-Lagrangeequations to be recast in the Euler-Poincar´e form using Marsden’s reduction theory(114).

Inducing the Metric Space on Anatomical Images

The group of geometric transformations are not directly observable. Rather imagesare observed via sensors which measure physical properties of the tissues. Carryingout Grenander’s metric pattern theory program, the space or orbit of anatomicalimagesI ∈ I must be made into a metric space. The metric distance betweenanatomical imageryI ∈ I is constructed from distances between the mappingsg ∈G. First, the anatomical orbit of all images is defined.

Definition 3.2The mathematical anatomy are functions I∈ I, I : x ∈ X 7→I(x), anorbit under the geometric transformations:I ={I ′:I ′(·)= I (g(·)), I ∈ I, g∈G}.This is a group action (113) onI with g · I= I ◦ gand group productg′ · g(·)= g◦g′(·)= g(g′(·)), which defines the equivalence relationI1 ∼ I2 if ∃g∈G such thatI1(·)= I2(g(·)), dividing I into disjoint orbits.

Now, there can be many high dimensional maps connectingI, I ′. For the in-duced functionρ on I to be a metric it is required thatρ is invariant to thisnonidentifiability; this it inherits from the left-invariance property ofρG.

Theorem 3.2The functionρ(·, ·): I × I→R+ between elements I, I ′ ∈ I de-fined as

ρ(I , I ′)2 = infg: ∂∂t g−1(t)=−Dg−1(t)v(t),I (g−1(·,1))=I ′(·),g−1(0)=id

∫ 1

0Ev(t) dt (5)

is a metric distance onI satisfying symmetry and the triangle inequality.

The metric property was proved in References (60, 71, 76). The property suf-ficient for ρ to be a metric is the left-invariance ofρ [in fact, all that is requiredis it be invariant to the stabilizer; see (71)]. The fact thatρG is left-invariant toGimplies that for allg∈G, ρ(I ◦ g, I ′ ◦ g)= ρ(I, I ′). This also implies any elementin the orbit can be taken as the template; all elements are equally good.

Expanding the Metric Space to IncorporatePhotometric Variation

Thus far, the metric depends only on the geometric transformations of the back-ground spaceX. Extend the construction of the metric to be image dependent fol-lowing Reference (71) by defining the group action to operate on the geometry andphotometric values allowing for the creation of new matter. For this, we introduce



an explicit dependence on time in the evolution of the image to accomodate pho-tometric variation via material transport.

Theorem 3.3Defining the image evolution in the orbit as J(y, t)= I(g−1(y, t), t),∂∂t g−1(y, t) = −Dg−1(y, t)v(y, t), then the functionρ(·, ·): I×I→R+ betweenelements I, I ′ ∈ I defined as

ρ(I , I ′)2 = infv(·),I (·):J(0)=I ,J(1)=I ′

∫ 1

0

(‖v(t)‖2L +

∥∥∥∥ ∂∂tJ(t)+∇ Jt (t)v(t)

∥∥∥∥2)

dt (6)

is a metric [ proven in (71)] satisfying symmetry and the triangle inequality.Defining

∇vE(y, t) = 2L†Lv(y, t)+ 2

(∂

∂tJ(y, t)+∇ Jt (y, t)v(y, t)

)∇ J(y, t), (7)

then the Euler Equation 4 holds with boundary term∇v E (·, 1)= 0 with thegeodesics for photometric evolution satisfying(proof in Appendix):

L†Lv(y, t)+(∂

∂tJ(·, t)+∇ Jt (·, t)v(·, t)

)∇ J(·, t) = 0 , (8)

∂

∂t

(∂ J(·, t)∂t

+∇ Jt (·, t)v(·, t))

+ div

(∂ J(·, t)∂t

v(·, t)+ (∇ Jt (·, t)v(·, t))v(·, t))= 0. (9)

EULER-LAGRANGE EQUATIONS FOR INEXACTIMAGE MATCHING AND GROWTH

A central problem in CA is essentially diffeomorphic image interpolation, i.e., toinfer the geometric image evolution that connects two elementsI0, I1 ∈ I underpure geometric evolution. For this a function of the path is defined; call itI0(g−1(t)),t ∈ [0, 1]. The goal is to construct the shortest length curveg(t), t ∈ [0, 1] whichminimizes the target norm squared‖I0(g−1(1))− I1‖2, as studied in Dupuis et al.(65). This is of course inexact matching, since there is a balance between metriclength of the path and target correspondence. We can view the image evolutiondefined by the geodesic as an interpolation between images via the geodesic.Because of the introduction of the free boundary, there is a boundary term whichis introduced.

Theorem 4.1 (Inexact Image Matching)The minimizer

infg(·): ∂

∂t g−1(t)=−Dg−1(t)v(t),g(0)=g0

∫ 1

0Ev(t)dt + ‖I1− I0(g−1(1))‖2 (10)



satisfies the Euler-Lagrange Equation 4 with boundary term(proof in Appendix):

∇vEv(·, 1)+ 2(I1(·)− I0(g−1(·, 1)))D(I0(g−1(·, 1)))t = 0, (11)

where D(I0 (g−1(·, t))) =∇ I t0(g−1(·, t)) Dg−1(·, t).

The above has no real-time variable, only connection or interpolation betweentwo images via the geodesic. Now assume that the observables define a runningnorm squared,‖I1(t)− I0(g−1(t))‖2, t ∈ [0, 1], corresponding to a time sequencethat might occur during space-time growth modelling. Then the problem is akinto finding the trajectory of diffeomorphisms creating observable time flow.

