christens en 1996 deform able templates using large deformation kinematics

8/8/2019 Christens en 1996 Deform Able Templates Using Large Deformation Kinematics

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 5, NO. 10, OCTOBER 1996 1435

Deformable Templates UsingLarge Deformation KinematicsGary E. Christensen, Member, ZEEE, Richard D. Rabbitt, and Michael I. Miller, Senior Member, ZEEE

Abstruct- A general automatic approach is presented foraccommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template imageinto alignment with a topologically similar target image. Localshape variability is accommodated by applying a vector-fieldtransformation to the underlying material coordinate system ofthe template while constraining the transformation to be smooth(globally positive definite Jacobian). Smoothness is guaranteedwithout specifically penalizing large-magnitude deformations ofsmall subvolumes by constraining the transformation on thebasis of a Stokesian limit of the fluid-dynamical Navier-Stokesequations. This differs fundamentally from quadratic penaltymethods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxesover time allowing large-magnitude deformations. Kinematicnonlinearities are inherently necessary to maintain continuityof structures during large-magnitude deformations, and areincluded in all results. After initial global registration, finalmappings are obtained by numerically solving a set of nonlinearpartial differential equations associated with the constrainedoptimization problem. Automatic regridding is performedby propagating templates as the nonlinear transformationsevaluated on a finite lattice become singular. Application of themethod to intersubject registration of neuroanatomicalstructuresillustrates the ability to account for local anatomical variability.

I. INTRODUCTION0 meet the greater challenges of observing and trackingT lobal biological shapes and structures, mathematicalrepresentations began to appear in the early 1980s under the

name of Grenandersglobal shape models. Applications haveappeared in human hands, cellular organelles, leaves, tracks,brains, and amoebas [11-[ 101. The global shape models repre-sent image ensembles via the construction and deformationof typical image templates. Individual variabilities associ-ated with the shapes of constituent biological structures aredescribed using probabilistically constrained transformationsapplied to the templates. The transformations generate a richfamily of shapes from a single template while maintaining its

Manuscript received October 25, 199 4; revised November 12, 1995. Thiswork was supported in part by the N SF under Presidential Young InvestigatorAward 8957206, the NIH under Grants R01-MH52158 and NIH-NCRR-RR01380, and by a grant from the Whitaker Foundation. The associate editorcoordinating the review of this manuscript and approving it for publicationwas Prof. John J. Weng.G. E. Christensen is with the Mallinckrodt Institute of Radiology, Wash-ington University School of Medicine, St. Louis, MO 63110 USA (e-mail:gec@ee. wustl.edu).R. D. Rabbitt is with the Department of Bioengineering, University of Utah,Salt Lake City, UT 84112 USA (e-mail: [email protected]).M . I. Miller is with the Department of Electrical Engineering, WashingtonUniversity, St. Louis, MO 63130 USA (e-mail: [email protected]).Publisher Item Identifier S 1057-7149(96)06029-0.

global properties. The ability to maintain, track, and recognizeglobal structures in the presence of relatively large local imagevariability is one of the major advances of these approachesover conventional pixel-based imaging paradigms.

It is well known that biological structures, such as humanneuroanatomies, vary in both global and local shape across apopulation. Although image volumes of two different normalbrains may contain the same global structures-white mat-ter, gray matter, ventricles, etc.-the internal structures willdiffer in shape, orientation, and fine structure. Our goal isto develop a method to represent the variabilities associatedwith individual neuroanatomies. The approach is to define theanatomical template as a three-dimensional (3-D) deformablecontinuum, with a deformation vector field applied throughoutthe continuum to provide a map from the individual anatomyback to the standard brain template. The vector field specifiesthe transformation.

We follow the approach defined in [6] and [111 to repre-sent human neuroanatomies via deformable templates. Thetemplate is essentially an electronic atlas or anatomy atlasand individual variation is generated by deformation of theatlas coordinate system. The anatomical template is a vectorfunction on the spatial domain Q c X3 where R is oftentaken to be the unit cube. Throughout, each template withinthe atlas is assumed to be an N (typically 256) gray-level spacecorresponding to one or several imaging modalities. Variation,s accommodated via the set of space-time transformationsh(., .) : 0 x [0 , T ]4 Q. These transformations are generatedfrom translations applied to the continuum evaluated on allpoints in Q. The time-dependent displacement field C(Z, t )is defined in terms of L(l,) by the expression z ( Z , t ) =2 - G(l, ) . The template is applied to individual subjectsor anatomies by constructing the time-dependent (real orcomputational) map from the data volume { S ( Z ) , C E R}back to points in the template to register the data with theatlas. The atlas and study (i.e., data) are said to be registeredwhen the transformation is found that minimizes a consistentnonnegative distance measure C ( , .) between the deformedatlas T [ Z- C(Z)] and the data S(a), ubject to constraintsimposed by mechanics.

In order to represent complex anatomical patterns such asventricles, cortical folds, etc., the transformations are repre-sented by vector fields. Because these vector fields are infinitedimensional, regularization is required to ensure preservationof various topological properties of the mapping [12]. Suchproperties include preservation of landmarks or local maximain the atlas images or connectedness of structures that would

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1436 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 5, NO . 10, OCTOBER 1996

be associated with a positive Jacobian, and other smoothnessfeatures such as continuity of the mapping.

Our approach herein is to derive partial differential equa-tions (PDEs) that the transformation fields must satisfy. ThePDEs are crafted using principles obtained directly fromcontinuum mechanics for deformable bodies. The deformationprocess does not attempt to model the physical process of braingrowth; we use mechanical analogies to enforce topologicalproperties that adhere to large deformation kinematics. Towardthis end, the classical conservation equations of mechan-ics have been modified to account for nonmass-conservingdeformations. This allows for the growth and shrinkage ofregions within the continuum. When combined with the var-ious constitutive laws (which have -been well studied incontinuum mechanics), we are provided with a mathematicalframework that constrains our deformations with smoothnessproperties of which we have complete control. For example,under circumstances in which we do not choose to force thedeformed template to stay close to its original configuration,the restricting force associated with large deformations can berelaxed while still maintaining smoothness of the deformationfield. This PDE formulation is consistent with our previouswork [2], [6], [ l l ] based on Bayesian estimation by viewingthe PDE as the minimizer of a posterior distribution (see thefollowing section on choosing the body force).

A . Relationship to Previous WorkThere has been a great deal of work done on multimodal-

ity image fusion, warping, and registration in a deformableanatomy context (see [13]-[15], for example). For much ofthe neuroanatomical work, the anatomical brain images beingmatched are assumed to be highly similar, requiring onlyglobal course features to be matched with rigid transformations(global rotation and translation) and/or simple scaling tomatch surface features andor boundaries. For such cases,investigators often define a small number of features, surfaceor internal fiducials, and/or landmarks that serve to drivethe registration, with high-dimensional transformations notrequired. The kinematics of the continuum and the associatedPDEs that are proposed herein do not play a significantrole for such transformations. Such low-dimensional rigidmotions and or scale transformations are straightforwardlyaccommodated in our method as initial conditions.

