voxel based versus region of interest analysis in diffusion tensor imaging of neurodevelopment there are two main methods of quantitative analysis in diffusion tensor imaging dti studies manual region of interest roi and automated voxel based the purpose of this study is to compare the results of each of these methods applied to the same data set linear correlative analysis was performed for mean diffusivity trace 3 adc and for fractional anisotropy fa versus age within 8 12 years n 32 and within 21 27 years n 28 as well as a group comparison spm analysis identified more structures changing with age partly due to the limited regions measured with roi analysis in general roi and voxel based analysis methods produced comparable results for widespread reductions of trace 3 adc and increases of fa with age particularly for group comparison the discrepancies i e missed regions were likely related to problems of spatial normalization for spm analysis and masking localized changes by averaging all the voxels within a region of interest for roi analysis these two analysis methods for dti offer complementary results but neither one yields the complete story of neurodevelopment

parallel adaptive gmres implementations for homotopy methods the success of probability one homotopy methods in solving large scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems iterative solution techniques such as gmres k favor parallel implementations however their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors one variation of gmres k considered here is to adapt the restart value k for any given problem and use householder reflections in the orthogonalization phase coupled with graph based matrix partitioning to achieve high accuracy and reduce the communication overhead this particular gmres implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability one homotopy algorithms occasionally unusually high accuracy ability to adapt to problems of widely varying difficulty and parallelism

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

conjugate gradient type methods for semilinear elliptic problems with symmetry we study block conjugate gradient methods in the context of continuation methods for bifurcation problems by exploiting symmetry in certain semilinear elliptic differential equations we can decompose the problems into small ones and reduce computational cost on the other hand the associated centered difference discretization matrices on the subdomains are nonsymmetric we symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices these properties allow the discrete pure mode solution paths branching from a multiple bifurcation point 0 m n of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point 0 1 1 of the counterpart of the reduced problem thus the structure of a multiple bifurcation is preserved in discretization while its treatment is reduced to those for simple bifurcation of problems on subdomains in particular we can adapt the continuation lanczos algorithm proposed in 1 to trace simple solution paths sample numerical results are reported

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

dr tamas diffeomorphic registration for tensor accurate alignment of anatomical structures in this work we propose dr tamas diffeomorphic registration for tensor accurate alignment of anatomical structures a novel framework for intersubject registration of diffusion tensor imaging dti data sets this framework is optimized for brain data and its main goal is to achieve an accurate alignment of all brain structures including white matter wm gray matter gm and spaces containing cerebrospinal fluid csf currently most dti based spatial normalization algorithms emphasize alignment of anisotropic structures while some diffusion derived metrics such as diffusion anisotropy and tensor eigenvector orientation are highly informative for proper alignment of wm other tensor metrics such as the trace or mean diffusivity md are fundamental for a proper alignment of gm and csf boundaries moreover it is desirable to include information from structural mri data e g t1 weighted or t2 weighted images which are usually available together with the diffusion data the fundamental property of dr tamas is to achieve global anatomical accuracy by incorporating in its cost function the most informative metrics locally another important feature of dr tamas is a symmetric time varying velocity based transformation model which enables it to account for potentially large anatomical variability in healthy subjects and patients the performance of dr tamas is evaluated with several data sets and compared with other widely used diffeomorphic image registration techniques employing both full tensor information and or dti derived scalar maps our results show that the proposed method has excellent overall performance in the entire brain while being equivalent to the best existing methods in wm

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

characterizing the shape of anatomical structures with poisson s equation poisson s equation a fundamental partial differential equation in classical physics has a number of properties that are interesting for shape analysis in particular the equipotential sets of the solution graph become smoother as the potential increases we use the displacement map the length of the streamlines formed by the gradient field of the solution to measure the complexity or smoothness of the equipotential sets and study its behavior as the potential increases we believe that this function complexity f potential which we call the shape characteristic is a very natural way to express shape robust algorithms are presented to compute the solution to poisson s equation the displacement map and the shape characteristic we first illustrate our technique on two dimensional synthetic examples and natural silhouettes we then perform two shape analysis studies on three dimensional neuroanatomical data extracted from magnetic resonance mr images of the brain in the first study we investigate changes in the caudate nucleus in schizotypal personality disorder spd and confirm previously published results on this structure in the second study we present a data set of caudate nuclei of premature infants with asymmetric white matter injury our method shows structural shape differences that volumetric measurements were unable to detect