Theorem 4.2 (Space-Time Growth)The minimizer

infg: ∂∂t g−1(t)=−Dg−1(t)v(t),g(0)=id,t ∈ [0,1]

∫ 1

0Ev(t)dt +

∫ 1

0‖I1(t)− I0(g−1(t))‖2 dt (12)

satisfies the Euler-Lagrange equation(proof in Appendix):

∂

∂t∇vEv(·, t)+ (Dv(·, t))t∇vEv(·, t)+ (D∇vEv(·, t))v(·, t)+ div v(·, t)∇vEv(·, t) = 2(I1(·, t)− I0(g−1(·, t)))D(I0(g−1(·, t)))t . (13)

COMPUTATIONAL IMAGE MATCHING

Beg’s Geometric Transformations via Inexact Matching

In this section, we examine the results of solving the Euler-Lagrange equations forgenerating the geodesics. Faisal Beg solves inexact image matching (Equation 10)via variations with respect to the velocity field exploiting the vector space structure.

Algorithm 5.1 Fixed points of the following algorithm(proof in supplemental ma-terial link in the online version of this chapter or at http://www.annualreviews.org)satisfy Equations 4 and 11. Initializevold= 0, choose constantε, then for all t ∈[0, 1],

Step 1:∂

∂tgnew(t) = vold(gnew(t), t),

∂

∂tg−1new(t) = −Dg−1new(t)vold(t),

χnew(t) = gnew(g−1new(t), 1),

Step 2: Computevnew(t) = vold(t)− ε∇vE(t), setvold← vnew, return to Step 1

(14)

where

∇vE(t) = vold(t)+ (L†L)−1(|Dχnew(t)|

×D(I0(g−1new(t)))t (I1(χnew(t)) −I0(g−1new(t)))). (15)

1 Jul 2002 17:2 AR AR164-16.tex AR164-16.sgm LaTeX2e(2002/01/18)P1: IBC


Here (L†L)−1 f=Kf where K is the Green’s kernel. The space-time solution ofEquation 13 has gradient

∇vE(t) = vold(t)+ (L†L)−1

(∫ 1

t|Dχnew(t ; τ )|D(I0(g−1new(t)))t

× (I1(χnew(t ; τ ), τ )− I0(g−1new(t))) dτ

)(16)

whereχ (t ; τ ) = g(g−1(t), τ )

(for proof see supplemental material link in the online version of this chapter orat http://www.annualreviews.org). Notice the smoothing nature of the space-timesolution.

Beg’s gradient descent algorithm discretizes Equation 15. At the fixed point, thediscrete version of the Euler-Lagrange equation is satisfied. Except where noted,the operator is chosen to beL=−50∇2+ 0.1 id and the flow timet∈ [0, 1] was dis-cretized into 20 timesteps. All geodesic solutions correspond to 10,000 iterations ofthe gradient algorithm. Results of the mapping of multiple shapes including the for-ward and inverse diffeomorphic matches for large deformation are shown in Figure2. Figure 3 shows the results of maps on the geodesic of the flows for the1

2C shape totheC shape (with 30 timesteps) and Heart1 to Heart2 experiments. Figure 4 showsresults from the Macaque brain and hippocampus section mapping experiments.

The Psychophysics of the Metric

Shown in Figure 5 are examples of metric computations in 2-D on high resolutionelectron-micrographs. The mitochondria are shown in increasing order of theirmetric distance. Here,L is−50∇ 2+ id. Notice how increasing the distance in themetric corresponds closely to one’s intuitive feel of closeness and similarity.

Space-Time Flows

The UCLA group of Thompson & Toga (97) has examined the instantaneousversion of the growth problem in which, at each time instant, the deformation fieldis estimated independent of previous time. Figure 6 shows maps of growth in thebrains of children, showing growth rates corresponding to the corpus callosum.The maps are made based on scanning the same child at age 7 and 11, and anotherchild at age 8 and 12. Pairs of 3-D scans acquired several years apart are firstrigidly registered. A 3-D deformation vector field based onL = −∇2 + c∇∇ t

is constructed at each time point with two components of the deformation fieldapplied to the regular grid overlaid on the anatomy. The Jacobian determinant,whose values are color coded on the deformations shows the local growth rate (redcolors, rapid growth).

Photometric Matching with Tumors

We examine photometric variation withL=∇2= ∂2

∂x21+ ∂2

∂x22

and ‖v(t)‖2L = 1α∑2

i=1‖∇2vi (t)‖2 from Equation 6, whereα is a positive parameter. Comparison



Figure 2 Row 1 showsI0 and row 2 showsI0 (g−1(·, 1)); row 3 showsI1, and row 4 showsI1

(g(·, 1)). Column 1 shows a diagonal shift, column 2 shows a dilation and contraction (scale),column 3 shows a rotational motion (the1

2C shape experiment), column 4 shows a sectionof Heart 1 matched to corresponding section of Heart 2, and columns 5 and 6 show sectionsof hippocampus in a young control to that with schizophrenia and with Alzheimer’s. Shownin the bottom row are the properties ofg−1

12 C

(·, 1) (panel 1), the deformation of the underlyingcoordinate space of the template,g1

2 C(·, 1) (panel 2), the deformation of the underlyingcoordinate space of the target, the Jacobian of the forward mapg(·, 1) (panel 3), and thenorm of the velocity field along the flow for 1, 100, 1,000, 5,000, 10,000 iterations of thealgorithm (panel 4). Notice the geodesic property at the convergence point corresponding tothe constant norm‖v(t)‖= constant.