Although the goal of image understanding is similar, thedeformable template in this work differs fundamentally fromdeformable contour shape models such as those used in[16]-[18] and the deformable surface shape models such asthose in [9] and [19]. In the previous approaches, deformablemodels correspond to parameterized contours and surfaces in a2-D or 3-D image. These tend to be low-dimensional models,the dimension determined by number of control points on thecurves or surfaces. The work proposed here requires volumetransformations proportional to the number of pixels in 2-D orvoxels in 3-D of the images being deformed.

The current work is perhaps most akin to the pioneeringwork in surface [20], [21] and solid deformation [22], [23],in which transformations are constructed to obey physical

laws. The computerized surface models of facial motion applylocal viscous damping. The deformable solid work for brainmatching uses linear elastic penalties. For all such approaches,quadratic penalties derived for beams or solids are always usedand the nonlinear kinematics of large deformations requiredto achieve a homeomorphism is ignored. Previous methods[2], [6], [24], [25] are limited in the sense already described.First, large magnitude displacements are severely penalized byregularization methods such as those based on linear elasticityor thin plates that develop restoring forces monotonicallyincreasing with strain. Except for the smallest deformations,such penalties prevent full deformation of the template intothe data. Secondly, these models are derived with smalldeformation approximations [26], [27] and are therefore notvalid for large distance, nonlinear deformations. As we shalldemonstrate, the transformations under these linear modelsdo not enforce such physical properties such as 1-1 andsmoothness of the transformation.

The contribution of this work is a new general approachfor accommodating the large-distance, nonlinear kinematicsrequired for transforming a template image into a target imagewhen refining an initial global transformation. This is achievedby modeling the template as a highly viscous fluid allowing forlarge-magnitude, nonlinear deformations of small subvolumes.The resulting optimization problem is solved on an adaptivecomputational mesh associated with the propagated template.This work builds on and extends on previous work reportedin [6], in which we describe an automatic, multiresolution ap-proach for accommodating complex shape variation based ona linear elastic model. Some preliminary results of the currentapproach were provided in [8] for comparison to 3-D elasticdeformations of neuroanatomical template images. The presentwork completes the presentation of the theory and providesdetails of the massively parallel computer implementation.

11. DILATATIONAL-VISCOUSLUID PDEA . Eulerian Reference Frame and a Frameworkfor Large Deformations

In describing large distance deformations, we use the Euler-ian reference frame, as depicted in Fig. 1.The displacementfield U(., t ) s defined as a map from points in the templateto fixed observation points in the deforming continuum, i.e., amass particle instantaneously located at 2 t time t originatedat point 2 - C ( 2 , t ) . The Eulerian description of materialdeformations specifies the time evolution of particle positionsand velocities as observed at fixed points. Because the dataS ( Z ) s constant over the minimization procedure and the factthat it is stored as discrete voxels, U(., t ) is parameterizedby its spatial samples located at the center coordinate of eachvoxel of S(2). patial tracking of the transformation in thismanner allows very fine (on the order of voxel volumes)resolution structural variability to be accommodated.

In the Eulerian frame, accounting for the difference betweenthe velocity v and the time rate of change of the deformationU is critical to using a fixed spatial grid to track the largedeformation of templates. The material derivative, denoted


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Fixed ReferenceFrame

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Point inthe Deformed

k

Point in theTemplateFig. 1. Eulerian frame showing coordinate system used to track the kinemat-ic s of the deformation. In the Eulerian frame, vector and scalar field variablesare associated with fixed spatial positions through which p articles move ratherthan with moving material particles. Th e point ?-.(., t z ) enotes the particlefrom the template which is resident at location .' at time t , .

d / d t , provides the time rate of change experienced by anelement of material instantaneously at point Z at time tand is defined by d / d t = a/& + v ; ( d / d x ; ) wherev' = [v1(5, t ) w3(?, t)lT.Using this, the velocityof an element of mass passing through 5 t time t is

d i id t

Q(Z, t )

'U=-diiu'

d t d X ;3=-+cvi-

i= lwhere u'= [u1(5,) us(?, )]'. The summationterm in (1) accounts for the kinematic nonlinearities of thedisplacement field U'. Note that for small deformations the totalmaterial derivative with respect to time t and partial derivativewith respect to time t are approximately equal.

u2(Z, )

B. Consewation o MomentumWe now derive the conservation of momentum equation,

which gives the relationship between forces applied to thecontinuum and the resulting deformation. Consider a fixedcontrol volume 0 with boundary dR and assume that a sourcesupplies or extracts mass from the volume at a rate of 7 per unitvolume. Distributed traction force 7 acts on the surface, anda distributed body force per unit volume acts on all materialelements within the body. Using i) Newton's second law inthe form of conservation of linear momentum, ii) the fact thatthe traction force 7' is related to the Cauchy stress tensor U by7 = T.n', where n' is the outward unit normal from the surface,iii) the divergence theorem, and iv) the Reynolds transporttheorem, modified to include a mass source (see Appendix

Fig. 2. The left column shows the template (top) and the study (bottom). Thetemplate was deformed into the study using an elastic-solid regularization (to pright) and a viscous-fluid regularization (bottom right).

A) , gives the differential form of conservation of momentum(2)

where p is the density of the material. The notation . Uis defined to be a 3 x 1 column vector with components[aU]i = E,"=,d T , j / & j ) where the components of T re[U]i j = Ti? for i, j =1, 2, 3.

The momentum equation provides the relationship betweenapplied body force g, the state of stress, and the resultingmaterial deformation. In the current approach, the body forceb' provides the connection between the sensor data and thedeforming continua. The final form of the momentum equation(2) and associated generalized Lagrangian depends upon theconstitutive behavior of the material, i.e., elastic, visco-elastic,Kelvin-Voight, etc.

-+v'd t- + v'q- V . U - 6= '.

