soliton dynamics in computational anatomy computational anatomy ca has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms among these geometric structures landmarks and image outlines in ca are shown to be singular solutions of a partial differential equation that is called the geodesic epdiff equation a recently discovered momentum map for singular solutions of epdiff yields their canonical hamiltonian formulation which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta the momentum map provides an isomorphism between landmarks and outlines for images and singular soliton solutions of the epdiff equation this isomorphism suggests a new dynamical paradigm for ca as well as new data representation

geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

hierarchical geodesic models in diffeomorphisms hierarchical linear models hlms are a standard approach for analyzing data where individuals are measured repeatedly over time however such models are only applicable to longitudinal studies of euclidean data this paper develops the theory of hierarchical geodesic models hgms which generalize hlms to the manifold setting our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms first individual level geodesics represent the trajectory of shape changes within individuals second a group level geodesic represents the average trajectory of shape changes for the population our proposed hgm is applicable to longitudinal data from unbalanced designs i e varying numbers of timepoints for individuals which is typical in medical studies we derive the solution of hgms on diffeomorphisms to estimate individual level geodesics the group geodesic and the residual diffeomorphisms we also propose an efficient parallel algorithm that easily scales to solve hgms on a large collection of 3d images of several individuals finally we present an effective model selection procedure based on cross validation we demonstrate the effectiveness of hgms for longitudinal analysis of synthetically generated shapes and 3d mri brain scans

clutter invariant atr one of the central problems in automated target recognition is to accommodate the infinite variety of clutter in real military environments the principle focus of our paper is on the construction of metric spaces where the metric measures the distance between objects of interest invariant to the infinite variety of clutter such metrics are formulated using second order random field models our results indicate that this approach significantly improves detection classification rates of targets in clutter

hilbert schmidt lower bounds for estimators on matrix lie groups for atr deformable template representations of observed imagery model the variability of target pose via the actions of the matrix lie groups on rigid templates in this paper we study the construction of minimum mean squared error estimators on the special orthogonal group so n for pose estimation due to the nonflat geometry of so n the standard bayesian formulation of optimal estimators and their characteristics requires modifications by utilizing hilbert schmidt metric defined on gl n a larger group containing so n a mean squared criterion is defined on so n the hilbert schmidt estimate hse is defined to be a minimum mean squared error estimator restricted to so n the expected error associated with the hse is shown to be a lower bound called the hilbert schmidt bound hsb on the error incurred by any other estimator analysis and algorithms are presented for evaluating the hse and the hsb in cases of both ground based and airborne targets

variational geometric and statistical methods for modeling brain anatomy and function we survey the recent activities of the odyssee laboratory in the area of the application of mathematics to the design of models for studying brain anatomy and function we start with the problem of reconstructing sources in meg and eeg and discuss the variational approach we have developed for solving these inverse problems this motivates the need for geometric models of the head we present a method for automatically and accurately extracting surface meshes of several tissues of the head from anatomical magnetic resonance mr images anatomical connectivity can be extracted from diffusion tensor magnetic resonance images but in the current state of the technology it must be preceded by a robust estimation and regularization stage we discuss our work based on variational principles and show how the results can be used to track fibers in the white matter wm as geodesics in some riemannian space we then go to the statistical modeling of functional magnetic resonance imaging fmri signals from the viewpoint of their decomposition in a pseudo deterministic and stochastic part that we then use to perform clustering of voxels in a way that is inspired by the theory of support vector machines and in a way that is grounded in information theory multimodal image matching is discussed next in the framework of image statistics and partial differential equations pdes with an eye on registering fmri to the anatomy the paper ends with a discussion of a new theory of random shapes that may prove useful in building anatomical and functional atlases

kernel bundle diffeomorphic image registration using stationary velocity fields and wendland basis functions in this paper we propose a multi scale multi kernel shape compactly supported kernel bundle framework for stationary velocity field based image registration wendland kernel bundle stationary velocity field wkb svf we exploit the possibility of directly choosing kernels to construct a reproducing kernel hilbert space rkhs instead of imposing it from a differential operator the proposed framework allows us to minimize computational cost without sacrificing the theoretical foundations of svf based diffeomorphic registration in order to recover deformations occurring at different scales we use compactly supported wendland kernels at multiple scales and orders to parameterize the velocity fields and the framework allows simultaneous optimization over all scales the performance of wkb svf is extensively compared to the 14 non rigid registration algorithms presented in a recent comparison paper on both mgh10 and cumc12 datasets the accuracy of wkb svf is improved when compared to other registration algorithms in a disease specific application for intra subject registration atrophy scores estimated using the proposed registration scheme separates the diagnostic groups of alzheimer s and normal controls better than the state of the art segmentation technique experimental results show that wkb svf is a robust flexible registration framework that allows theoretically well founded and computationally efficient multi scale representation of deformations and is equally well suited for both inter and intra subject image registration

