on a class of nonlinear dynamical systems the structure of a differential operator in the application of the decomposition method abstract a suitable expresion of a differential operator is deduced allowing the explicit calculation of the general term f h in the power series expansion of a function f x t to be utilized for solving nonlinear differential equations

polynomial nonlinearities in differential equations abstract adomian and his collaborators have found solutions for nonlinear stochastic or deterministic in the limiting case differential equations involving polynomial non linearities using the first author s a n polynomials and recently developed methods of calculating these polynomials it becomes very easy to write solutions for non linear terms such as y m for positive integers m the cases of negative integer m or even decimal powers will appear elsewhere

statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

on the tricomi problem the tricomi equation is solved by the decomposition method 6 8 on the half space x y 0 and also for suitable boundary conditions given on the boundary of the hyperbolic elliptic domain

on the convergence region for decomposition solutions abstract the author s decomposition method using his a n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear or nonlinear stochastic operator equations providing a computable accurate solution which converges rapidly in computation the above is sufficient for a rapid test of convergence region

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

intrinsic and extrinsic analysis in computational anatomy we present intrinsic and extrinsic methods for studying anatomical coordinates in order to perform statistical inference on random physiological signals f across clinical populations in both intrinsic and extrinsic methods we introduce generalized partition functions of the coordinates psi x x epsilon m which are used to construct a random field of f on m as statistical model in the intrinsic analysis such partition functions are built intrinsically for individual anatomical coordinate based on courant s theorem on nodal analysis via self adjoint linear elliptic differential operators in contrast the extrinsic method needs only one set of partition functions for a template coordinate system and then applies to each anatomical coordinate system via diffeomorphic transformation for illustration we apply both intrinsic and extrinsic methods to a clinical study cortical thickness variation of the left cingulate gyrus in schizophrenia both methods show that the left cingulate gyrus tends to become thinner in schizophrenia relative to the healthy control population however the intrinsic method increases the statistical power

time sequence diffeomorphic metric mapping and parallel transport track time dependent shape changes abstract serial mri human brain scans have facilitated the detection of brain development and of the earliest signs of neuropsychiatric and neurodegenerative diseases monitoring disease progression and resolving drug effects in clinical trials for preventing or slowing the rate of brain degeneration to track anatomical shape changes in serial images we introduce new point based time sequence large deformation diffeomorphic metric mapping ts lddmm to infer the time flow of within subject geometric shape changes that carry known observations through a period its euler lagrange equation is generalized for anatomies whose shapes are characterized by point sets such as landmarks curves and surfaces the time dependent momentum obtained from the ts lddmm encodes within subject shape changes for the purpose of across subject shape comparison we then propose a diffeomorphic analysis framework to translate within subject deformation in a global template without incorporating across subject anatomical variations via parallel transport technique the analysis involves the retraction of the within subject time dependent momentum along the ts lddmm trajectory from each time to the baseline the translation of the momentum in a global template and the reconstruction of the ts lddmm trajectory starting from the global template

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

development of a boundary element method for time dependent planar thermoelasticity abstract a new boundary element method is developed for two dimensional quasistatic thermoelasticity this time domain formulation involves only surface quantities consequently volume discretization is completely eliminated and the method becomes a viable alternative to the usual finite element approaches after presenting a brief overview of the governing equations boundary integral equations for coupled quasistatic thermoelasticity are derived by starting with existing fundamental solutions along with an appropriate reciprocal theorem details of a general purpose numerical implementation are then discussed next boundary element methods for the two more practical theories uncoupled quasistatic and steady state thermoelasticity are developed directly from limiting forms of the coupled formulation several numerical examples are provided to illustrate the validity and attractiveness of the boundary element approach for this entire class of problems

nonlinear plasma response abstract the decomposition method g adomian stochastic systems academic press new york london 1983 g adomian stochastic systems ii in press and earlier work of adomian referenced therein allows solution without linearization of problems normally requiring the self consistent field technique the latter technique can provide a sufficiently accurate approximation in weakly coupled plasma but the decomposition method of adomian provides a superior and physically preferable methodology for the general case