between two imagesI and I ′ is performed by minimizing over all paths satisfy-ing v(x, t)= 0, t∈ [0, 1], x∈ ∂X, andJ(0)= I, J(1)= I ′ as in Reference (71); seealgorithm in Supplemental Material link in the online version of this chapteror at http://www.annualreviews.org. Figure 7 shows results demonstrating thetransformation processI(g−1(·, t), t) as functions of timet ∈ [0, 1] between twoimagesI andI ′. The top two rows show the process of either creating photometricor geometric change. Depending on the choice of the parameterα, the process



Figure 3 Rows 1 and 3 show images along the geodesic at time points 0 (panel 1),t3 (panel2), t9 (panel 3),t12 (panel 4),t15 (panel 5), andt18 (panel 6). The metric distance from the startpoint is shown below each panel. Rows 2 and 4 show the vector fields at three different timepoints on the geodesic,v(·, t0), v(·, t10), v(·, t19). The heart data (rows 3, 4) were taken fromthe laboratory of Dr. Raimond Winslow.

allows for the creation of pixel luminance yielding a transformation that looks likefading with almost no geometric deformation at all (α small), or the process willintroduce a large deformation of the grid resembling an explosion (α large). Thebottom row of Figure 7 is the result of matching sections of a brain with a tumorshape. The rightmost panel shows the centered black spot.

FINITE DIMENSIONAL SHAPE SPACES

A special class of images corresponding to finite dimensional landmark shapeshas been extensively studied by Bookstein (16, 29, 30) and Kendall (115) andcolleagues. Fix an integer number of landmarksN> 0 and denote byIN= (Rd)N

the space of landmarks parameterizing the finiteN-shapes through the points thatidentify themIN= (x1, . . ., xN), I ′N = (x′1, . . ., x′N). The natural distances betweenshapesρ: IN × IN→ R+ are the lengths the paths through which the points must



Figure 4 Row 1 shows the maps along the geodesic connecting Macaque1 to Macaque2;row 2 shows the velocity fields along the geodesic for the 2-D matching experiments. Thedata were taken from the laboratory of Dr. David Van Essen. Rows 3 and 4 show similarresults for the hippocampus mappings along the geodesic connecting the young control tothe patient with Alzheimer’s disease. Row 4 shows velocity fields along the geodesic for the2-D matching experiments. The data were taken form the laboratory of Dr. John Csernansky.

travel to correspond. Distance is measured through a quadratic form determiningthe length of the tangent elements as the particles flow.

Definition 6.1For N-shapes IN, I ′N with trajectories g(t)= (g(x1, t), . . ., g(xN, t)), in

Rd, g(xi, 0)= xi, g(xi, 1)= x′i of length∫ 1

0‖ ∂g(t)∂t ‖2Q(g(t)) dt, then the geodesic dis-

tance is

ρ(IN, I ′N)2= infg(·):g(xn,0)=xn,g(xn,1)=x′n

∫ 1

0

N∑i j=1

v(g(xi , t), t)t Q(g(t))i j v(g(xj, t), t) dt,

(17)

where Q(g(t)) is an Nd×Nd positive definite matrix with d× d blocks Q(g(t))i j

with Nd×Nd inverse K(g(t))= (Q(g(t)))−1, with d× d blocks(K(g(t)))i j .



Figure 5 The top row shows the mitochondria subshapes for matching from thelaboratory of Dr. Jeffrey Saffitz. Rows 2, 3, 4, and 5 give below each panel the metricfor the geodesic distances to indicate which shapes are close and far. Notice how in-creasing the distance in the metric corresponds to closeness and similarity.



Figure 7 The first row corresponds to smallα; the second row corresponds to largeα. The left column shows start and end imagesI0, I1; the middle column shows thegeometric deformations on the grid determined by mappingg−1(x, 1); the right columnshows the image value change. The last row shows a time series of geometric andphotometric transformations in generation of tumor in brain tissue. Results from Miller& Younes (71).

Diffeomorphic Landmark Matching in EuclideanSpace and the Sphere

MATCHING ON THE CUBE [0, 1]d The Joshi metric for diffeomorphic matching oflandmarks turns out to reduce to the dense matching diffeomorphism only involvingthe paths of the landmarks. These are the Joshi diffeomorphism splines as firstshown in (69). Assume the the quadratic form‖v(t)‖2L admitsd× d Green’s kernelK so that for allf in the space‖ f ‖Lfinite,

〈K (x, ·), f 〉L =∫

XL†LK(x, y) f (y) dy= f (x). (18)

Theorem 6.1 (Joshi (69))For N-shapes IN, I ′N ∈RdN, ∂∂t g(x, t)= v(g(x, t), t),

g(xn, 0)= xn, g(xn, 1)= x′n the optimizing diffeomorphism satisfying

infv:R1

0‖v(·,t)‖2dt<∞

∫ 1

0‖v(·, t)‖2Ldt +

N∑n=1

‖yn − g(xn, 1)‖2Rd

σ 2, (19)



with Nd×Nd Green’s kernel K(g(t)), inverse Q(g(t))=K−1(g(t)) where

K (g(t)) =

K (g(x1, t), g(x1, t)) · · · K (g(x1, t), g(xN, t))...

......