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

parametric surface diffeomorphometry for low dimensional embeddings of dense segmentations and imagery in the field of computational anatomy biological form including our focus neuroanatomy is studied quantitatively through the action of the diffeomorphism group on example anatomies a technique called diffeomorphometry here we design an algorithm within this framework to pass from dense objects common in neuromaging studies binary segmentations structural images to a sparse representation defined on the surface boundaries of anatomical structures and embedded into the low dimensional coordinates of a parametric model our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points as well as images through functional composition with the inverse this allows us to index the diffeomorphisms with respect to two dimensional surface geometries like subcortical gray matter structures but explicitly map onto cost functions determined by noisy 3 dimensional measurements we consider models generated from empirical covariance of training data as well as bandlimited laplace beltrami eigenfunction models when no such data is available we show applications to noisy or anomalous segmentations and other typical problems in neuroimaging studies we reproduce statistical results detecting changes in alzheimer s disease despite dimensionality reduction lastly we apply our algorithm to the common problem of segmenting subcortical structures from t1 mr images

tensor products of division algebras and fields abstract this paper began as an investigation of the question of whether d 1 f d 2 is a domain where the d i are division algebras and f is an algebraically closed field contained in their centers we present an example where the answer is no and also study the picard group and brauer group properties of f 1 f f 2 where the f i are fields finally as part of our example we have results about division algebras and brauer groups over curves specifically we give a splitting criterion for certain brauer group elements on the product of two curves over f

registration of anatomical images using paths of diffeomorphisms parameterized with stationary vector field flows computational anatomy aims for the study of variability in anatomical structures from images variability is encoded by the spatial transformations existing between anatomical images and a template selected as reference in the absence of a more justified model for inter subject variability transformations are considered to belong to a convenient family of diffeomorphisms which provides a suitable mathematical setting for the analysis of anatomical variability one of the proposed paradigms for diffeomorphic registration is the large deformation diffeomorphic metric mapping lddmm in this framework transformations are characterized as end points of paths parameterized by time varying flows of vector fields defined on the tangent space of a riemannian manifold of diffeomorphisms and computed from the solution of the non stationary transport equation associated to these flows with this characterization optimization in lddmm is performed on the space of non stationary vector field flows resulting into a time and memory consuming algorithm recently an alternative characterization of paths of diffeomorphisms based on constant time flows of vector fields has been proposed in the literature with this parameterization diffeomorphisms constitute solutions of stationary odes in this article the stationary parameterization is included for diffeomorphic registration in the lddmm framework we formulate the variational problem related to this registration scenario and derive the associated euler lagrange equations moreover the performance of the non stationary vs the stationary parameterizations in real and simulated 3d mri brain datasets is evaluated compared to the non stationary parameterization our proposal provides similar results in terms of image matching and local differences between the diffeomorphic transformations while drastically reducing memory and time requirements

a nonconservative lagrangian framework for statistical fluid registration safira in this paper we used a nonconservative lagrangian mechanics approach to formulate a new statistical algorithm for fluid registration of 3 d brain images this algorithm is named safira acronym for statistically assisted fluid image registration algorithm a nonstatistical version of this algorithm was implemented where the deformation was regularized by penalizing deviations from a zero rate of strain in the terms regularizing the deformation included the covariance of the deformation matrices and the vector fields q here we used a lagrangian framework to reformulate this algorithm showing that the regularizing terms essentially allow nonconservative work to occur during the flow given 3 d brain images from a group of subjects vector fields and their corresponding deformation matrices are computed in a first round of registrations using the nonstatistical implementation covariance matrices for both the deformation matrices and the vector fields are then obtained and incorporated separately or jointly in the nonconservative terms creating four versions of safira we evaluated and compared our algorithms performance on 92 3 d brain scans from healthy monozygotic and dizygotic twins 2 d validations are also shown for corpus callosum shapes delineated at midline in the same subjects after preliminary tests to demonstrate each method we compared their detection power using tensor based morphometry tbm a technique to analyze local volumetric differences in brain structure we compared the accuracy of each algorithm variant using various statistical metrics derived from the images and deformation fields all these tests were also run with a traditional fluid method which has been quite widely used in tbm studies the versions incorporating vector based empirical statistics on brain variation were consistently more accurate than their counterparts when used for automated volumetric quantification in new brain images this suggests the advantages of this approach for large scale neuroimaging studies