volumetric transformation of brain anatomy presents diffeomorphic transformations of three dimensional 3 d anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery these transformations are generated in a hierarchical manner accommodating both global and local anatomical detail the initial low dimensional registration is accomplished by constraining the transformation to be in a low dimensional basis the basis is defined by the green s function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator the high dimensional large deformations are vector fields generated via the mismatch between the template and target image volumes constrained to be the solution of a navier stokes fluid model as part of this procedure the jacobian of the transformation is tracked insuring the generation of diffeomorphisms it is shown that transformations constrained by quadratic regularization methods such as the laplacian biharmonic and linear elasticity models do not ensure that the transformation maintains topology and therefore must only be used for coarse global registration

displacement and temperature discontinuity boundary integral equation and boundary element method for analysis of cracks in three dimensional isotropic thermoelastic media abstract the displacement and temperature discontinuity boundary integral equation and boundary element method are developed for the analysis of cracks in three dimensional isotropic thermoelastic media the fundamental solutions for unit point displacement and temperature discontinuities are derived and the displacement and temperature discontinuity boundary integral equations are obtained for an arbitrarily shaped planar crack the displacement and temperature discontinuity boundary integral equation method is developed to analyze singularities of near crack border fields and the stress and heat flux intensity factors are derived in terms of displacement and temperature discontinuities across crack faces the fundamental solutions for a constant triangular element are obtained by integrating the fundamental solutions for unit point displacement and temperature discontinuities on the basis of these solutions the displacement and temperature discontinuity boundary element method is proposed as an application elliptical cracks are analyzed to validate the developed method

landmark matching via large deformation diffeomorphisms this paper describes the generation of large deformation diffeomorphisms spl phi spl omega 0 1 sup 3 spl rlhar2 spl omega for landmark matching generated as solutions to the transport equation d spl phi x t dt spl nu spl phi x t t t spl isin 0 1 and spl phi x 0 x with the image map defined as spl phi spl middot 1 and therefore controlled via the velocity field spl nu spl middot t t spl isin 0 1 imagery are assumed characterized via sets of landmarks x sub n y sub n n 1 2 n the optimal diffeomorphic match is constructed to minimize a running smoothness cost spl par l spl nu spl par sup 2 associated with a linear differential operator l on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks both inexact and exact landmark matching is studied here given noisy landmarks x sub n matched to y sub n measured with error covariances spl sigma sub n then the matching problem is solved generating the optimal diffeomorphism spl phi spl circ x 1 spl int sub 0 sup 1 spl nu spl circ spl phi spl circ x t t dt x where spl nu spl circ spl middot argmin sub spl nu spl middot spl int sub 1 sup 1 spl int sub spl omega spl par l spl nu x t spl par sup 2 dxdt spl sigma sub n 1 sup n y sub n spl phi x sub n 1 sup t spl sigma sub n sup 1 y sub n spl phi x sub n 1 conditions for the existence of solutions in the space of diffeomorphisms are established with a gradient algorithm provided for generating the optimal flow solving the minimum problem results on matching two dimensional 2 d and three dimensional 3 d imagery are presented in the macaque monkey

automatic segmentation of the left ventricle in cardiac mr and ct images this paper describes a segmentation technique to automatically extract the myocardium in 4d cardiac mr and ct datasets the segmentation algorithm is a two step process the global localization step roughly localizes the left ventricle using techniques such as maximum discrimination thresholding and connected component analysis the local deformations step combines em based region segmentation and dijkstra active contours using graph cuts spline fitting or point pattern matching the technique has been tested on a large number of patients and both quantitative and qualitative results are presented