C. The Fluid AnalogWe choose a constitutive law that allows for large, nonlinear

deformations while maintaining a continuous homeomorphicmap with smooth deformations of the template, This is analternative constitutive law to Hooke's law used in linear elas-ticity, one corresponding to a viscous-fluid with the propertythat the stress within the deformed configuration is allowedto relax with time. This implies that the mechanical energy(penalty) does not (necessarily) increase with the magnitudeof the pointwise deformation. A Navier-Poisson Newtonianfluid model is used, for which the Cauchy stress tensor Uis related to the rate of deformation tensor D by U =[X(trD) - p ] f + 2pD, where p and X are the viscosityconstants, p is F e presyre, and the rate of deformation tensoris D = l/Z[VG + (VG)']. Making these substitutions in(2) gives the modified Navier-Stokes equation p V 2 G+ ( A +p)$($. G)+G(u' ) = $p+p(dv'/dt)+v'q. A simplified modelis obtained for v ey low Reynold's number flow where thepressure gradient V p nd the inertial terms p ( d C / d t ) + 7q are


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neglected, such that (2) becomespv2v+ (A +,)a($.)+ G(G) = 0 (3)

where V2 = V TV is the Laplacian operator. Without loss ofgenerality, we assume that (3) is defined on R = [O , lI3 ={the unit cube} with the boundary conditions G(Z, t )= 0 forZ E d R and all t . The notation dR s defined as the boundaryof R . Notice that when these boundary conditions are insertedinto (l), he displacement on the boundary of R s zero, i.e.,8R s mapped to dR under the fluid transformation.

The template is deformed by solving (3) subject to a bodyforce that drives the template viscously into registrationwith the study. Two mechanisms are clearly identified. Thefirst, represented by pV2G, is associated with constant volumeviscous flow of the template. The second, represented by( A +p )$(e.>, is nonzero when regions of the template growo,r dissolve. Using continuity [(14), Appendix A] we note thatV .v = q / p - d p / d t , so the second term in (3) is related todilatation caused by the mass injection per unit volume 77 andthe rate of change of density d p l d t . In the present work, weare specifically interested in allowing growth and shrinkageof local regions of the template and hence . G in generalis nonzero. For the Navier-Poisson fluid employed here, thedilatational viscous stress resisting the rate of change of localvolume is controlled by [A + 2 / 3 ) p ] ,whereas the shear stressbetween adjacent regions is controlled only by p.D. Th e Body Force

A crucial part of the specification of the PDE in (2) is theforce that drives the deformation of the template to match theconfiguration of the data. This force is induced by adoptingthe Bayesian view of the transformations, in which there existsa prior distribution on the transformations and the solutioncorresponds to a maximization of the posterior distributionrelating the transformation fields to the data as collected bythe sensor. The posterior is taken in Gibbs form with the totalpotential associated with the sum of the statistical model ofthe sensor and the prior distribution. The PDE formulationis then the variational maximizer (minimizer) of the Gibbspotential. The forcing function in the PDE is then none otherthan the variation of the sensor statistical likelihood functionwith respect to the vector displacement field [2], [6], [l l] .

This view is based on the existence of a variational principlefrom which the field equations can be induced; such principlesare well known for conservative systems [28], [29] and link thePDE formulation as the minimizer (maximizer) of some poten-tial (see Section V). Variational formulations have also beendeveloped for numerous nonconservative systems, including afluid model consistent with that addressed herein [30].

In this work, we use a Gaussian sensor model (likelihood)C ( T ( 4 , G ( Z , t ) )= - l T ( Z - G ( 2 , t ) )- S( Z ) I 2d Z (4);so

where a is a constant. For magnetic resonance imaging (MRI)this appears to be reasonably appropriate [31]; for positronemission tomography (PET) and charge coupled device (CCD)

h100 h

hhhA4BPk

CAa,Xa3

- 3

5 0 -

---L

k L - * - - y Y P - / P t A L0 1 1 1 1 1 1 1 1 I0 50 100

x pixelsFig. 3 . Vector map of the viscous fluid transformation for the wedge-shapedpatch to C experiment. The vectors shown here are only a subset of the full12 8 x 128 vector field estimated in this experiment.

appropriate [32], [33]. Others have investigated using thecorrelation [23]. Taking the variation of this cost function withrespect to the displacement field yields the body force

tG [ Z , G ( Z , t ) ]= -a(??[& G(Z, ) ]- S (Z ) ) v q - q~ , q .5 )The body force in ( 5 ) is composed of two terms. The

term $5!13-q~,~) , nterpreted as $ E T ( [ ) valuated at [ =Z - G(Z, ) , s the gradient of the template and has largestvalues at the edges of the structures in the template. Thisterm determines the directions of the local deformation forcesapplied to the template. The second term T ( Z - G ( Z , t ) ) S ( Z )is the difference in intensity between the deformed templateand the study. This term locally weights the first term. Itis large when there is a large intensity mismatch and smallotherwise. Notice that this second term causes the body forceto be locally zero in areas where the deformed template islocally aligned with the study. The body force is globally zerowhen the deformed template and study are globally aligned.

The nonlinear deformation force given in ( 5 ) is evaluateddirectly and is not approximated. In contrast, others assumesmall deformations [23], [36] and linearize the body forceusing a Taylor series approximation. But since the motivationof the work presented herein is to allow large distance,nonlinear deformations, the small deformation assumption isnot valid and linearization is not applicable.

Although the body force in ( 5 ) is a local operator, itgenerates the long distance forces required for deforming thetemplate because the deforming continuum is coupled togetherby (3). The nonlinear dependence of the body force on U isrequired so that the body force remains consistent with the

data image reconstruction, Poisson models would be more deformed template


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111. NUMERICAL OLUTIONThe solution of the viscous-fluid PDE given by ( l ) , 3 ) ,and( 5 ) includes nonlinearities associated with the body force and

kinematic nonlinearities associated with the material deriva-tive. In order to solve these nonlinear equations, the algorithmemploys Euler integration in time combined with successiveoverrelaxation in the spatial domain. The nonlinear PDE isdecomposed into a set of linear equations on the velocity fieldU(Z, t ) for each instant of time t , and then integrated usingthe material derivative nonlinear relationship between the Uand U fields.Algorithm I :

Initialize t = 0 and G(Z, 0 ) = 0.Using (9, alculate the body force ;(a, C(Z, ) )givenIf b(2,C(2, t ) ) s below a threshold for all 2 R or themaximum number of iterations is reached, then STOP.Solve the linear PDE (3) for the instantaneous velocity.((., t ) at time t assuming that .(Z, t ) = 0, 2 E dR.This is done using successive overrelaxation (SOR) [8].Perform explicit Euler integration by calculatingC(Z, t + A) from the discretized version of thematerial derivative (1) given by C ( Z , t + A ) =Let t = t + A and go to Step 2.

qg, ) .

q.,)+A q Z , t )- A zI;(Z, t ) [ d C ( 2 , ) / a x ; ] .Notice in Step 4 that the nonlinearity introduced by the

body force term in (3) is handled by fixing g(2, T(., t ) )attime t. Since ;(a, C(2, t ) ) s fixed for time t , solving for theinstantaneous velocity is a linear problem. Also note in Step 4that the assumption the velocity is zero on the boundary of Rimplies that the displacement field is also zero on the boundaryof 0. This assumption ensures that boundary of the deformeddomain maps back onto the boundary of the original domainR through the identity mapping.