geodesic regression and the theory of least squares on riemannian manifolds this paper develops the theory of geodesic regression and least squares estimation on riemannian manifolds geodesic regression is a method for finding the relationship between a real valued independent variable and a manifold valued dependent random variable where this relationship is modeled as a geodesic curve on the manifold least squares estimation is formulated intrinsically as a minimization of the sum of squared geodesic distances of the data to the estimated model geodesic regression is a direct generalization of linear regression to the manifold setting and it provides a simple parameterization of the estimated relationship as an initial point and velocity analogous to the intercept and slope a nonparametric permutation test for determining the significance of the trend is also given for the case of symmetric spaces two main theoretical results are established first conditions for existence and uniqueness of the least squares problem are provided second a maximum likelihood criteria is developed for a suitable definition of gaussian errors on the manifold while the method can be generally applied to data on any manifold specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age

an elasticity based covariance analysis of shapes we introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects based on the notion of nonlinear elastic deformations from one shape to another a suitable linearization of geometric shape variations is introduced once such a linearization is available a principal component analysis can be investigated this requires the definition of a covariance metric an inner product on linearized shape variations the resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation the underlying elasticity concept represents an alternative to riemannian shape statistics in this paper we compare a standard l 2 type covariance metric with a metric based on the hessian of the nonlinear elastic energy furthermore we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity for the built in pairwise elastic registration a relaxed model formulation is employed which allows for a non exact matching shape contours are approximated by single well phase fields which enables an extension of the method to a covariance analysis of image morphologies the model is implemented with multilinear finite elements embedded in a multi scale approach the characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2d and 3d

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

dynamic programming generation of curves on brain surfaces dynamic programming algorithms are presented for automated generation of length minimizing geodesics and curves of extremal curvature on the neocortex of the macaque and the visible human probabilistic models of curve variation are constructed in terms of the variability in speed curvature and torsion in the frenet representation

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

factorization theory from commutative to noncommutative settings abstract we study the non uniqueness of factorizations of non zero divisors into atoms irreducibles in noncommutative rings to do so we extend concepts from the commutative theory of non unique factorizations to a noncommutative setting several notions of factorizations as well as distances between them are introduced in addition arithmetical invariants characterizing the non uniqueness of factorizations such as the catenary degree the invariant and the tame degree are extended from commutative to noncommutative settings we introduce the concept of a cancellative semigroup being permutably factorial and characterize this property by means of corresponding catenary and tame degrees also we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization applying the abstract machinery we develop we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero sum sequences over a ray class group we also determine catenary degrees and the permutable tame degree for the semigroup of non zero divisors of the ring of n n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup

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

population shape regression from random design data regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent regressor variable and in particular it is applicable to the study of anatomical growth and shape change when the underlying process can be modeled by parameters in a euclidean space classical regression techniques hardle applied nonparametric regression 1990 wand and jones kernel smoothing 1995 are applicable and have been studied extensively however recent work suggests that attempts to describe anatomical shapes using flat euclidean spaces undermines our ability to represent natural biological variability fletcher et al ieee trans med imaging 23 8 995 1005 2004 grenander and miller q appl math 56 4 617 694 1998 in this paper we develop a method for regression analysis of general manifold valued data specifically we extend nadaraya watson kernel regression by recasting the regression problem in terms of frechet expectation although this method is quite general our driving problem is the study anatomical shape change as a function of age from random design image data we demonstrate our method by analyzing shape change in the brain from a random design dataset of mr images of 97 healthy adults ranging in age from 20 to 79 years to study the small scale changes in anatomy we use the infinite dimensional manifold of diffeomorphic transformations with an associated metric we regress a representative anatomical shape as a function of age from this population

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

multi contrast multi scale surface registration for improved alignment of cortical areas the position of cortical areas can be approximately predicted from cortical surface folding patterns however there is extensive inter subject variability in cortical folding patterns prohibiting a one to one mapping of cortical folds in certain areas in addition the relationship between cortical area boundaries and the shape of the cortex is variable and weaker for higher order cortical areas current surface registration techniques align cortical folding patterns using sulcal landmarks or cortical curvature for instance the alignment of cortical areas by these techniques is thus inherently limited by the sole use of geometric similarity metrics magnetic resonance imaging t1 maps show intra cortical contrast that reflects myelin content and thus can be used to improve the alignment of cortical areas in this article we present a new symmetric diffeomorphic multi contrast multi scale surface registration mmsr technique that works with partially inflated surfaces in the level set framework mmsr generates a more precise alignment of cortical surface curvature in comparison to two widely recognized surface registration algorithms the resulting overlap in gyrus labels is comparable to freesurfer most importantly mmsr improves the alignment of cortical areas further by including t1 maps as a first application we present a group average t1 map at a uniquely high resolution and multiple cortical depths which reflects the myeloarchitecture of the cortex mmsr can also be applied to other mr contrasts such as functional and connectivity data