on the error propagation of semi lagrange and fourier methods for advection problems in this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired the numerical methods considered are based on the fast fourier transform polynomial interpolation semi lagrangian methods using a lagrange or spline interpolation and a discontinuous galerkin semi lagrangian approach which is conservative and has to store more than a single value per cell we demonstrate by carrying out numerical experiments that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation based semi lagrangian methods for the discontinuous galerkin semi lagrangian method however we find that the characteristic property of semi lagrangian error estimates namely the fact that the error increases proportionally to the number of time steps is not observed we provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi lagrangian methods the method based on the fast fourier transform is exact but due to round off errors susceptible to a linear increase of the error in the number of time steps we show how to modify the cooley tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps finally we show for a simple model that our conclusions hold true if the advection solver is used as part of a splitting scheme

variable precision registration via wavelets optimal spatial scales for inter subject registration of functional mri abstract the detection of significantly activated brain regions in multi subject functional magnetic resonance imaging fmri studies almost invariably entails the coregistration of individual subjects data in a standard space here we investigate how sensitivity to detect loci of generic activation in such studies may be conditioned by the precision of anatomical registration we describe a novel algorithm implemented in the wavelet domain for inhomogeneous deformation of individual images to match a template the algorithm matches anatomical features in a coarse to fine fashion first minimising a cost function in terms of relatively coarse spatial features and then proceeding iteratively to match the images in terms of progressively more detailed anatomical features applying the method to data acquired from two groups of 12 healthy volunteers with mean age 27 and 70 years respectively during performance of a paired associate learning task we show that geometrical overlap between template and individual images is monotonically improved compared to an affine transform by additional inhomogeneous deformations informed by more detailed features likewise sensitivity to detect activated voxels can be substantially improved by a factor of 4 or more if wavelet mediated deformations informed by medium sized anatomical features are applied in addition to a preliminary affine transform however sensitivity to detect activated voxels was reduced by over registering data or matching anatomical features at the finest scales of the wavelet transform the benefits of variable precision registration are particularly salient for data acquired in older subjects which showed evidence of greater inter subject anatomic variability and generally required more extensive local deformation to achieve a satisfactory match to the template image we conclude that major benefits in sensitivity to detect functional activation in multi subject fmri studies can be attained with an inhomogeneous deformation applied over appropriate spatial scales

effects of spatial transformation on regional brain volume estimates spatial transformation of mr brain images is a standard tool used in automated anatomical parcellation and other quantitative and qualitative methods to assess brain tissue volume composition and distribution despite widespread use the quantitative effects of spatial transformation on regional brain volume estimates have been little studied we report on the effects of transformation on regional brain volumes of 38 17m 21f manually parcellated brains after tracing in native space regions of interest were transformed using a classic piecewise linear talairach transformation tal or a nonlinear registration air 5th order nonlinear algorithm 158 parameters to one of three talairach based templates 1 tal50 constructed from 50 talairach transformed normal brains 2 the mni 305 atlas 3 ia38 constructed from mni305 transformed scans of the 38 subjects used in this study native volumes were compared to the transformed volumes we found that 1 significant group level differences can be obtained in transformed data sets that are in the opposite direction of effects obtained in native space 2 the effects of transformation are heterogeneous across brain regions even after covarying for total brain volume and age 3 volumetric intra class correlations between native and transformed brains differ by registration method and template choice region and tissue type and 4 transformed brains produced hippocampus and corpus callosum volume proportions that were significantly different from those obtained in native space our results suggest that region based volumetric differences uncovered by spatial transformation based methods should be replicated in native space brains and that meta analyses should take into account whether volumes are determined using spatially transformed images and or specific automated methods

spatial normalization and averaging of diffusion tensor mri data sets diffusion tensor magnetic resonance imaging dt mri is unique in providing information about both the structural integrity and the orientation of white matter fibers in vivo and through tractography revealing the trajectories of white matter tracts dt mri is therefore a promising technique for detecting differences in white matter architecture between different subject populations however while studies involving analyses of group averages of scalar quantities derived from dt mri data have been performed as yet there have been no similar studies involving the whole tensor here we present the first step towards realizing such a study i e the spatial normalization of whole tensor data sets the approach is illustrated by spatial normalization of 10 dt mri data sets to a standard anatomical template both qualitative and quantitative approaches are described for assessing the results of spatial normalization techniques are then described for combining the spatially normalized data sets according to three definitions of average i e the mean median and mode of a distribution of tensors the current absence of and hence need for appropriate statistical tests for comparison of results derived from group averaged dt mri data sets is then discussed finally the feasibility of performing tractography on the group averaged dt mri data set is investigated and the possibility and implications of generating a generic map of brain connectivity from a group of subjects is considered