A. Solution on a Finite Spatial GridThe viscous fluid PDE is solved on a discrete lattice

associated with 0.For large curved deformations, the transfor-mation evaluated at these spatial grid points becomes singularover time even though the transformation evaluated on thecontinuum would not. To circumvent this problem, a methodof regridding the template is used by generating a new, orpropagated template, whenever the discretized transformationapproaches local singularity. This is accomplished as follows.

The discretized displacement field is propagated throughtime using the procedure in the previous section until the Jaco-bian of the transformation indicates that the transformation isapproaching singularity. When the magnitude of the Jacobiandrops below a certain value, the computation is stopped and anew propagated template is generated equal to the deformedtemplate at the previous instant. In the current implemen-tation, a new template was propagated if the magnitude ofthe Jacobian dropped below a value of 0.5. A larger valuecauses more templates to be propagated, which will increasenumerical precision errors of the transformation due to theincreased number of concatenated transformations that need to

be interpolated to obtain the total transformation. Conversely,a smaller propagation threshold causes less templates to bepropagated. This may increase the numerical precision errorsdue to interpolating transformations that are nearly singularin places. The algorithm is restarted using the new template.When mapped back to the original template, this is equivalentto defining a new nonhomogeneous computation grid on theundeformed continua. Initial conditions for the propagatedtemplate are set to match the final state of the previoustemplate, i.e., the instantaneous velocity remains the same andthe displacement field corresponding to the new template is setto zero. The total transformation is continuously tracked viathe concatenation of the displacement fields associated withall of the propagated templates.

We note that the above procedure assures that the concate-nated transformation of the template into the study preservesthe topology of the template. This is because each of the prop-agated template transformations preserve topology (due to thefact that the Jacobian of the transformation is positive globally)and the concatenation of topology-preserving transformationsproduces a topology-preserving transformation.

The method of propagating templates (i.e., regridding) isformalized in the following way. Let T ( j ) , = 0, 1, 2 . . .denote the sequence of propagated templat2s with associateddisplacement field fi(j), so that T ( j ) ( Z U ( j I ( 2 , i ) ) s thedeformation of the jt h template. Next, define t,, as the timethat template T ( j - l ) is propagated to template T ( j ) uch thatt,, = t o = 0 < t,, < . . . < t p M .ach propagated templateT(j)-except T(O),which is equal to the original templateT-is defined as the deformation of the previous template~ ( j - l )t time t p 3 , s follows:

for 0 5 i 5 N where pa and r; a E,=,j(Z, ;)(a/axj).This definition ensures that each propagated displacement fieldsatisfies the discrete apjroximation of the material derivative(1) and that T ( j ) ( Z - U ( j ) ( Z , p , ) )= T ( j ) ( Z ) t propagationtime t p , The displacement field i2 associated with the originaltempla? T is defined in terms of the propagated displacementfields U( j )by

Aq.,i ) =8 0 ) 2,i ) , O l i l P l ,fi(j)(,jZ,t ; )

+ f i ( j - 1 ) ( 2 - fiq.,i ) , ,,), p j < 2 5 pj+1.(8)


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Equation (8) is a consistency condition ensuring T(O)(2i 5 p l and that T ( j ) ( 2 6( j ) (2 ,i ) )= T(Z- 6( j ) (2 ,i )-p j + l . Incorporating (6)-(8) into Algorithm 1 gives the follow-ing algorithm.Algorithm 2:

.!7(')(2,i ) ) = T ( 2- m ( Z , t i ) )= T ( 2- C(2, i ) ) ,0 56(j-l)(z 9 q 2 , i), t p J )= T ( 2- C(Z, i ) ) , j < i I

+Let t = 0, i = 0, T( ' ) (2) T ( Z ) , nd U(' ) (? , 0) = 0.Calculate the body force $(Z,i(Z)(Z, t ) )= -(T(i )[Z-If g(Z, G ( i ) ( Z , ) ) s below a threshold for all 2 E Ror the maximum number of iterations is reached, thenSTOP.Solve the linear PDE (3) for the instantaneous velocityv'(Z, t ) at time t assuming that v'(2, ) = 0, 2 E 80using SOR [SI.Calculate the perturbation of the displacement field

Choose a real-time step size A which is a function ofIf the Jacobian of Z - U ( i ) ( Z , )- AZ(Z) is less than0.5, then propagate to template i +1 using T(i+l)(Ic)T ( ~ ) ( zCf(i)(2,) ) ,6('+')(~,) = 0, set i = i + 1,and go to Step 5. Otherwise, uptate the ith displacementfield using 6( i ) (Z , + A) = U ( i ) ( Z , )+ A g ( Z ) , sett = t + A, and go to Step 2.

6 ( i ) ( Z , ) ]- S(Z))?T(i)I,--~~~~(,-,~).

A(?) = v'(2, )- E&, Ui(2, ) ai3"(2, ) / a X ; ] .Y = max,-n l l ~ (Z ) l l . +

The next theorem states that under sufficiently smooth con-ditions on the transformation and velocity fields, the solutiongenerated by Algorithm 2 solves the fluid PDE given by (l),(3), and (5) as t;+l - t; goes to zero for all 0 5 i < N .This, of course, proves that the regridding method allowslarge curved deformations to be calculated on a discrete gridwhile maintaining a diffeomorphism (i.e., avoiding a singulartramformation).

Theorem 3.1: Let 6(j) C2(!2) nd v' 6 C ' ( 0 ) whereR C R3. Then Algorithm 2 satisfies the fluid PDE givenby (l), ( 3 ) , and (5) in the sense that for times ti witht o = 0 < tl < . . . < tN < T,,,, the following systemof equations are satisfied:

I O , i = O

Awhere ri = E:=,j(Z,i ) (a /ax j ) or 0I5 N .Pro08 Equation (9) is satisfied by Algorithm 2 if thereal-time integration (10) is satisfied. Thus, it is only necessaryto show that the real-time integration of Algorithm 2 given by(7) and (8) implies (10). We now show by induction that (7)and (8) satisfy (10).Basis: Substituting (7) into (8) gives (10) for times t o =t,, Ii Ip l .

Induction: Assume that Algorithm 2 satisfies (10) for timet; and show that (10) is satisfied for t;+l.Substituting (7) into(8) gives

G(2, i+ l )= W ( Z , i )+ ( ti+l - i ).[qz,i)- r i i P ( z , i)]+ Cf(j-l)(2- d q 2 , i )-. [q z , i)- ri6(j)(z,i)], p , ) .- i )

Using the fact that 8 - l ) E Cz(R), e can write the Taylorseries expansion of 8 j - l ) about the point 2 (2,; ) sG(Z, +I ) =6 q Z , i )+ (ti+l- i)

. [ v ' ( ~ ,i) rizW(z, i)]+ 8 - 1 ) [2 6(j)2,i), ,J- ( t .+ l - qGC ( j - 1 ) [Z-O(d(Z, ti), P3]

. [I- V U-(qZ,t


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The background in all of the experiments was assumed to beblack (intensity zero). This requirement is due to the fact thatthe velocity of the displacement field along the boundary ofthe domain is zero in (3). Thus, the displacement along theboundary of the domain is zero.