prior based segmentation and shape registration in the presence of perspective distortion challenging object detection and segmentation tasks can be facilitated by the availability of a reference object however accounting for possible transformations between the different object views as part of the segmentation process remains difficult recent statistical methods address this problem by using comprehensive training data other techniques can only accommodate similarity transformations we suggest a novel variational approach to prior based segmentation using a single reference object that accounts for planar projective transformation generalizing the chan vese level set framework we introduce a novel shape similarity measure and embed the projective homography between the prior shape and the image to segment within a region based segmentation functional the proposed algorithm detects the object of interest extracts its boundaries and concurrently carries out the registration to the prior shape we demonstrate prior based segmentation on a variety of images and verify the accuracy of the recovered transformation parameters

a unified approach to fast image registration and a new curvature based registration technique abstract image registration is central to many challenges in medical imaging today it has a vast range of applications the purpose of this note is twofold first we review some of the most promising non linear registration strategies currently used in medical imaging we show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment second we introduce within the variational framework a new non linear registration model based on a curvature type smoother we show that affine linear transformations belong to the kernel of this regularizer as a result the approach becomes more robust against poor initializations of a pre registration step furthermore we develop a stable and fast implementation of the new scheme based on a real discrete cosine transformation we demonstrate the advantages of the new technique for synthetic data sets and present an application of the algorithm for registering mr mammography images

factorizations in upper triangular matrices over information semialgebras abstract an integral domain or a commutative cancellative monoid is atomic if every nonzero nonunit element factors into atoms and it satisfies the accp if every ascending chain of principal ideals eventually stabilizes the interplay between these two properties has been investigated since the 1970s an atomic domain or monoid satisfies the finite factorization property ffp if every element has only finitely many factorizations and it satisfies the bounded factorization property bfp if for each element there is a bound for the number of atoms in each of its factorizations these two properties have been systematically studied since they were introduced by anderson anderson and zafrullah in 1990 noetherian domains satisfy the bfp while dedekind domains satisfy the ffp it is well known that for commutative cancellative monoids and in particular for multiplicative monoids of integral domains ffp bfp accp atomic for n 2 we show that each of these four properties transfers back and forth between an information semialgebra s certain commutative cancellative semiring and the multiplicative monoid t n s consisting of n n upper triangular matrices over s we also show that a similar transfer behavior takes place if one replaces t n s by its submonoid u n s consisting of upper triangular matrices with units along their main diagonals as a consequence we find that the atomic chain ffp bfp accp atomic also holds for the two classes comprising the noncommutative monoids t n s and u n s finally we construct various rational information semialgebras to verify that in general none of the established implications is reversible

strongly consistent estimation of the sample distribution of noisy continuous parameter fields the general problem of defining and determining the sample distribution in the case of continuous parameter random fields is addressed defining a distribution in the case of deterministic functions is straightforward based on measures of sublevel sets however the fields we consider are the sum of a deterministic component nonrandom multidimensional function and an i i d random field an attempt to extend the same notion to the stochastic case immediately raises some fundamental difficulties we show that by uniformly sampling such random fields the difficulties may be avoided and a sample distribution may be compatibly defined and determined not surprisingly the obtained result resembles the known fact that the probability distribution of the sum of two independent random variables is the convolution of their distributions finally we apply the results to derive a solution to the problem of deformation estimation of one and multidimensional signals in the presence of measurement noise

invariant higher order variational problems we investigate higher order geometric k splines for template matching on lie groups this is motivated by the need to apply diffeomorphic template matching to a series of images e g in longitudinal studies of computational anatomy our approach formulates euler poincar theory in higher order tangent spaces on lie groups in particular we develop the euler poincar formalism for higher order variational problems that are invariant under lie group transformations the theory is then applied to higher order template matching and the corresponding curves on the lie group of transformations are shown to satisfy higher order euler poincar equations the example of so 3 for template matching on the sphere is presented explicitly various cotangent bundle momentum maps emerge naturally that help organize the formulas we also present hamiltonian and hamilton ostrogradsky lie poisson formulations of the higher order euler poincar theory for applications on the hamiltonian side

large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type in large deformation diffeomorphic metric mapping lddmm the diffeomorphic matching of images are modeled as evolution in time or a flow of an associated smooth velocity vector field v controlling the evolution the initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image thus methods such as principal component analysis pca of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to small deformation assumption in the analysis of linear displacements we apply this approach to a study of dementia of the alzheimer type dat the left hippocampus in the dat group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality further pca of the initial momentum leads to correct classification of 12 out of 18 dat subjects and 22 out of 26 control subjects