K (g(xN, t), g(x1, t)) · · · K (g(xN, t), g(xN, t))

is given by

v(·, t) =N∑

i=1

K (g(xi , t), ·)N∑

n=1

Q(g(t))in g(xn, t)

and

g(xn, ·)n = 1, . . . , N = argming(xn, ·)n = 1, . . . , N

∫ 1

0

∑i j

g(xi , t)t Q(g(t))i j g(xj , t) dt

+N∑

n=1

‖yn − g(xn, 1)‖2Rd

σ 2

(proof in supplemental material link in the online version of this chapter or athttp://www.annualreviews.org).

Joshi (69) examines the matching problem in Euclidean subsets ofRn follow-ing the gradient of the cost in the planen= 2, and cubic volumesn= 3 choos-ing operators constructed from the LaplacianL= diag(−∇2+ cI), studying bothexact and inexact landmark matching. InR3, the 3× 3 diagonal blocksK(x, y)= diag[κe−

√1c ‖y−x‖] with κ normalizingκ

∫e−√

1c ‖y−x‖ dy = 1. The top row of

Figure 8 shows the matching for the “OVAL” and “S” (panels 1 and 2) from Joshi’sgradient algorithm (69). The corresponding points were used as landmarks withvariancesσ 2= 0.1 (panel 3) andσ 2= 10 (panel 4).

Matching on the SphereS2: Investigators in CA, beginning with the earliest workof Dale & Sereno (34), have introduced transformations on the sphere to study theneocortex. Bakircioglu et al. (70) constructed diffeomorphisms on the sphere viamaps that are constrained to satisfy flow equations∂

∂t g(x, t) = v(g(x, t), t), g(x, 0)= x, t ∈ [0, 1], x ∈ S2, v(x, t)= ∑2

i=1 νi (x, t) Ei(x); whereE1(·), E2(·) arecoordinate frames on the sphere from stereographic projection.

For smoothing, the spherical Laplacian operator is used (L= ∂2

∂φ2 + cotφ ∂∂φ+

1sin2φ

∂2

∂θ2 ), which has as eigenfunctions the spherical harmonics (116) inducing the2N× 2N diagonal block matrix

K (g(t))i j = diag

[ ∞∑n=1

1

n2(n+ 1)22n+ 1

4πPn(cos(9(g(xi , t), g(xj , t)))

],

with the spherical solid angle9(·, ·) and Legendre polynomialsPn(x)= 12nn!

dn

dxn

(x2− 1)n.



Given N-landmarksxn, yn, n= 1, . . ., N ⊂ S2, with distance on the sphere9 and coordinate frames on the sphereE1, E2, the optimal diffeomorphism sat-isfying ∂

∂t g(·, t) = ∑2i=1 νi (g(·, t), t)Ei (g(·, t)), g(x, 0)= x, x∈S2, minimizing∑2

i=1

∫S2× [0,1] ∇2|ν i(x(θ , ψ), t)|2 sinψdψdθ is given by

infg:g(t)=P2

i=1νi (g(t),t)Ei (g(t))

∫ 1

0

∑i j

ν(xi , t)t (K (g(t))−1)i j ν(xj , t) dt

+N∑

n=1

92(yn, g(xn, 1))

σ 2n

.

The results of computations of spherical deformations are shown in the bottomrow of Figure 8.

Mapping the Cerebral Cortex

To understand individual variations in the cortical topography, the Van Essen grouphas been using large deformations to establish correspondences between the cor-tical maps of various individual cortical surfaces (66, 100, 101). The left columnof Figure 9 shows a surface-based atlas of the macaque visual cortex. The toppanel shows the macaque surface in 3-D and the bottom panel shows the flatatlas. The right column of Figure 9 shows results from flat mapping transforma-tions of the macaque. The top row of Figure 10 shows an interspecies compar-ison via cortical surface deformation between the macaque and visible human(100).

Landmark Matching Via Other Metric Distances

EUCLIDEAN DISTANCE AND BOOKSTEIN THIN-PLATE SPLINE DISTANCE Euclideandistance betweenN-shapes inRd measures distance according to straight linepaths between the objects. Choosing the quadratic form to be a constant indepen-dent of the path, the metric distance becomesρ(IN, I ′N)2= ∑N

i j=1 (xi−x′i )t(K−1)i j

(xj−x′j ).Bookstein’s (16, 29, 30) thin-plate spline landmark matching measure approx-

imates the tangent flow of the landmark trajectories by the tangent at the origin.This is clearly not symmetric, nor does it satisfy the triangle inequality. However,for small deformations, this is a powerful methodology that approximates the dif-feomorphic flow metric of Joshi in the tangent space by the metric at the origin.Assume Green’s operator as thedN× dN block matrixK(g(0)) evaluated at theorigin of the flow withd× d blocksK(g(0))i j . The approximate distance reducesto ρ(IN, I ′N)2 =∑N

i j=1(xi − x′i )t (K (g(0))−1)i j (xj − x′j ).

KENDALL’S SIMILITUDE INVARIANT DISTANCE Kendall (115) defines the distancebetween sets of N-shapes invariant to uniform scale, rotation, and translation. De-fine the affine similitudes to be matricesA= sO, s∈ (0,∞), O∈SO(n), with their



action on the shapes, the scale-rotation-translation of each point of the shape:AI+ a= (Ax1+ a, . . ., AxN+ a)t. The Kendall invariant distance requires the ma-tricesK(IN) have the property that, for all (A, a), K(AIN+ a)=AK(IN)At.