Appropriate values of the viscosity coefficients X and pdepend upon the particular imaging modality. Combining (3)and ( 5 ) , we find the parameters X/a and p / a have units ofseconds where we have assumed that the image data isnormalized and nondimensional. For the present numericalresults we take X/a = 0.0 and p / a = 0.01 s. No attempthas been made to optimize these parameters or to study theinfluence of inhomogeneous properties on the results. Thesestudies are left for future work.

A. The Patch to C ExperimentThe first experiment shows how a small patch of material

can deform into a longer curved patch of the same material.Fig. 2 shows a wedge-shaped template (top left) and a C-shaped study (bottom left) used in this experiment. Thedimension of these images is 128x128 pixels. The templatewas initially aligned with the study so that they overlap. Ifthis initial alignment was not performed, the algorithm wouldstick in a local minimum with the patch shrinking into asmall point in order to minimize the cost (4).

Shown in the top right panel is the result of deforming thetemplate into the study using the linear-elastic regularization(see Section V for details). Notice that the linear-elasticregularization method prevented the template from deforminginto the study. The bottom right panel shows the result ofthe viscous-fluid regularization, which allows the templateto more completely match the study. The difference in areaof these objects is accommodated by the algorithm becauseconservation of mass is not enforced. The dilation of thematerial can be measured by evaluating the mass source termin (3).

A vector representation of the displacement field for thefluid transformation is shown in Fig. 3. Each displacementvector points from the original location of a material particleto its final location. Notice that the displacement vectorsterminate on a regular grid but point to the very irregular setof corresponding locations in the textbook. This illustrates thatthe Eulerian framework for the displacement field provides themapping from the study back to the original textbook.

The method of propagating templates used for solving thefluid PDE on a discrete grid is illustrated in Fig. 4(cf., Section111). The top row shows propagated templates T(4),T(12),T ( l S ) ,nd T(30)from left to right) that were used to deformthe wedgeLshaped patch into the C. The second row of thisfigure illustrates the transformation applied to a rectangulargrid for each propagated template. Notice that because the fluidtransformation is continuous and globally one-to-one that thegrid lines are continuous and do not cross over one another.The nonuniformity in the thickness of the deformed grid linesresults from the fact that the original grid is expanded andcompressed during the transformation. Notice the extent ofthe curved grid lines. This solution emphasizes the nature

of the kinematics as manifest via the material derivative foraccommodating nonlinear kinematic trajectories.

The third row of Fig. 4 shows the Jacobians of the trans-formations3- i(-)(Z, tp ,)hat were used to determine thepropagation times tp t for i = 4, 12, 18, and 30. Because theJacobian of each propagated transformation3- i(-)(Z, t,,)is greater than 0.5 for all x E 0, the concatenation of thepropagated transformations is continuous and globally one-to-one. The bottom row of Fig. 4 shows the Jacobian of thetransformation h ( Z , t p , ) = Z - C ( Z , tp , )evaluated at thepropagation times i = 4, 12, 18, and 30. Notice that theJacobian of z(Z, tp , ) s the Jacobian of the composition ofthe first i - 1 propagated transformations, corresponding tothe product of the Jacobians of the first i - 1 propagatedtransformations.

+

B. 2 - 0 Macaque Monkey C ryosection Data ExperimentThe fluid transformation accommodates variation of struc-

tures with length scales on the order of the image resolution.This is precisely the kind of detail and accommodation ofvariability that we require for the cryosection brain data shownin Fig. 5.The top-left and top-middle panels of Fig. 5 showoptical cryosection images of the visual cortex of two differentmacaque monkey brain slices. This cryosection data is atapproximately l o x the resolution of MRI data and it allowsus to see the exquisite detail of the cortical mantle. This isillustrated via the dark gray folds interfacing to the whitematter.

The result of using the fluid model to deform the templateinto the study is shown in the top-right panel. The templatewas first globally deformed into the study using the elastic-ity basis from [6]. Subsequent to the basis transformation,the fluid transformation was applied. Notice the similarityof the target (middle panel) and the transformed template(right panel). The left panel in the middle row shows theresult of smoothly deforming a rectangular grid using thisdeformation. The 2- and y-components of the displacementfield are shown in the middle-middle and right-middle panels,respectively. Intensities that are close to white correspond topositive displacements; dark intensities correspond to negativedisplacements. The low spatial frequency intensity changes indisplacements cause global shape changes and resulted fromthe elastic-solid transformation. The high spatial frequencyintensity changes correspond to local shape changes resultingfrom the fluid transformation.

The fact that the transformation is smooth allows for themapping of the anatomical information-known a priori forthe template-to the target. The bottom row, left panel shows agray/white matter segmentation of the template. The transfor-mation of the template segmentation is shown in the bottom-right panel. This automatic segmentation of the study showsexcellent agreement with a hand segmentation of the studyshown in the bottom-middle panel. Discrepancies between thehand and automatic segmentation of the study are due, in part,to errors in the hand segmentation. Since the accuracy of thehand segmentation is not known, no attempt was made in thispreliminary study to quantify differences between anatomicsegmentations and hand segmentations.


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Fig. 4. The columns correspond to propagated templates: 4, 12, 18, an d 30. The top row shows the propagated templates, the top middle row shows the timeprogression of the fluid transformation applied to a rectangular grid, the bottom middle row shows the Jacobians of the previous propagated transformations,and the bottom row shows the Jacobians of the transformations back to the original template.

C. 3-0 Human M R I Experiment correspond to the template (left), the deformed template afterAs a final experiment, we demonstrate the fluid model ap-

plied to high-resolution 3-D MRI data of the human head. Thisexperiment demonstrates that 3-D transformations are requiredto accommodate complex 3-D anatomical shape variation. TheMRI data used was collected using an MPRAGE sequence. Itwas downsampled from 256 x 256 x 128 voxels at l x l x1.25 mm 3 and symmetrically padded by zeros to produce 128x 128 x 100 voxels at 2 mm3. The range of intensities inthe study image was preprocessed by histogram-matching itto the template.

The template was deformed into the shape of the study byfirst using the 3-D elastic basis method described in [8] to ac-commodate the global shape differences. This transformationwas then refined using the 3-D fluid model to accommodatethe local shape differences. The result of this experiment isshown in Fig. 6. The rows from top to bottom correspondto axial slices 48, 55, and 60, respectively, and the columns

the 3-D elastic basis method (middle left), the deformedtemplate after the 3- D fluid method (middle right), and thestudy (right). Notice that, in both the global and local stagesof the deformation, variation is accommodated in 3-D and isindependent of the slice orientation. This is evident by the factthat structures appear and disappear from a fixed observationslice. The change of shape of the ventricles in the top row isa good illustration of this.