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

cortical cartography using the discrete conformal approach of circle packings cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain metric based approaches are the most common but suffer from high distortions conformal or angle based algorithms are supported by a comprehensive mathematical theory the conformal approach that uses circle packings is versatile in the manipulation and display of results in addition it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods in this paper we begin with a brief description of cortical flat mapping from data acquisition to map displays including a brief review of past flat mapping approaches we then describe the mathematics of conformal geometry and key elements of conformal mapping we introduce the mechanics of circle packing and discuss its connections with conformal geometry using a triangulated surface representing a cortical hemisphere we illustrate several manipulations available using circle packing methods and describe the associated ensemble conformal features ecfs we conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy

a primal sketch of the cortex mean curvature a morphogenesis based approach to study the variability of the folding patterns in this paper we propose a new representation of the cortical surface that may be used to study the cortex folding process and to recover some putative stable anatomical landmarks called sulcal roots usually buried in the depth of adult brains this representation is a primal sketch derived from a scale space computed for the mean curvature of the cortical surface this scale space stems from a diffusion equation geodesic to the cortical surface the primal sketch is made up of objects defined from mean curvature minima and saddle points the resulting sketch aims first at highlighting significant elementary cortical folds second at representing the fold merging process during brain growth the relevance of the framework is illustrated by the study of central sulcus sulcal roots from antenatal to adult age some results are proposed for ten different brains some preliminary results are also provided for superior temporal sulcus

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex application to the planum temporale dynamic programming is used to define boundaries of cortical submanifolds with focus on the planum temporale pt of the superior temporal gyrus stg which has been implicated in a variety of neuropsychiatric disorders to this end automated methods are used to generate the pt manifold from 10 high resolution mri subvolumes roi masks encompassing the stg a procedure to define the subvolume roi masks from original mri brain scans is developed bayesian segmentation is then used to segment the subvolumes into cerebrospinal fluid gray matter gm and white matter wm 3d isocontouring using the intensity value at which there is equal probability of gm and wm is used to reconstruct the triangulated graph representing the stg cortical surface enabling principal curvature at each point on the graph to be computed dynamic programming is used to delineate the pt manifold by tracking principal curves from the retro insular end of the heschl s gyrus hg to the stg along the posterior stg up to the start of the ramus and back to the retro insular end of the hg a coordinate system is then defined on the pt manifold the origin is defined by the retro insular end of the hg and the y axis passes through the point on the posterior stg where the ramus begins automated labeling of gm in the stg is robust with l 1 distances between bayesian and manual segmentation in the range 0 001 0 12 n 20 pt reconstruction is also robust with 90 of the vertices of the reconstructed pt within about 1 voxel n 20 from semiautomated contours finally the reliability index based on interrater intraclass correlation for the surface area derived from repeated reconstructions is 0 96 for the left pt and 0 94 for the right pt thus demonstrating the robustness of dynamic programming in defining a coordinate system on the pt it provides a method with potential significance in the study of neuropsychiatric disorders

reliable and efficient pattern matching using an affine invariant metric we present a new pattern similarity measure that behaves well under affine transformations our similarity measure is useful for pattern matching since it is defined on patterns with multiple components satisfies the metric properties is invariant under affine transformations and is robust with respect to perturbation and occlusion we give an algorithm based on hierarchical subdivision of transformation space which minimises our measure under the group of affine transformations given two patterns in addition we present results obtained using an implementation of this algorithm

a nonlinear programming heuristic for computing optimal link capacities in a multi hour alternate routing communications network we consider the problem of determining the optimal number of communications channels in a telecommunications network traffic between pairs of nodes is assumed to vary with the time of day traffic between two nodes can flow over a one link direct path or if no direct channels are available over a two link alternate route the problem is to compute the optimal number of channels on each link to minimize cost subject to a constraint that enough trunks are present so that the probability of no idle channel does not exceed a specified amount the mathematical formulation of this problem known as the multi hour engineering problem yields an integer nonlinear programming problem nlp we apply recursive quadratic programming with exact penalty function epf line searches to solve the continuous nlp and use the special problem structure to provide a starting point and epf parameter a simple heuristic is used to obtain integral number of channels satisfying desired blocking objectives the method successfully solved a set of randomly generated test problems with up to 35 variables and 432 nonlinear constraints