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

a performance comparison of semi lagrangian discontinuous galerkin and spline based vlasov solvers in four dimensions abstract the purpose of the present paper is to compare two semi lagrangian methods in the context of the four dimensional vlasov poisson equation more specifically our goal is to compare the performance of the more recently developed semi lagrangian discontinuous galerkin scheme to cubic spline interpolation which is widely used in eulerian vlasov simulation to that end we perform simulations for nonlinear landau damping and a two stream instability and provide benchmarks for the selalib and sldg codes both on a workstation and using mpi on a cluster in addition we will present results for the graphic processing unit gpu implementation contained in sldg we find that the semi lagrangian discontinuous galerkin scheme shows a moderate improvement in run time for nonlinear landau damping and a substantial improvement for the two stream instability it should be emphasized that these results are markedly different from results obtained in the asymptotic regime which favor spline interpolation thus we conclude that the traditional approach of evaluating numerical methods is misleading even for short time simulations in addition the absence of any global communication in the semi lagrangian discontinuous galerkin method gives it a decisive advantage for scaling to more than 256 cores

accuracy of the discrete fourier transform and the fast fourier transform fast fourier transform fft based computations can be far more accurate than the slow transforms suggest discrete fourier transforms computed through the fft are far more accurate than slow transforms and convolutions computed via fft are far more accurate than the direct results however these results depend critically on the accuracy of the fft software employed which should generally be considered suspect popular recursions for fast computation of the sine cosine table or twiddle factors are inaccurate due to inherent instability some analyses of these recursions that have appeared heretofore in print suggesting stability are incorrect even in higher dimensions the fft is remarkably stable

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

adaptive control of a class of slowly time varying plants abstract an adaptive control system of the type considered earlier by the author which is designed to be stable under the assumption that the unknown plant parameters are constant is shown to maintain stability when the plant parameters are slowly time varying

discriminative segmentation based evaluation through shape dissimilarity segmentation based scores play an important role in the evaluation of computational tools in medical image analysis these scores evaluate the quality of various tasks such as image registration and segmentation by measuring the similarity between two binary label maps commonly these measurements blend two aspects of the similarity pose misalignments and shape discrepancies not being able to distinguish between these two aspects these scores often yield similar results to a widely varying range of different segmentation pairs consequently the comparisons and analysis achieved by interpreting these scores become questionable in this paper we address this problem by exploring a new segmentation based score called normalized weighted spectral distance nwsd that measures only shape discrepancies using the spectrum of the laplace operator through experiments on synthetic and real data we demonstrate that nwsd provides additional information for evaluating differences between segmentations which is not captured by other commonly used scores our results demonstrate that when jointly used with other scores such as dice s similarity coefficient the additional information provided by nwsd allows richer more discriminative evaluations we show for the task of registration that through this addition we can distinguish different types of registration errors this allows us to identify the source of errors and discriminate registration results which so far had to be treated as being of similar quality in previous evaluation studies

a splitting algorithm for vlasov simulation with filamentation filtration a fourier fourier transformed version of the splitting algorithm for simulating solutions of the vlasov poisson system of equations is introduced it is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm it is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle in cell counterpart it is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results

incorporating prior knowledge into image registration the first step in the spatial normalization of brain images is usually to determine the affine transformation that best maps the image to a template image in a standard space we have developed a rapid and automatic method for performing this registration which uses a bayesian scheme to incorporate prior knowledge of the variability in the shape and size of heads we compared affine registrations with and without incorporating the prior knowledge we found that the affine transformations derived using the bayesian scheme are much more robust and that the rate of convergence is greater

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

multiple bifurcations generated by mode interactions in a reaction diffusion problem abstract we study multiple bifurcations in a system of reaction diffusion equations defined on a unit square with robin boundary conditions first we investigate linear stabilities of the system at the uniform steady state solution then we discuss how multiple bifurcations can be generated by mode interactions of the system and how these multiple bifurcations can be preserved in the associated discrete system a continuation unsymmetric lanczos algorithm is described to trace discrete solution curves numerical experiments on the brusselator equations are reported

voxel based morphometry should not be used with imperfectly registered images abstract john ashburner and karl friston 2000 introduced a standardized method of voxel based morphometry vbm for comparisons of local concentrations of gray matter between two groups of subjects segmented images of gray matter from grossly normalized high resolution images are smoothed and their group differences analyzed by the now conventional voxelwise worsley approach to gaussian random fields of differences this comment concerns an unfortunate interaction between the algorithm s spatial normalization and voxelwise comparison steps whereby several obvious quantitative confounds are injected at the core of the inference engine the authors put forward specifically the statistics of the resulting voxelwise comparisons are uninformative about group differences wherever the spatial normalization algorithm has failed to register on any robustly appearing image gradient the method of ashburner and friston is defensible only far from all image gradients