V. DISCUSSIONA . Linear Elasticity

For purpose of comparison to previous work (see [2], 6],for example), it is interesting to examine Gaussian priorscorresponding to such penalties as linearized elasticity. Fora linear elastic solid the Cauchy stress tensor U relates tothe strain tensor E by Hookes law, U = Xo(trE ) I + 2 p 0 E


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Fig. 5. Top-left panel shows the monkey visual cortex cryosection template and the top-middle panel shows the second monkey slice. The top-right panelshows the result of fluidly deforming the top-left into the top-middle anatomy. The middle row, left panel, shows the fluid transformation applied to therectangular grid. The z- nd y-components of the estimated transformation are shown in the middle row, middle and right panels, respectively. The bottomrow shows a gray /white matter hand segmenta tion of the template (left) and study (middle). Shown in the bottom-right panel is an autom atic segmentationof the study that was genera ted by applying the fluid transformation to the tem plate segmenta tion.

where A and bo are Lam6 elastic material constants and Iis the identity tensor. For small deformations the linear strainbecomes E = ;[e.'+(e.')'].Substituting this into Hooke'slaw and the momentum equation ( 2 ) gives

b0V2u'+A, +,,)?(a .) + :(a) = 0 (11)where we have neglected the inertial term, p ( d d / d t ) + 37.

It is straightforward to directly connect the PDE to theBayesian posterior. For example, the momentum (3) and (17)arise via minimization of a Lagrangian according to

C[T(Z- G ) , S ( Z ) ] Z d t (12)1where L* is the generalized Lagrangian energy density asso-ciated with constraints imposed by the linearized mechanics.For conservative systems, such as linear elasticity, L* equalsthe total kinetic energy minus the potential energy [29], 1341,and for nonconservative systems, such as viscous fluids, L* ismore complex [30]. Choosing

3 3

i=l j= 1+"[y++U j ( Z )4 d X i

substituting this energy density into (12), and setting the firstvariation equal to zero reproduces the momentum (11) forlinear elasticity. Consider, for example the case of 3-D linear-elastic solid [27] where C is the energy of the Gibbs potentialassociated with the Gaussian cost between the study andthe deformed template. Computing the first variation of thefunctional (12) shows that the forcing term 6 n the PDE [cf.,(2), (3), (1 l) ] is none other than the variation of the likelihoodwith respect to the transformation (5).

Since the PDE associated with linear elasticity has stressthat depends only on the instantaneous strain independent ofthe path or history of the deformation, a conservative systemresults that can be represented as a variational problem thatinvolves no real time. The path dependent stress-strain rela-tionships such as required for the large-deformation viscous-fluid model presented here requires a formulation in space xtime.

B. Viscous Fluid Versus Linear E lasticityIn previous work [6], linear-elasticity regularization was

effectively used to enforce smoothness of the spatially sam-pled transformation. However, linear-elasticity penalizes largedistance deformations thereby preventing the textbook fromcompletely deforming into the study. As Fig. 2demonstrates,the viscous fluid regularization does not have this property.This shortcoming in the elastic map is due to the quadratic


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Fig. 6. 3-D deformation of a 128 x 128 x 100 MRI data set at 2 mm resolution. The rows (top to bottom) correspond to transverse slices 48, 55 , and60, respectively. The left column shows the undefomed template, the left-center column shows the 3-D elastic basis transformed template, the right-centercolumn shows the 3-D fluid transformation that refined the elastic basis transformation, and the right column shows the target data set. Notice that the elasticbasis transformation accommodated the global shape variation while the fluid transformation accommodated the local variation.

growth of the elasticity penalty and the inability of the linearelastic theory to address large magnitude deformations. Theviscous-fluid regularization suffers from no such difficultiesbecause the restoring forces relax over time, while at the sametime accounting for the large distance kinematic nonlinearities.

The advantage of the fluid model for deforming oneanatomy into another is demonstrated in Fig. 7. The rowsfrom top to bottom correspond to a fluid, a strong elastic, anda weak elastic transformation of the monkey visual cortex.The left column shows the deformed template, the middlecolumn shows the difference image between the deformedtemplate and the study, and the right column shows the pointsin white where the Jacobian of the transformation is singular.Notice that the fluid model (top row) allowed the template todeform into the study as shown by the near-black differenceimage while maintaining a positive Jacobian. Conversely, thedifference image for the transformation with the large elasticpenalty (middle row) shows that the template was preventedfrom fully deforming into the study. The all-black Jacobianimage for this transformation confirms that the large elastictransformation also has a positive Jacobian. The difference

image for the weak elastic penalty (bottom row) shows thatwhile the template was not held back as much as in the strongelastic case, it was still held back more than in the fluidcase. In addition, the white points in its Jacobian image showthat the transformation went singular and does not reflect aone-to-one mapping of volumes, areas, or lines [39].

The linear elasticity prior was derived assuming smallangles of rotation and small linear deformations. Becauseit is assumed that the angles of rotation are small and thedeformations are small and linear, large deformations cannotbe accommodated with the linear PDE. This shortcomingof the linear elastic theory could be removed by using ahyperelastic energy form [ 3 5 ] .Accounting for the nonlinearkinematics would enforce a diffeomorphism, but would con-tinue to penalize large magnitude deformations due to theelastic strain energy.

C. Parallel Compu ter ImplementationThe elastic-solid basis and the fluid transformation algo-

rithms were implemented on the massively parallel DECmpp12000Sf lode l 200 (MasPar). This computer is a 128x64


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E xper im en t

2D monkey3D h u m a n

1445

d a t a to ta ls ize es t imate d i t rs increas ing bas is t ime

12 8 x 128 242 200 20 33 sec64 x 64 x 50 192 100 40 7.9 mi n

# of p a r a m . # of # of i t rs before

Fig. 7. The left column shows a fluid (top), a strong elastic (middle), and a weak elastic (bottom) transformation. The middle column shows the differenceimages between the deformed templa te and the study. The right column show s the negative Jacobian points of the deformation (right), respectively (cf. Fig. 5).

Experiment

patch to C2D monkey3D human

TABLE IELASTIC ASIS ROGRAMEXECUTIONIMES N DECMPP12OOOSxda ta # of param # of # P D E # of totalsize estimated iters itrs regrids time

12 8 x 128 6.6 x lo4 600 250 33 7.1 min320 X 256 3.3 x l o5 200 250 12 12.6 min

128 x 12 8 x 100 9.8 x l o6 500 200 118 9.4 hoursmesh-connected single-instruction-multiple-data (SIMD) ar-chitecture, and is well suited for solving the elastic and fluidpartial differential equations. Tables I and I1 show the timingresults for the previous experiments. The second column ofTable I1 shows the image sizes of the template and study im-ages for each experiment. The number of parameters that wereestimated for each experiment is shown in the third column.The number of relaxation iterations required to numericallysatisfy (11) at each time step is shown in column five. Thesesame algorithms would take over lo x longer to compute on aSilicon Graphics Power Challenge with one R8000 processorthan on the 128 x 64 MasPar [40].