geodesic estimation for large deformation anatomical shape averaging and interpolation the goal of this research is to promote variational methods for anatomical averaging that operate within the space of the underlying image registration problem this approach is effective when using the large deformation viscous framework where linear averaging is not valid or in the elastic case the theory behind this novel atlas building algorithm is similar to the traditional pairwise registration problem but with single image forces replaced by average forces these group forces drive an average transport ordinary differential equation allowing one to estimate the geodesic that moves an image toward the mean shape configuration this model gives large deformation atlases that are optimal with respect to the shape manifold as defined by the data and the image registration assumptions we use the techniques in the large deformation context here but they also pertain to small deformation atlas construction furthermore a natural inherently inverse consistent image registration is gained for free as is a tool for constant arc length geodesic shape interpolation the geodesic atlas creation algorithm is quantitatively compared to the euclidean anatomical average to elucidate the need for optimized atlases the procedures generate improved average representations of highly variable anatomy from distinct populations

nonlinear differential equations with negative power nonlinearities abstract differential equations involving the term y m where m is a positive integer are solved by the decomposition method 1 3 441 452

large deformation diffeomorphic metric curve mapping we present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping lddmm scheme for computing an optimal transformation between two curves embedded in euclidean space d curves are first represented as vector valued measures which incorporate both location and the first order geometric structure of the curves then a hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves we describe a discretized version of this in which discrete sequences of points along the curve are represented by vector valued functionals this gives a convenient and practical way to define a matching functional for curves we derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in 2 and 3 behaviors of the curve mapping are discussed using 2d curves the applications to shape classification is shown and experiments with 3d curves extracted from brain cortical surfaces are presented

the dual boundary element method for transient thermoelastic crack problems abstract a two dimensional boundary element formulation which does not require domain discretization and allows a single region analysis is presented for transient thermoelastic crack problems the formulation is based on the dual boundary element method temperature and displacement equations are used on one crack surface flux and traction equations on the other time is modelled with linear time elements stress intensity factors are calculated using the path independent integral several crack problems are solved and the results are compared where possible with existing solutions

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia spatial normalization is a crucial step in assessing patterns of neuroanatomical structure and function associated with health and disease errors that occur during spatial normalization can influence hypothesis testing due to the dimensionalities of mapping algorithms and anatomical manifolds landmarks curves surfaces volumes used to drive the mapping algorithms the primary aim of this paper is to improve statistical inference using multiple anatomical manifolds and large deformation diffeomorphic metric mapping lddmm algorithms we propose that combining information generated by the various manifolds and algorithms improves the reliability of hypothesis testing we used this unified approach to assess variation in the thickness of the cingulate gyrus in subjects with schizophrenia and healthy comparison subjects three different lddmm algorithms for mapping landmarks curves and triangulated meshes were used to transform thickness maps of the cingulate surfaces into an atlas coordinate system we then tested for group differences by combining the information from the three types of anatomical manifolds and lddmm mapping algorithms the unified approach provided reliable statistical results and eliminated ambiguous results due to surface mismatches subjects with schizophrenia had non uniform cortical thinning over the left and right cingulate gyri especially in the anterior portion as compared to

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

registration of longitudinal brain image sequences with implicit template and spatial temporal heuristics abstract accurate measurement of longitudinal changes of brain structures and functions is very important but challenging in many clinical studies also across subject comparison of longitudinal changes is critical in identifying disease related changes in this paper we propose a novel method to meet these two requirements by simultaneously registering sets of longitudinal image sequences of different subjects to the common space without assuming any explicit template specifically our goal is to 1 consistently measure the longitudinal changes from a longitudinal image sequence of each subject and 2 jointly align all image sequences of different subjects to a hidden common space to achieve these two goals we first introduce a set of temporal fiber bundles to explore the spatial temporal behavior of anatomical changes in each longitudinal image sequence then a probabilistic model is built upon the temporal fibers to characterize both spatial smoothness and temporal continuity finally the transformation fields that connect each time point image of each subject to the common space are simultaneously estimated by the expectation maximization em approach via the maximum a posterior map estimation of the probabilistic models promising results have been obtained in quantitative measurement of longitudinal brain changes i e hippocampus volume changes showing better performance than those obtained by either the pairwise or the groupwise only registration methods