global voxel and cluster tests by theory and permutation for a difference between two groups of structural mr images of the brain the authors describe almost entirely automated procedures for estimation of global voxel and cluster level statistics to test the null hypothesis of zero neuroanatomical difference between two groups of structural magnetic resonance imaging mri data theoretical distributions under the null hypothesis are available for 1 global tissue class volumes 2 standardized linear model analysis of variance anova and ancova coefficients estimated at each voxel and 3 an area of spatially connected clusters generated by applying an arbitrary threshold to a two dimensional 2 d map of normal statistics at voxel level the authors describe novel methods for economically ascertaining probability distributions under the null hypothesis with fewer assumptions by permutation of the observed data nominal type i error control by permutation testing is generally excellent whereas theoretical distributions may be over conservative permutation has the additional advantage that it can be used to test any statistic of interest such as the sum of suprathreshold voxel statistics in a cluster or cluster mass regardless of its theoretical tractability under the null hypothesis these issues are illustrated by application to mri data acquired from 18 adolescents with hyperkinetic disorder and 16 control subjects matched for age and gender

convergence analysis of a discontinuous galerkin strang splitting approximation for the vlasov poisson equations a rigorous convergence analysis of the strang splitting algorithm with a discontinuous galerkin approximation in space for the vlasov poisson equations is provided it is shown that under suitable assumptions the error is of order mathcal o tau 2 h q h q tau where tau is the size of a time step h is the cell size and q the order of the discontinuous galerkin approximation in order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed

weighted fourier series representation and its application to quantifying the amount of gray matter we present a novel weighted fourier series wfs representation for cortical surfaces the wfs representation is a data smoothing technique that provides the explicit smooth functional estimation of unknown cortical boundary as a linear combination of basis functions the basic properties of the representation are investigated in connection with a self adjoint partial differential equation and the traditional spherical harmonic spharm representation to reduce steep computational requirements a new iterative residual fitting irf algorithm is developed its computational and numerical implementation issues are discussed in detail the computer codes are also available at http www stat wisc edu mchung softwares weighted spharm weighted spharm html as an illustration the wfs is applied in quantifying the amount of gray matter in a group of high functioning autistic subjects within the wfs framework cortical thickness and gray matter density are computed and compared

multi contrast large deformation diffeomorphic metric mapping for diffusion tensor imaging diffusion tensor imaging dti can reveal detailed white matter anatomy and has the potential to detect abnormalities in specific white matter structures such detection and quantification are however not straightforward the voxel based analysis after image normalization is one of the most widely used methods for quantitative image analyses to apply this approach to dti it is important to examine if structures in the white matter are well registered among subjects which would be highly dependent on employed algorithms for normalization in this paper we evaluate the accuracy of normalization of dti data using a highly elastic transformation algorithm called large deformation diffeomorphic metric mapping after simulation based validation of the algorithm dti data from normal subjects were used to measure the registration accuracy to examine the impact of morphological abnormalities on the accuracy the algorithm was also tested using data from alzheimer s disease ad patients with severe brain atrophy the accuracy level was measured by using manual landmark based white matter matching and surface based brain and ventricle matching as gold standard to improve the accuracy level cascading and multi contrast approaches were developed the accuracy level for the white matter was 1 88 0 55 and 2 19 0 84 mm for the measured locations in the controls and patients respectively

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated

morphological appearance manifolds in computational anatomy groupwise registration and morphological analysis abstract existing approaches to computational anatomy assume that a perfectly conforming diffeomorphism applied to an anatomy of interest captures its morphological characteristics relative to a template however the amount of biological variability in a groupwise analysis renders this task practically impossible due to the nonexistence of a single template that matches all anatomies in an ensemble even if such a template is constructed by group averaging procedures consequently anatomical characteristics not captured by the transformation and which are left out in the residual image are lost permanently from subsequent analysis if only properties of the transformation are examined this paper extends our recent work makrogiannis s verma r davatzikos c 2007 anatomical equivalence class a computational anatomy framework using a lossless shape descriptor ieee trans biomed imag 26 4 619 631 on characterizing subtle morphological variations via a lossless morphological descriptor that takes the residual into account along with the transformation since there are infinitely many transformation residual pairs that reconstruct a given anatomy we treat them as members of an anatomical equivalence class aec thereby forming a manifold embedded in the space spanned by transformation residual this paper develops a unique and optimal representation of each anatomy that is estimated from the corresponding aecs by solving a global optimization problem this effectively determines the optimal template and transformation parameters for each individual anatomy and eliminates respective confounding variation in the data it therefore constitutes the second novelty in that it represents a group wise optimal registration strategy that individually adjusts the template and the smoothness of the transformation according to each anatomy experimental results support the superiority of our morphological analysis framework over conventional analysis and demonstrate better diagnostic accuracy