Theorem 6.2 [Kendall (115)]Defining the mean shape position asg(t) = 1N

∑Nn=1

g(xn, t), with K(g(t))= σ 2(g(t))id, where id is the Nd×Nd identity matrix andσ 2(g(t))= ∑N

i=1‖g(xi, t) − g(t)‖2, with ρ0 a distance onR+×Rd, the geodesic

connecting IN, I ′N minimizing∑N

i=1

∫ 10

1σ 2(g(t))‖ ∂g

∂t (xi , t)‖2 dt has distance(proofin Appendix)

ρ(IN, I ′N)2 = ρ0((σ (g(0)), g(0)), (σ (g(1)), g(1)))2

+(

arccosN∑

i=1

⟨g(xi , 0)− g(0)

σ (g(0)),

g(xi , 1)− g(1)

σ (g(1))

⟩)2

(20)

with Kendall’s similitude invariant distance given by

ρ(IN, I ′N) = inf{ρ(AIN + a, I ′N),A similitude,a ∈ Rd}. (21)

Kendall’s distance ˜ρ in Equation 21 requires computing the minimum ofρ(sOIN+a, I ′N), for s> 0, O ∈ SO(d) anda ∈ Rd. Defining the normalized landmarks byγ (xi , t) = g(xi,t)−g(t)

σ (g(t)) , then because the action ofs anda does not affectγ (xi, ·);one can select them in order to cancel the distanceρ0 without changing the secondterm, implying that ˜ρ(IN, I ′N) is the minimum of arccos

∑Ni=1〈γ i(xi, 0),Oγ i(xi, 1)〉

whenO varies inSO(d). Whend= 2, there is an explicit solution ˜ρ(IN, I ′N)=arccos|∑N

i=1〈γi (0, xi ), γi (1, xi )〉|.

PROBABILISTIC MEASURES OF VARIATIONAND STATISTICAL INFERENCE

For constructing probability measures of anatomical variation the Grenander schoolcharacterizes shape as Gaussian fields indexed over the manifolds on which the vec-tor fields are defined (1, 2, 64, 118, 118a). Associate with the diffeomorphic mapsg: I→ I ′ the vector fields modulo the identity map as a 3-dimensional Gaussianvector field{U(x),= g(x)− x, x∈M} on the smooth sub-manifold of the full brainvolumeM⊂ X. Expand the Gaussian field using a complete orthonormal basis onthe background spaceM; the U-field becomesU(·)= ∑k ukψk(·), whereuk areindependent Gaussian random variables with fixed means and covariances, andψk a complete orthonormal base. Complete orthonormal bases are constructed viacalculation of empirical covariances and their eigenfunctions.

The Csernansky group has been quantifying the variation of the shape of the hip-pocampus subvolumes in brains via magnetic resonance imagery (61, 62, 119, 120).CA methods have been used to identify deformations in the shape of the hippocam-pus that strongly discriminate subjects with schizophrenia from matched controls(119) and quantify the mildest forms of Alzheimer’s Disease (AD) (120). The



statistical covariation of hippocampus shape from the populations are groupedinto n1 controls andn2 test cases. The maps are represented via then-vectorsof coefficientsU= (u1, . . ., un)t, whereU(x)= ∑n

k=1 ukψk(x). Wang et al. (121)have shown that logistic regressions based onχ2 scoring of the groups of coef-ficients indicate that 3–5 basis functions are sufficient for describing group dif-ferences. TheU1, U2-vectors are modelled as Gaussian, with empirical meansÛ1= 1

n1

∑n1i=1 U1

i ,Û2= 1

n2

∑n2j=1 U2

j , and common covariances:

∑= 1

n1+ n2− 2

(n1∑

i=1

(U1

i − Û1)(

U1i − Û1

)t + n2∑j=1

(U2

j − Û2)(

U2j − Û2

)t

).

(22)

The hypothesis test of the two group means with unknown but common covariancefor the null hypothesis,H0: U1 = U2, has the Hotelling’sT2 statistic givenby T2 = n1n2

n1+n2( Û1 − Û2)T 6−1( Û1 − Û2). The sample meansU1 and Û2 are

therefore normally distributed with meansU1, U2, and common covariance1n16,

with√

n1n2/(n1+ n2)( Û1− Û2) normally distributed, covariance6 under the nullhypothesis. Following (122, p. 109), (n1+ n2−2)6 is distributed as6n1+n2

i=1−2 XiXt

iwhereXi is distributed according toN (0, 6). Thus,T2 has anF distribution, andthe null hypothesisH0 is rejected with a significance levelα if

T2 ≥ (n1+ n2− 2)K

n1+ n2− K − 1F∗K ,n1+ n2− K − 1(α), (23)

where F∗K ,n1+n2−K−1(α) denotes the upper 100α% point of the FK ,n1+n2−K−1

distribution, andK is the total number of basis functions used in calculating theT2 statistics. The log-likelihood ratio for hypothesis testing is

3 = −1

2(Ui − Û schiz)T

∑−1(Ui − Û schiz)

+ 1

2(Ui − Û ctrl)T

∑−1(Ui − Û ctrl). (24)

The vector coefficients represent the coefficients for the principal components usedin the study. UnderH0, H1 the log-likelihood ratio3 (3 < 0,≥ 0, respectively)has sufficient statistic Gaussian distributed with means variancesU0, σ

20 , U1, and

varianceσ 21 , respectively.