bem analysis of fracture problems in three dimensional thermoelasticity using j integral abstract this paper describes the formulation and numerical implementation of the j integral applied to three dimensional thermoelasticity using a boundary element technique the mixed mode stress intensity factors has been evaluated through a decomposition technique in this technique the j integral is split into j i j ii and j iii associated to the three basic modes of fracture the decomposition technique is compared to the crack opening displacement technique good accuracy was obtained in the mixed mode test examples

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

tensor based cortical surface morphometry via weighted spherical harmonic representation we present a new tensor based morphometric framework that quantifies cortical shape variations using a local area element the local area element is computed from the riemannian metric tensors which are obtained from the smooth functional parametrization of a cortical mesh for the smooth parametrization we have developed a novel weighted spherical harmonic spharm representation which generalizes the traditional spharm as a special case for a specific choice of weights the weighted spharm is shown to be the least squares approximation to the solution of an isotropic heat diffusion on a unit sphere the main aims of this paper are to present the weighted spharm and to show how it can be used in the tensor based morphometry as an illustration the methodology has been applied in the problem of detecting abnormal cortical regions in the group of high functioning autistic subjects

evolutions equations in computational anatomy one of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs notably in the brain or the heart over the last decade our group has progressively developed several approaches for this problem all related to the riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act several important shape evolution equations that are now used routinely in applications have emerged over time our goal in this paper is to provide an overview of these equations placing them in their theoretical context and giving examples of applications in which they can be used we introduce the required theoretical background before discussing several classes of equations of increasingly complexity these equations include energy minimizing evolutions deriving from riemannian gradient descent geodesics parallel transport and jacobi fields

the emerging discipline of computational functional anatomy computational functional anatomy cfa is the study of functional and physiological response variables in anatomical coordinates for this we focus on two things i the construction of bijections via diffeomorphisms between the coordinatized manifolds of human anatomy and ii the transfer group action and parallel transport of functional information into anatomical atlases via these bijections we review advances in the unification of the bijective comparison of anatomical submanifolds via point sets including points curves and surface triangulations as well as dense imagery we examine the transfer via these bijections of functional response variables into anatomical coordinates via group action on scalars and matrices in dti as well as parallel transport of metric information across multiple templates which preserves the inner product

transient behaviour of semilinear stochastic systems with random parameters abstract the transient solution of a class of nonautonomous stochastic differential equations with given random initial conditions is studied the considered evolution equation is characterized by the presence of a deterministic linear term and of a random nonlinear term whose parameters can be modeled by random processes of rice noise type analytical approximated expressions for the first and second order moments and for the probability density of the solution process are derived and are applied to determine the statistical properties of a class of stochastic nonlinear oscillators

on composite nonlinearities and the decomposition method abstract accurate convergent computable solutions using the decomposition method have been demonstrated in and papers for wide classes of nonlinear and or stochastic differential partial differential or algebraic equations it is shown specifically in this paper that composite nonlinearities of the form nx n0 n1 n2 x appearing in such equations where the ni are nonlinear operators can also be handled with the adomian an polynomials