fluid registration of diffusion tensor images using information theory we apply an information theoretic cost metric the symmetrized kullback leibler skl divergence or j divergence to fluid registration of diffusion tensor images the difference between diffusion tensors is quantified based on the skl divergence of their associated probability density functions pdfs three dimensional dti data from 34 subjects were fluidly registered to an optimized target image to allow large image deformations but preserve image topology we regularized the flow with a large deformation diffeomorphic mapping based on the kinematics of a navier stokes fluid a driving force was developed to minimize the j divergence between the deforming source and target diffusion functions while reorienting the flowing tensors to preserve fiber topography in initial experiments we showed that the skl divergence based on full diffusion pdfs is adaptable to higher order diffusion models such as high angular resolution diffusion imaging hardi the skl divergence was sensitive to subtle differences between two diffusivity profiles showing promise for nonlinear registration applications and multisubject statistical analysis of hardi data

an efficient method for determining economical configurations of elementary packet switched networks the packet switched network design problem can be formulated as a capacity and flow assignment cfa problem the cfa problem is investigated for an elementary network consisting of one tandem switch and n local switches it is regarded as the structural unit of a hierarchical network it is assumed that any line speed is available and the cost of each line is a linear function of its speed with a fixed charge for installation this cfa problem is shown to be equivalent to a 0 1 integer programming problem with a discontinuous cost function a threshold rule and a row wise or column wise improvement rci iteration are proposed to solve the problem the threshold rule assigns all the traffic between two local switches to a direct route if the required traffic exceeds a predetermined threshold value and otherwise to a tandem route the rci iteration searches the vertices of the unit cube of 2n dimensional euclidean space by a procedure roughly like the simplex method whenever the network has external traffic direct application of the threshold rule ensures a global optimum when there is no external traffic a simple modification of the rci iteration yields almost a global optimum within 2n steps

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

the geometric median on riemannian manifolds with application to robust atlas estimation one of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images the study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space which map one anatomy to another it is now well established that representing the geometry of shapes or images in euclidian spaces undermines our ability to represent natural variability in populations in our previous work we have extended classical statistical analysis techniques such as averaging principal components analysis and regression to riemannian manifolds which are more appropriate representations for describing anatomical variability in this paper we extend the notion of robust estimation a well established and powerful tool in traditional statistical analysis of euclidian data to manifold valued representations of anatomical variability in particular we extend the geometric median a classic robust estimator of centrality for data in euclidean spaces we formulate the geometric median of data on a riemannian manifold as the minimizer of the sum of geodesic distances to the data points we prove existence and uniqueness of the geometric median on manifolds with non positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds generalizing the weiszfeld procedure for finding the geometric median of euclidean data we present an algorithm for computing the geometric median on an arbitrary manifold we show that this algorithm converges to the unique solution when it exists in this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images using this approach we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images

model driven harmonic parameterization of the cortical surface hip hop in the context of inter subject brain surface matching we present a parameterization of the cortical surface constrained by a model of cortical organization the parameterization is defined via an harmonic mapping of each hemisphere surface to a rectangular planar domain that integrates a representation of the model as opposed to previous landmark based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso coordinate axis in the model this strategy overcomes some limitation to sulcus based registration techniques such as topological variability in sulcal landmarks across subjects experiments on 62 subjects with manually traced sulci are presented and compared with the result of the freesurfer software the evaluation involves a measure of dispersion of sulci with both angular and area distortions we show that the model based strategy can lead to a natural efficient and very fast less than 5 min per hemisphere method for defining inter subjects correspondences we discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

nonrigid brain mr image registration using uniform spherical region descriptor there are two main issues that make nonrigid image registration a challenging task first voxel intensity similarity may not be necessarily equivalent to anatomical similarity in the image correspondence searching process second during the imaging process some interferences such as unexpected rotations of input volumes and monotonic gray level bias fields can adversely affect the registration quality in this paper a new feature based nonrigid image registration method is proposed the proposed method is based on a new type of image feature namely uniform spherical region descriptor usrd as signatures for each voxel the usrd is rotation and monotonic gray level transformation invariant and can be efficiently calculated the registration process is therefore formulated as a feature matching problem the usrd feature is integrated with the markov random field labeling framework in which energy function is defined for registration the energy function is then optimized by the expansion algorithm the proposed method has been compared with five state of the art registration approaches on both the simulated and real 3 d databases obtained from the brainweb and internet brain segmentation repository respectively experimental results demonstrate that the proposed method can achieve high registration accuracy and reliable robustness behavior