Wang et al. (121) have found that scale and volume are not powerful discrim-inants of group difference in the schizophrenic and normal populations; how-ever, shape difference is. Figure 11 examines results from AD, normal aging andschizophrenia. The top row shows the difference of hippocampal surface patternsbetween the controls and targets groups (left: AD, middle: normal aging, right:schizophrenia) visualized asz-scores on the mean surface.

Joshi showed that Fisher’s method of randomization can be used to derive adistribution-free estimate of the level of significance of the difference. For all per-mutations of the given two groups, the means and covariances are calculated from



Monte Carlo simulations generating 10,000 uniformly distributed random per-mutations. The collection ofT2 statistics from each permutation gives rise to anempirical distributionF(·) estimatingF(·) in Equation 23 usingFK ,n1+n2−K−1=n1+ n2− K − 1(n1+ n2− 2)K T2. The null hypothesis that the two groups have equal means isrejected whenp= ∫∞T2 F( f ) df falls below a predefined significance level. TheGaussian assumption for the coefficient vectors are valid since the empirical dis-tribution of theF statistics follows theF-distribution curve.

SUMMARY

This paper reviews recent developments in the formulation of metric spaces forstudying biological shapes in computational anatomy. We expect that such a for-mulation will provide the fundamental basis for future developments in the quan-tification of growth and form.

ACKNOWLEDGMENTS

We would like to acknowledge Richard Rabbitt for introducing the concept offlows. We would like to acknowledge Ulf Grenander and David Mumford for theircontinued involvement in the mathematical formulation of the emerging disci-pline of computational anatomy. We would like to acknowledge Drs. John Cser-nansky and David van Essen of Washington University for their neuro-scientificcollaborations in CA. We would like to acknowledge Faisal Beg and Tilak Rat-nanather for their involvement in the development of the manuscript. This workwas supported by the National Institute of Health grants 5 RO1 MH60974-07, PO1AG03991-16, 1 RO1 MH60883-01, 1 RO1 MH62626-01, P41 RR15241, and 1P20 MH6211300A1 and Office of Naval Research grant N00014-00-1-0327.

The Annual Review of Biomedical Engineeringis online athttp://bioeng.annualreviews.org

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APPENDIX

Proof of Theorem 3.1:To derive Euler-Lagrange equations for the velocity exam-ine perturbations on the group elements and velocity fields,g→ g+ εη, v→ v +εψ . For exact correspondence of the group elements,η(0)= η(1)= 0, as depictedin panel 2 of Figure 1. For inexact image matching, onlyη(0)= 0 with η(x, t)= 0,∀x∈ ∂X, ψ(x, t)= 0, ∀x∈ ∂X. If η is a perturbation ofg, define theGateaux dif-ferentialof E(vg) : G → R+ in the directionη to be the limit, as the perturbationtends to 0. Also ifψ is a perturbation ofv, define theGateaux differentialof E(gv) :V → R+ in the directionψ to be the limit, as the perturbation tends to 0.

Lemma A.1 The variations of the velocity and group element viaψ , η are givenby

ψ(x, t) = ∂ηvg(x, t) = limε→0

1

ε(vg+εη(x, t)− vg(x, t))

= d

dtη(g−1(x, t), t)− Dvg(x, t)η(g−1(x, t), t) (25)

η(x, t) = ∂ψgv(x, t) = limε→0

1

ε(gv+εψ (x, t)− gv(x, t))

= Dgv(x, t)∫ t

0Dgv(x, u)−1ψ(gv(x, u), u) du. (26)

Proof of Lemma: Defining the notationx ' y to mean limε→0x− yε= 0, then

d

dt(g+ εη)(x, t) = vg(g(x, t), t)+ ε d

dtη(x, t)

= vg+εη(g(x, t)+ εη(x, t), t) (27)

(a)' vg(g(x, t), t)+ εDvg(g(x, t), t)η(x, t)+ ε∂ηvg(g(x, t), t), (28)

with (a) following from the definition of∂ηvg, the total derivative and theo(ε)equality. Equating Equations 27 and 28 gives the first result of Equation 25. Equa-tion 26 follows from the linearity ofη givenψ fixed in Equation 25:

∂

∂tη(x, t) = Dv(g(x, t), t)η(x, t)+ ψ(g(x, t), t)

= ∂

∂tDg(x, t)(Dg(x, t))−1η(x, t)+ ψ(g(x, t), t) (29)

since

∂

∂tDg(x, t) = Dv(g(x, t), t)Dg(x, t). (30)

Thenη in Equation 26 satisfies the derivative Equation 29, completing the proofof the Lemma.



Consider the energyE(vg) defined for all time-dependent diffeomorphismsg(·, t)solving dg

dt = v(g, t). Now evaluating the variation ofE(vg)=∫ 1

0 Evg(t) dt withrespect to perturbationsg→ g+ εη gives

∂ηE(vg) = limε→0

1

ε(E(vg+εη)− E(vg))

(a)=∫ 1

0〈∇vE(·, t), d

dtη(g−1(·, t), t)− Dv(·, t)η(g−1(·, t), t))〉 dt (31)

(b)= 〈∇vE(·, t), η(g−1(·, t), t)〉|10−∫ 1

0

⟨d

dt(|Dg(·, t)|∇vE(g(·, t), t)), η(·, t)

⟩dt

−∫ 1

0〈|Dg(·, t)|∇vE(g(·, t), t), Dv(g(·, t), t)η(·, t)〉 dt, (32)

with (a) coming from Equation 25, and (b) the change of variablesy= g(x, t),with integration by parts eliminating the time derivative ofη. The first term is theboundary term which is zero for exact matching. This gives the terms in the innerproduct forming the Euler-Lagrange functional variation:

− d

dt(∇vE(g(x, t), t)|Dg(x, t)|)− (Dv(g(x, t), t))t∇vE(g(x, t), t)|Dg(x, t)|

(33)

= −(∂

∂t∇vE

)(g(x, t), t)|Dg(x, t)| − (D∇vE)(g(x, t), t)v(g(x, t), t)|Dg(x, t)|

− ∂

∂t(log |Dg(x, t)|)∇vE(g(x, t), t)|Dg(x, t)|

− (Dv(g(x, t), t))t∇vE(g(x, t), t)|Dg(x, t)|. (34)

Now the identity for the derivative of the log-determinant is given by log|A+ εB| =log|A| + ε traceA−1B+ o(ε), and the time derivative of the Jacobian∂

∂t Dg(x, t) isgiven by Equation 30 so that

∂

∂tlog |Dg(x, t)| = trace ((Dg(x, t))−1Dv(g(x, t), t)Dg(x, t))

= traceDv(g(x, t), t) = div v(g(x, t), t). (35)

Substituting along withy= g(x, t) into Equation 34 gives the Euler-Lagrangeequation (Equation 4) completing the first part of the proof.

Proof of Theorem 3.3: Define the energyE(gv, J)= ∫ 10(‖v(t) ‖2L + ‖ ∂ J

∂t (t) +∇ Jt (t)v(t) ‖2) dt . This has to be minimized jointly inv andJ. The gradient inv isgiven by∇vE = 2L†Lv + 2( ∂

∂t J(t) + ∇Jt (t)v(t))∇J(t). The nonboundary termfor the perturbation is identical to the above, giving Equation 4. The boundary



condition att= 1 applies sinceg(1) is free at the endpoint, so the value of theperturbationη(·, 1) is nonzero, and the boundary term must be taken into accountfrom Equation 32, giving〈∇vE(1),η(g−1(1), 1)〉=0 so that∇vE(1)= 0. The vari-ation inJ follows by introducingZ= ∂ J

∂t + ∇Jtv, and computing the perturbationby φ with v fixed:

limε→0

E(gv, J + εφ)− E(gv, J)

ε

= 2∫ 1

0

⟨Z(t),

∂

∂tφ(t)+∇φt (t)v(t)

⟩dt

= −2∫ 1

0

⟨∂

∂tZ(t), φ(t)

⟩dt − 2

∫ 1

0〈∇(Z(t)v(t)), φ(t)〉 dt.

Proof of Inexact Image Matching Theorem 4.1:To compute the boundary termEquation 11 which holds for inexact matching at the endpointt= 1, the variationof the inverse map∂ηg−1 in a perturbation is required:

y = (g+ εη)((g+ εη)−1(y, t), t)

' g((g+ εη)−1(y, t), t)+ εη((g+ εη)−1(y, t), t)

= y+ Dg(g−1(y, t))((g+ εη)−1(y, t)− g−1(y, t))+ εη(g−1(y, t), t),

implying

limε→0

1

ε((g+ εη)−1(y, t)− g−1(y, t))

(a)= −Dg−1(y, t)η(g−1(y, t), t), (36)

with (a) fromDg−1v (y, t)= (Dgv(g−1

v (y, t), t))−1. The variation of the second match-ing term becomes

∂η‖I1− I0(g−1)‖2 = −2⟨I1− I0(g−1(1)),∇ I t

0(g−1(1))∂ηg−1(1)

⟩, (37)

(a)= 2⟨I1− I0(g−1(1)),∇ I t

0(g−1(1))Dg−1(1)η(g−1(1), 1)⟩, (38)

(b)= 2〈I1− I0(g−1(1))D(I0(g−1(1)))t , η(g−1(1), 1)〉, (39)

with (a) using Equation 36 and (b) collecting termsD(I0 (g−1(y, 1)))=∇ I t0 (g−1(y,

1))Dg−1(y, 1). At t= 1 withg−1(1) free the boundary term Equation 32 is nonzero;adding it to Equation 39 gives Equation 11.

Proof of Space-Time Growth Theorem 4.2:Substitution ofy= g(x, t) in Equa-tion 34 gives the variation of the‖v(t)‖2L to be:

−∫ 1

0

⟨∂

∂t∇vEv(·, t)+ (Dv(·, t))t∇vEv(·, t)+ (D∇vEv(·, t))v(·, t)

+ div v(·, t)∇vEv(·, t), η(g−1(t), t)

⟩dt. (40)



The variation of the second term under the perturbation ofg by η follows fromusing Equation 36:

−2∫ 1

0

⟨(I1(t)− I0(g−1(t)),∇ I t

0(g−1(t))∂ηg−1(t)

⟩dt (41)

= 2∫ 1

0

⟨(I1(t)− I0(g−1((t), t),∇ I t

0(g−1(t))Dg−1(t)η(g−1(t), t)⟩dt (42)

(a)= 2∫ 1

0〈(I1(t)− I0(g−1(t))D(I0(g−1(t)))t , η(g−1(t), t)〉 dt, (43)

with (a) writing D(I0(g−1(t)))=∇ I t0(g−1(t))Dg−1(t). Collecting the two compo-

nents of the gradient term from Equations 40 and 43 gives the proof.