high resolution random mesh algorithms for creating a probabilistic 3d surface atlas of the human brain abstract striking variations exist across individuals in the internal and external geometry of the brain such normal variations in the size orientation topology and geometric complexity of cortical and subcortical structures have complicated the problem of quantifying deviations from normal anatomy and of developing standardized neuroanatomical atlases this paper describes the design implementation and results of a technique for creating a three dimensional 3d probabilistic surface atlas of the human brain we have developed implemented and tested a new 3d statistical method for assessing structural variations in a database of anatomic images the algorithm enables the internal surface anatomy of new subjects to be analyzed at an extremely local level the goal was to quantify subtle and distributed patterns of deviation from normal anatomy by automatically generating detailed probability maps of the anatomy of new subjects connected systems of parametric meshes were used to model the internal course of the following structures in both hemispheres the parieto occipital sulcus the anterior and posterior rami of the calcarine sulcus the cingulate and marginal sulci and the supracallosal sulcus these sulci penetrate sufficiently deeply into the brain to introduce an obvious topological decomposition of its volume architecture a family of surface maps was constructed encoding statistical properties of local anatomical variation within individual sulci a probability space of random transformations based on the theory of gaussian random fields was developed to reflect the observed variability in stereotaxic space of the connected system of anatomic surfaces a complete system of probability density functions was computed yielding confidence limits on surface variation the ultimate goal of brain mapping is to provide a framework for integrating functional and anatomical data across many subjects and modalities this task requires precise quantitative knowledge of the variations in geometry and location of intracerebral structures and critical functional interfaces the surface mapping and probabilistic techniques presented here provide a basis for the generation of anatomical templates and expert diagnostic systems which retain quantitative information on intersubject variations in brain architecture

multi structure network shape analysis via normal surface momentum maps article i nfo we present a shape analysis pipeline for the assessment of anatomical variations in subcortical networks in mr images the shape analysis pipeline injects the global shape properties of the cfa subcortical template into the subcortical parcellations generated from freesurfer via large deformation diffeomorphic metric mapping lddmm examples are shown for this injection in several subcortical structures whose raw mr images were sampled from the database of open access series of imaging studies oasis the shape analysis is performed on random field representation of the template surface momentum maps that encode the shape variation of subcortical structure targets of each individual subject relative to the template the momentum maps have the optimum property that they are supported only on the boundary of the subcortical structures with the direction normal to the subcortical nuclei boundary thereby reducing the dimension of shape variation significantly a two level statistical model was built on these momentum maps to assess anatomical connectivity among the subcortical structures on the basis of similar surface deformation compression or expansion results in the study of healthy aging on the hippocampus amygdala network indicate the anatomical connectivity between the basolateral complex of the amygdala and the subiculum of the hippocampus on the basis of shape compression in healthy elders relative to young adults

computation of thermal stresses in quasistatic non stationary thermoelasticity using boundary elements in this paper we present the boundary integral formulation for the solution of quasi static problems of two dimensional uncoupled thermoelasticity a detailed scheme for the numerical solution of the boundary integral equations and computation of internal stresses by regularized integral representation is developed in time formulation the method is tested in a numerical example

diffeomorphic metric surface mapping in subregion of the superior temporal gyrus this paper describes the application of large deformation diffeomorphic metric mapping to cortical surfaces based on the shape and geometric properties of subregions of the superior temporal gyrus in the human brain the anatomical surfaces of the cortex are represented as triangulated meshes the diffeomorphic matching algorithm is implemented by defining a norm between the triangulated meshes based on the algorithms of vaillant and glaunes the diffeomorphic correspondence is defined as a flow of the extrinsic three dimensional coordinates containing the cortical surface that registers the initial and target geometry by minimizing the norm the methods are demonstrated in 40 high resolution mri cortical surfaces of planum temporale pt constructed from subsets of the superior temporal gyrus stg the effectiveness of the algorithm is demonstrated via the euclidean positional distance distance of normal vectors and curvature before and after the surface matching as well as the comparison with a landmark matching algorithm the results demonstrate that both the positional and shape variability of the anatomical configurations are being represented by the diffeomorphic maps

stochastic model for colored noise abstract the input output point of view for the ito equation to avoid stochastic integration is related to the stochastic operator approach for a physically reasonable robust mathematical model

on a class of nonlinear stochastic dynamical systems analysis of the transient behaviour abstract a class of nonlinear dynamical systems with random parameters and initial conditions is considered the transient time evolution of the mean value of the dependent variable is analyzed both by means of the stochastic averaging method and by adomian s decomposition method a proof of the equivalence of the decomposition method to the time series approximation of the dynamical response is provided an application is made to the van der pol equation and finally a comparison between the two previously discussed methods is given