nonrigid image registration with crystal dislocation energy the goal of nonrigid image registration is to find a suitable transformation such that the transformed moving image becomes similar to the reference image the image registration problem can also be treated as an optimization problem which tries to minimize an objective energy function that measures the differences between two involved images in this paper we consider image matching as the process of aligning object boundaries in two different images the registration energy function can be defined based on the total energy associated with the object boundaries the optimal transformation is obtained by finding the equilibrium state when the total energy is minimized which indicates the object boundaries find their correspondences and stop deforming we make an analogy between the above processes with the dislocation system in physics the object boundaries are viewed as dislocations line defects in crystal then the well developed dislocation energy is used to derive the energy assigned to object boundaries in images the newly derived registration energy function takes the global gradient information of the entire image into consideration and produces an orientation dependent and long range interaction between two images to drive the registration process this property of interaction endows the new registration framework with both fast convergence rate and high registration accuracy moreover the new energy function can be adapted to realize symmetric diffeomorphic transformation so as to ensure one to one matching between subjects in this paper the superiority of the new method is theoretically proven experimentally tested and compared with the state of the art syn method experimental results with 3 d magnetic resonance brain images demonstrate that the proposed method outperforms the compared methods in terms of both registration accuracy and computation time

3 d deformable image registration a topology preservation scheme based on hierarchical deformation models and interval analysis optimization this paper deals with topology preservation in three dimensional 3 d deformable image registration this work is a nontrivial extension of which addresses the case of two dimensional 2 d topology preserving mappings in both cases the deformation map is modeled as a hierarchical displacement field decomposed on a multiresolution b spline basis topology preservation is enforced by controlling the jacobian of the transformation finding the optimal displacement parameters amounts to solving a constrained optimization problem the residual energy between the target image and the deformed source image is minimized under constraints on the jacobian unlike the 2 d case in which simple linear constraints are derived the 3 d b spline based deformable mapping yields a difficult until now unsolved optimization problem in this paper we tackle the problem by resorting to interval analysis optimization techniques care is taken to keep the computational burden as low as possible results on multipatient 3 d mri registration illustrate the ability of the method to preserve topology on the continuous image domain

piecewise linear multicriteria programs the continuous case and its discontinuous generalization in this paper we study piecewise linear multicriteria programs that is multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint we obtain an algebraic representation of a semi closed polyhedron and apply it to show that the image of a semi closed polyhedron under a continuous linear function is always one semi closed polyhedron we establish that the weak pareto solution point set of a piecewise linear multicriteria program is the union of finitely many semi closed polyhedra we propose an algorithm for finding the pareto point set of a continuous piecewise linear bi criteria program and generalize it to the discontinuous case we apply our algorithm to solve the discontinuous bi criteria portfolio selection problem with an l risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

computational anatomy shape growth and atrophy comparison via diffeomorphisms computational anatomy ca is the mathematical study of anatomy i in i i alpha o g an orbit under groups of diffeomorphisms i e smooth invertible mappings g in g of anatomical exemplars i alpha in i the observable images are the output of medical imaging devices there are three components that ca examines i constructions of the anatomical submanifolds ii comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g in g defining the shape or geometry of the anatomical manifolds and iii generation of probability laws of anatomical variation p on the images i for inference and disease testing within anatomical models this paper reviews recent advances in these three areas applied to shape growth and atrophy

generalized tensor based morphometry of hiv aids using multivariate statistics on deformation tensors this paper investigates the performance of a new multivariate method for tensor based morphometry tbm statistics on riemannian manifolds are developed that exploit the full information in deformation tensor fields in tbm multiple brain images are warped to a common neuroanatomical template via 3 d nonlinear registration the resulting deformation fields are analyzed statistically to identify group differences in anatomy rather than study the jacobian determinant volume expansion factor of these deformations as is common we retain the full deformation tensors and apply a manifold version of hotelling s test to them in a log euclidean domain in 2 d and 3 d magnetic resonance imaging mri data from 26 hiv aids patients and 14 matched healthy subjects we compared multivariate tensor analysis versus univariate tests of simpler tensor derived indices the jacobian determinant the trace geodesic anisotropy and eigenvalues of the deformation tensor and the angle of rotation of its eigenvectors we detected consistent but more extensive patterns of structural abnormalities with multivariate tests on the full tensor manifold their improved power was established by analyzing cumulative value plots using false discovery rate fdr methods appropriately controlling for false positives this increased detection sensitivity may empower drug trials and large scale studies of disease that use tensor based morphometry