VI. SUMMARYIn this paper, we have presented a general approach for

accommodating local shape variations by transforming the co-ordinate system of a template image into that of a target image.


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U fields that cross and break up any region of the template).We have studied the constrained registration problem viasolutions of associated PDEs. For the results presented here,the PDEs have been solved through direct numerical meanson a massively parallel SIMD machine. We also note thatdue to the complexity of transformations that accommodatelarge-magnitude, nonlinear kinematics, further analysis andcharacterization of this new algorithm is required. We arecurrently working on ways to do this.

APPENDIXFUNDAMENTAL CONSERVATION LAWS F MECHANICS

In the field of continuum mechanics it is generally assumedthat a control volume can be selected such that mass withinthe volume is conserved. For deformation of image templates,however, it is often desirable to allow the addition of mass(growth) or the subtraction of mass (shrinkage) on a local level.In order to allow for this we have reformulated the basic lawsto account for a mass creation term. Derivation of the basiclaws follows the classic method (reviewed by [37]) modifiedto include a spatially dependent mass source.A . Conservation of Mass with a Mass Source

Consider a fixed volume R bounded by surface dR.Asource of mass supplies or extracts mass from the volumeat a rate of q per unit volume. By conservation of mass therate of mass change within the volume is equal to the rate ofmass source minus the flux of mass through the surface. Usingthe divergence theorem, we obtain

where d l d t is the material derivative, and the above equationis simply the differential form of the continuity equation witha mass source term.B. Reynolds Transport Theorem with a Mass Source

In determining the equations describing conservation ofmomentum, it is necessary to determine the rate of change of afield variable associated with the material. The rate of changeof the product $ p associated with the material instantaneouslyin a fixed volume R s equal to the rate of increase of y!Jp insideR plus the rate outward flux of $p carried by mass transportthrough the volume surface d o . Using the divergence theoremand applying the continuity equation with a mass source term(14) we find

Equation (15) is a generalized version of the Reynolds trans-port theorem including a mass source term q.C. Conservation o Momentum

Consider a fixed control volume R with boundary aR.Traction stress 7 acts on the surface and a distributed body

force b per unit volume acts on all material elements withinthe body. Applying Newtons Second La w in the form ofconservation of linear momentum yields

-5t /JJ pvdV = J/;dA +JJJ i;dV (16)n an n

The traction force ?is related to the stress tensor U by 7 = T-nwhere n is the outward unit normal from the surface. Usingthis, the divergence theorem, and the generalized Reynoldstransport theorem gives the equation for conservation of mo-mentum with a mass source

ACKNOWLEDGMENTThe authors are grateful to the following: S. Nadel, M.D.,

formerly of Duke University, for providing the 2-D MRI data:D. C. Van Essen, M.D., Ph.D., of the Department of Anatomyand Neurobiology at Washington University, for providing thecryosection data; and M. W. Vannier, M.D., of the Departmentof Radiology at Washington University, for providing the 3-DMRI data.

REFERENCESU. Grenander, Y. Chow, and D. Keenan, HANDS: A Pa ttem TheoreticStudy of Biological Shapes.Y. Amit, U. Grenander, and M. Piccioni, Structural image restorationthrough deformable templates, J. Amer. Stat. Assoc., vol. 86, no. 414,pp. 376-387, June 1991.A. Knoerr, Global models of natural boundaries, Pattem Anal. Rep.no . 148, Div. Appl. Math., Brown. Univ., 1988.U. Grenander and M. I. Miller. (1991) Jump-diffusion processes fo rabduction and recognition of biological shapes. Available: Electron.Signals Syst. Research Lab., Dept. of Elect. Eng., Washington Univ.,St. Louis, MO.A. Srivastava, M. I. Miller, and U. Grenand er, Jump-diffusion processesfor object tracking and direction finding, in Proc. 29rh Annual Aller tonCon$ Comm., Contr. Comput., Univ. Illinois, Urbana-Champaign, IL,1991, pp. 563-570.M . I. Miller, G. E. Christensen, Y. Amit, and U. Grenander, Mathe-matical textbook of deformable neuroanatomies, Proc. Nut Acad. Sci.vol. 90, no. 24, Dec. 1993.G. E. Christensen, R. D. Rabbitt, and M. I. Miller, A deformableneuroanatomy textbook based on viscous fluid mechanics, in Proc.27th Annual Con$ Inform. Sci. Syst., J. Prince and T. Runolfsson, Eds.,Department of Electrical Engineering, The Johns Hopkins University,Baltimore, MD, Mar. 24 2 6 , 1993, pp. 211-216.~, 3-D brain mapping using a deformable neuroanatomy, PhysicsM ed . B i d , vol. 39, pp. 609-618, 1994.S . Joshi et al. Global shape models for optical sectioning microscopy,J. Optic. Soc., Oct. 1992.M . I. Miller et al.,Mitochondria, membranes and amoebae: 1, 2, and 3dimensional shape models, in Statistics and Imagin g, vol. 11,K. Mardia,Ed. Abingdon: Carfax, 1994.U. Grenander, M. I. Miller, and G. E. Christensen, Deformable anatom-ical data bases using global shape models: a position paper for the1992 electronic imaging of the human body workshop, in Proc. Coop.Working GYP.Whole Body 3- 0 Electron. Imaging Human Body, CrewSystem Ergonomics Information Analysis Center, Wright Patterson AirForce Base, OH, Mar. 9-11, 1 992.R. C. Buck, Advanced Calculus, 3rd ed. New York: McGraw -Hill,1978.P. E. Undrill et al.,Integrated presentation of 3d data derived frommulti-sensor imagery and anatomical atlases using a parallel processingsystem, in Proc. SPIE, San Diego, CA, 1992, vol. 1653.C. A. Pelizzari et al.,Accurate three-dimensional registration of CT,PET, andlor MR images of the brain, J. Comput. Assisted Tomog., vol.13, no . 1, pp. 20-26, 19 89.