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

large deformation diffeomorphic metric mapping of vector fields this paper proposes a method to match diffusion tensor magnetic resonance images dt mris through the large deformation diffeomorphic metric mapping of vector fields focusing on the fiber orientations considered as unit vector fields on the image volume we study a suitable action of diffeomorphisms on such vector fields and provide an extension of the large deformation diffeomorphic metric mapping framework to this type of dataset resulting in optimizing for geodesics on the space of diffeomorphisms connecting two images existence of the minimizers under smoothness assumptions on the compared vector fields is proved and coarse to fine hierarchical strategies are detailed to reduce both ambiguities and computation load this is illustrated by numerical experiments on dt mri heart images

multi manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces cortical surface based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex one of the main challenges in the cortical surface based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals in this paper we introduce a multi manifold large deformation diffeomorphic metric mapping mm lddmm algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms we present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin we solve for trajectories geodesics of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme the mm lddmm algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space which provides a natural scheme for shape deformation average and template or atlas generation we applied the mm lddmm algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects the estimated template shape reflects regions which are highly variable across these subjects compared with existing single manifold lddmm algorithms such as the lddmm curve mapping and the lddmm surface mapping the mm lddmm mapping provides better results in terms of surface to surface distances in five predefined regions

the dual boundary element method effective implementation for crack problems summary the present paper is concerned with the effective numerical implementation of the two dimensional dual boundary element method for linear elastic crack problems the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation both crack surfaces are discretized with discontinuous quadratic boundary elements this strategy not only automatically satisfies the necessary conditions for the existence of the finite part integrals which occur naturally but also circumvents the problem of collocation at crack tips crack kinks and crack edge corners examples of geometries with edge and embedded crack are analysed with the present method highly accurate results are obtained when the stress intensity factor is evaluated with the j integral technique the accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed mode conditions

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

cortical surface shape analysis based on spherical wavelets in vivo quantification of neuroanatomical shape variations is possible due to recent advances in medical imaging and has proven useful in the study of neuropathology and neurodevelopment in this paper we apply a spherical wavelet transformation to extract shape features of cortical surfaces reconstructed from magnetic resonance images mris of a set of subjects the spherical wavelet transformation can characterize the underlying functions in a local fashion in both space and frequency in contrast to spherical harmonics that have a global basis set we perform principal component analysis pca on these wavelet shape features to study patterns of shape variation within normal population from coarse to fine resolution in addition we study the development of cortical folding in newborns using the gompertz model in the wavelet domain which allows us to characterize the order of development of large scale and finer folding patterns independently given a limited amount of training data we use a regularization framework to estimate the parameters of the gompertz model to improve the prediction performance on new data we develop an efficient method to estimate this regularized gompertz model based on the broyden fletcher goldfarb shannon bfgs approximation promising results are presented using both pca and the folding development model in the wavelet domain the cortical folding development model provides quantitative anatomic information regarding macroscopic cortical folding development and may be of potential use as a biomarker for early diagnosis of neurologic deficits in newborns

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data this paper proposes an original approach for the statistical analysis of longitudinal shape data the proposed method allows the characterization of typical growth patterns and subject specific shape changes in repeated time series observations of several subjects this can be seen as the extension of usual longitudinal statistics of scalar measurements to high dimensional shape or image data the method is based on the estimation of continuous subject specific growth trajectories and the comparison of such temporal shape changes across subjects differences between growth trajectories are decomposed into morphological deformations which account for shape changes independent of the time and time warps which account for different rates of shape changes over time given a longitudinal shape data set we estimate a mean growth scenario representative of the population and the variations of this scenario both in terms of shape changes and in terms of change in growth speed then intrinsic statistics are derived in the space of spatiotemporal deformations which characterize the typical variations in shape and in growth speed within the studied population they can be used to detect systematic developmental delays across subjects in the context of neuroscience we apply this method to analyze the differences in the growth of the hippocampus in children diagnosed with autism developmental delays and in controls result suggest that group differences may be better characterized by a different speed of maturation rather than shape differences at a given age in the context of anthropology we assess the differences in the typical growth of the endocranium between chimpanzees and bonobos we take advantage of this study to show the robustness of the method with respect to change of parameters and perturbation of the age estimates