Proof of Theorem 6.2: Define normalized landmarksγ (xn, t)= g(xn,t)−g(t)σ (g(t)) ,

g(xn, t)= σ (g(t)) γ (xn, t) + g(t), then

∂g

∂t(xn, t) = dσ (g(t))

dtγ (xn, t)+ σ (g(t))

∂γ

∂t(xn, t)+ dg

dt, (44)

implying∫ 1

0 ‖ ∂g(t)∂t ‖2Q(g(t)) dt equals

N∑n=1

(∫ 1

0

1

σ 2(g(t))

(dσ (g(t)

dt

)2

|γ (xn, t)|2 dt

+∫ 1

0

1

σ 2(g(t))

dg

dt

2

dt +∫ 1

0

∥∥∥∥∂γ∂t(xn, t)

∥∥∥∥2

dt

+ 2∫ 1

0

1

σ (g(t))

dσ (g(t)

dtγ (xn, t)

dγ (xn, t)

dt+ 2

∫ 1

0

1

σ (g(t))2

dg(t)

dtγ (xn, t) dt

+ 2∫ 1

0

1

σ (g(t))g(t)

N∑n=1

dγ (xn, t)

dtdt

)

(a)=∫ 1

0

1

σ 2(g(t))

(dσ (g(t)

dt

)2

dt + N∫ 1

0

1

σ 2(g(t))

dg

dt

2

dt

+∫ 1

0

N∑n=1

∥∥∥∥dγ

dt(xn, t)

∥∥∥∥2

dt, (45)

with (a) following from‖γ (t)‖2= 1, implying the 4th termddt‖γ (t)‖2 = 0, the

5th term has mean zero, and the final term is the derivative of the mean, whichis zero. Denote byρ0((σ (g(0)), g(0)), (σ (g(1)), g(1)))2 the minimum of the firsttwo terms; the minimum of the last term can be explicitly computed (because(γ (., x1), . . ., γ (·, xN)) and belongs to a sphere of dimensionN− 2 and is given by



the length of the great circle in the sphere which connects the extremities of thepath, namely (arccos

∑Nn=1〈γ (0, xn), γ (1, xn)〉)2 so,

ρ(IN, I ′N)2 = ρ0((σ (g(0)), g(0)), (σ (g(1)), g(1)))2

+(

arccosN∑

n=1

〈γ (0, xn), γ (1, xn)〉Rd

)2

. (46)

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stgr

owth

isfo

und

cons

iste

ntly

,acr

oss

ages

7–13

,in

the

isth

mus

,are

gion

ofth

eco

rpus

callo

sum

that

carr

ies

fiber

sto

area

sof

the

cere

bral

cort

exth

atsu

ppor

tla

ngua

gefu

nctio

n,an

dar

eas

ofth

ete

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ro-p

arie

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orte

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tass

ocia

tive

and

mat

hem

atic

alth

inki

ng.R

esul

tsta

ken

from

Tho

mps

onet

al.(

97).


Fig

ure

8(T

op

row)

Pan

els

1–2

show

the

two

test

patte

rns

and

the

mat

chin

g(p

anel

s3–

4)us

ing

varia

nces

ofσ

2=

0.1,

10,

resp

ectiv

ely.

Res

ults

from

(69)

.(Bo

tto

mro

w)P

anel

5sh

ows

sphe

rical

defo

rmat

ions

driv

enby

thre

ela

ndm

arks

alon

gth

elo

ngitu

delin

eθ=π/3

map

ped

toth

eco

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ong

the

line

θ=π/4.

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ice

the

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ract

ion

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elo

ngitu

delin

esbe

twee

nθ=π/3

andθ=π/4.

Pan

el6

show

sa

sinu

soid

ofla

ndm

arks

onth

esp

here

with

the

resu

lting

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rmat

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nin

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otic

eth

edr

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form

atio

nof

sphe

rical

coor

dina

tes.

Res

ults

take

nfr

omB

akirc

iogl

uet

al.(

70).


Figure 9 Two columns show the Van Essen surface-based atlas of macaque visualcortex. Left column shows the macaque atlas (top row) and flattened version (bottomrow). PanelA shows the flat map. Shading indicates cortical geography, black linesindicate the landmark contours used to constrain the deformation (along sulcal fundiand along the map perimeter). PanelB shows the pattern of grid lines after deformationinto register with the target atlas map in panelD. PanelC shows the vector field forselected grid points (at intervals of 10 map-mm). Arrow bases indicate grid positionsin the source map, and arrow tips indicate the location of grid points in the deformedsource map. PanelD shows the map of geography (shading) and target registrationcontours (black lines) on the atlas map. PanelE shows contours from the deformedsource map (black lines). Results from (100).

Figure 10 (Top row) Panel 1 shows 3-D Visible Human Male, panel 2 shows the land-marks on macaque flat maps, panel 3 shows the landmarks on the human flat maps, andpanel 4 shows the boundaries of deformed macaque visual areas (black lines) superim-posed on the fMRI activation pattern from an attentional task from Corbetta et al. (117)after deformation to the Visible Man atlas. (Bottom row) Panel 5 shows the sphericalmap of the macaque visual cortex. Panel 6 shows the spherical map of the deformedmacaque visual areas, along with the deformed latitude and longitude isocontours.Panel 7 shows the deformed macaque visual areas with the latitude and longitude linesof the Visible Human spherical map. Results taken from Van Essen et al. (100).


Fig

ure

11To

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ace

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001

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df=

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anel

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Josh

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ian

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valid

.

o metrics and euler-lagrange quations of …alain trouve´ universit´e paris 13, institut galil ee,...

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