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[I51 P. T. Fox, J. S. Perlmutter, and M. E. Raichle, A stereotactic method ofanatomical localization for positron emission tomography, J. Comput.Assisted Tomog., vol. 9, no. 1, pp. 141-153, 1985.[16] U. Grenander and M. I. Miller, Representations of knowledge incomplex systems, J. Royal Stat. Soc. B, vol. 56, no. 3, 1994.[I71 A. Hill, T. F. Cootes, C. J. Taylor, and K. Lindley, Medical imageinterpretation: A generic approach using deformable templates, Med.Informat., vol. 19, no. 1, pp. 47-59, 1994.[I81 T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, Activeshape models-their training and applica tion, Comput. Vision ImageUnderstand., vol. 61, no. 1, pp. 38-59, 1995.[19] D. Metaxas and D. Terzopoulos, Shape and nonrigid motion estimationthrough physics-based synthesis, IEEE Trans. Pattern Anal. MachineIntell., vol. 15, no. 6, pp. 58C591, 1993.[20] R. Szeliski, Bayesian Modeling of Uncertainy in Low-Level Vision.Boston: Kluwer, 1989.[21] D. Terzopoulos and K. Waters, Physically based facial modeling,analysis, and animation, J. Visual. Comput. Animation, vol. 1, pp.73-80, 1990.[22] R. Bajcsy, R. Lieberson, and M. Reivich, A computerized system forthe elastic matching of deformcd radiographic images to idealized atlasimages, J. Comput. Assisted Tomog., vol. 7, no. 4, pp. 618-625, 1983.[23] R. Bajcsy and S. Kovacic, Multiresolution elastic matching, Comput. ,Vision, Graphics, Image Processing, vol. 46, pp. 1-21, 1989.[241 G. Wahba, Spline Models fo r Observational Datu. Philadelphia, PA:Soc. Industrial Applied Math., 1990.

[25] F. L. Bookstein, Morphometric Tools or Landmark Data. Cambridge,U K Cambridge Univ. Press, 1991.[26] S. Timoshenko, Theory o Elasticity. New York M cGraw-Hill, 1934.[271 L. A. Segel, Mathematics Applied to Continuum Mechanics. Ne wYork Dover, 1987.[28] L. Meirovitch, Analytical Methods in Wbrations. New York McMil-lan, 1967.[29] J. N. Reddy, Energy and Variational Methods in Applied Mechanics.New York Wiley-Interscience, 1984.[30] L. Mittag, M. J. Stephen, and W. Yourgrau, Variational principles inhydrodynamics, in Variational Principles in Dynamics and QuantumTheory, W. Yourgrau and S. Mandlestam, Eds. New York: Dove r,1979.[31] E. R. McVeigh, R. M. Hcnkelman, and M. J. Bronskill, Noise andfiltration in magnetic resonance imaging, Med. Physics, vol. 12, no. 5,pp. 586591, 1985.[32] D. L. Snyder, Jr., L. J. Thomas, and M. M. Ter-Pogossian, A mathe-matical model for positron emission tomography systems having time-of-flight measurements, IEEE Trans. Nuclear Science, vol. NS-28, pp.3575-3583, 1981.[33] D. L. Snyder, A. M. Hammoud, and R. L. White, Image recoveryfrom data acquired with a charge-coupled-device camera, J.Optic. Soc.Amer. A, vol. 10, pp. 1014-1023, 1 993.[34] B. D. Vujanovic and S. E. Jones, V ariational methods in noncon serva-tive phenomena, in Mathematics Science Engineering, vol. 182. NewYork: Academic, 1989.[35] J. C. Simo, A framework for finite strain elastoplasticity based onmaximum dissipation and the multiplicative decomposition: part I,continuum formulation, Comautat. MethodsAvvl. Mechanics Enc. vol._66, pp. 199-219, 1988.[361 B. Hom, Robot Vision. Cambridge, MA: MIT Press, 1986.[37] L. E. Malvern, Introduction to t& Mechanics o a Continuous Medium.Englewood Cliffs, NJ: Prentice-Hall, 1969.[38] J. C. Strikwerda, Finite Difference Schemes and Partial DifferentialEquations.[39] G. E. Christensen et al.: Topological properties of smooth anatomicmaps, in Information Processing in Medical Imaging, vol. 3 , Y. Bizais,C. Braillot, and R. DiPaola, Eds. Norwell, MA: Kluwer, June 1995,pp. 101-112.1401 G. E. Christensen, M. I. Miller, U. Grenander, and M. W. Vannier,Individualizing Neuroanatomical Atlases Using a Massively ParallelComputer, IEEE Computer, pp. 32-38, Jan. 1996.

Pacific Grove, CA: Wadsworth and Brooks/Cole, 1989.

Gary E. Christensen (S86-M95) received theB.S. degrees in electrical engineering and computerscience, (magna cum laude) from Washington Uni-versity, St. Louis, MO, in 1988, and the M.S. andD.Sc. degrees in electrical engineering from Wash-ington University in 1989 and 1994, respectively.In 1994, he joined the faculty of the Washing-ton University Schoo l of Medicine, where he iscurrently an Assistant Professor in the Mallinck-rodt Institute of Radiology and the Department ofSurgery. He is also the Director of the CraniofacialImaging Laboratory, St. Louis Childrens Hospital, Washington UniversityMedical Center. His primary research interests include image processing, 3-D visualization, medical imaging, computer architecture, parallel computing,and deformable shape models.

Richar- D. Rabbitt earned the B.S. and M.S.degrees in mechanical engineermg from MichiganState University, East Lansing, MI, in 1980 and1982, respectively. In 1986, he was awarded thePh.D. in mechanics from Rensselaer PolytechnicInstitute, Troy, NY, where he also studied as apostdoctoral research associate in applied mathe-matics.From 1987 to 1994, he was an Assistant Professorof mechanical engineenng at Washington Univer-sity, St. Louis, MO. He was selected a PresidentialYoung Investigator by the National Science Foundation in 1988 for his workon the biomechanics of hearing. He is currently an Associate Professorof bioengineering at the University of Utah, Salt Lake City, with primaryspecialization in the biomechanics of soft tissues.

Michael I. Miller (SM95) received the B.S. degre ein electrical engineering from the State Universityof New York, Stony Brook, in 1976. He receivedthe M.S. degree in electrical engineering in 1978and the Ph.D. in biomedical engineering in 1983,both from Johns Hopkins University, Baltimore,M D.In 1983, he joined the Electrical EngineeringDepartment, Washington University, St. Louis, MO,where he is a Professor of Electrical Engineering.He is also a full urofessor at the MallinckrodtInstitute of Radiology at Washington University, and at the Institute forBiomedical Computing. His current research interests include automatedtarget recognition, medical imaging, computational linguistics, pattem the-ory, stochastic processes, and computational neuroscience. Dr. Miller is theDirector of the Army Center for Imaging Science, a consortium of universitiesinvolving Washington University, Harvard University, Massachusetts Instituteof Technology, and the University of Texas.He is a recipient of a Presidential Young Investigator Award, and currentlyholds the Newton R. and Sarah L. Wilson Professorship in BiomedicalEngineering.

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christens en 1996 deform able templates using large deformation kinematics

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