coupled differential equations and coupled boundary conditions abstract it is demonstrated that the high accuracy for approximations requiring only a few terms which is typical of the decomposition method for nonlinear stochastic operator equations or special cases linear or deterministic holds for coupled equations and coupled boundary conditions as well

on matching brain volumes abstract to characterize the complex morphological variations that occur naturally in human neuroanatomy so that their confounding effect can be minimized in the identification of brain structures in medical images a computational framework has evolved in which individual anatomies are modeled as warped versions of a canonical representation of the anatomy known as an atlas to realize this framework the method of elastic matching was invented for determining the spatial mapping between a three dimensional image pair in which one image volume is modeled as an elastic continuum that is deformed to match the appearance of the second volume in this paper we review the seminal ideas underlying the elastic matching technique consider the practical implications of an integral formulation of the approach and explore a more general bayesian interpretation of the method in order to address issues that are less naturally resolved within a continuum mechanical setting such as the examination of a solution s reliability or the incorporation of empirical information that may be available about the spatial mappings into the analysis

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

a fast diffeomorphic image registration algorithm abstract this paper describes dartel which is an algorithm for diffeomorphic image registration it is implemented for both 2d and 3d image registration and has been formulated to include an option for estimating inverse consistent deformations nonlinear registration is considered as a local optimisation problem which is solved using a levenberg marquardt strategy the necessary matrix solutions are obtained in reasonable time using a multigrid method a constant eulerian velocity framework is used which allows a rapid scaling and squaring method to be used in the computations dartel has been applied to intersubject registration of 471 whole brain images and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects

smooth functional and structural maps on the neocortex via orthonormal bases of the laplace beltrami operator functional and structural maps such as a curvature cortical thickness and functional magnetic resonance imaging mri maps indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies due to the highly convoluted nature of the cerebral cortex and image quality these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system in this paper we generalized the spline smoothing problem wahba 1990 from a sphere to any arbitrary two dimensional 2 d manifold with boundaries we first seek a numerical solution to orthonormal basis functions of the laplace beltrami lb operator with neumann boundary conditions for a 2 d manifold m then solve the spline smoothing problem in a reproducing kernel hilbert space r k h s of real valued functions on manifold m with kernel constructed from the basis functions the explicit discrete lb representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete lb orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem and then smoothed functions in r k h s can be represented as a linear combination of the basis functions we demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing we present applications of our approach to smoothing sulcal mean curvature cortical thickness and functional statistical maps on submanifolds of the neocortex

solving nonlinear differential equations with decimal power nonlinearities abstract the decomposition method stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press is applied to the solution of nonlinear differential equations of the form ly ny x t where l is a linear differential operator and ny is a nonlinear term of the form y with a decimal number

the tps direct transport a new method for transporting deformations in the size and shape space modern shape analysis allows the fine comparison of shape changes occurring between different objects very often the classic machineries of generalized procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks however if size and shape data are structured in different groups thus constituting different morphological trajectories a data centering is needed if one wants to compare solely the deformation representing the trajectories to do that inter individual variation must be filtered out this maneuver is rarely applied in studies using simulated or real data a geometrical procedure named parallel transport that can be based on various connection types is necessary to perform such kind of data centering usually the levi civita connection is used for interpolation of curves in a riemannian space it can also be used to transport a deformation we demonstrate that this procedure does not preserve some important characters of the deformation even in the affine case we propose a novel procedure called tps direct transport which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools we recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study