well posedness of two nonrigid multimodal image registration methods image registration methods may be designed as solutions of minimization problems on a set of geometric deformations in the nonrigid case solving these problems often means computing the steady state of a system of evolution equations involving the gradient of the error criterion defined from the euler lagrange equations of the corresponding minimization problem the well posedness of the registration methodrequires showing the existence of a solution to the minimization problem as well as that of a stable solution of the evolution equations derived from it we provide such proofs in the case where the error criterion is derived from two different statistical similarity measures global mutual information and local cross covariance we also describe our numerical implementation for solving the corresponding evolution equations and show examples of registrations of real 2d and 3d images achieved with these algorithms the proofs are quite general and can be applied to most of the known nonrigid image

mind demons symmetric diffeomorphic deformable registration of mr and ct for image guided spine surgery intraoperative localization of target anatomy and critical structures defined in preoperative mr ct images can be achieved through the use of multimodality deformable registration we propose a symmetric diffeomorphic deformable registration algorithm incorporating a modality independent neighborhood descriptor mind and a robust huber metric for mr to ct registration the method called mind demons finds a deformation field between two images by optimizing an energy functional that incorporates both the forward and inverse deformations smoothness on the integrated velocity fields a modality insensitive similarity function suitable to multimodality images and smoothness on the diffeomorphisms themselves direct optimization without relying on the exponential map and stationary velocity field approximation used in conventional diffeomorphic demons is carried out using a gauss newton method for fast convergence registration performance and sensitivity to registration parameters were analyzed in simulation phantom experiments and clinical studies emulating application in image guided spine surgery and results were compared to mutual information mi free form deformation ffd local mi lmi ffd normalized mi nmi demons and mind with a diffusion based registration method mind elastic the method yielded sub voxel invertibility 0 008 mm and nonzero positive jacobian determinants it also showed improved registration accuracy in comparison to the reference methods with mean target registration error tre of 1 7 mm compared to 11 3 3 1 5 6 and 2 4 mm for mi ffd lmi ffd nmi demons and mind elastic methods respectively validation in clinical studies demonstrated realistic deformations with sub voxel tre in cases of cervical thoracic and lumbar spine

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

smoothing under diffeomorphic constraints with homeomorphic splines in this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms this diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time dependent vector field this solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques this method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost in a one dimensional setting incorporating diffeomorphic constraints is equivalent to imposing monotonicity thus as an illustration it is shown that such a monotone spline can be used to make monotone any unconstrained estimator of a regression function and that this monotone smoother inherits the convergence properties of the unconstrained estimator some numerical experiments are proposed to illustrate its finite sample performances and to compare them with another monotone estimator we also provide a two dimensional application on the computation of diffeomorphisms for landmark and image matching

statistical properties of jacobian maps and the realization of unbiased large deformation nonlinear image registration maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging mri scans using nonlinear image registration the resulting changes are commonly analyzed using tensor based morphometry to make inferences about anatomical differences often based on the jacobian map which estimates local tissue gain or loss here we provide rigorous mathematical analyses of the jacobian maps and use them to motivate a new numerical method to construct unbiased nonlinear image registration first we argue that logarithmic transformation is crucial for analyzing jacobian values representing morphometric differences we then examine the statistical distributions of log jacobian maps by defining the kullback leibler kl distance on material density functions arising in continuum mechanical models with this framework unbiased image registration can be constructed by quantifying the symmetric kl distance between the identity map and the resulting deformation implementation details addressing the proposed unbiased registration as well as the minimization of symmetric image matching functionals are then discussed and shown to be applicable to other registration methods such as inverse consistent registration in the results section we test the proposed framework as well as present an illustrative application mapping detailed 3 d brain changes in sequential magnetic resonance imaging scans of a patient diagnosed with semantic dementia using permutation tests we show that the symmetrization of image registration statistically reduces skewness in the log jacobian map

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

group actions homeomorphisms and matching a general framework this paper constructs metrics on the space of images cal i defined as orbits under group actions cal g the groups studied include the finite dimensional matrix groups and their products as well as the infinite dimensional diffeomorphisms examined in trouve 1999 quaterly of applied math and dupuis et al 1998 quaterly of applied math left invariant metrics are defined on the product cal g times cal i thus allowing the generation of transformations of the background geometry as well as the image values examples of the application of such metrics are presented for rigid object matching with and without signature variation curves and volume matching and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another

the variational particle mesh method for matching curves diffeomorphic matching only one of several names for this technique is a technique for non rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time series of vector fields one seeks the flow between two topologically equivalent curves or surfaces which minimizes some metric defined on the vector fields i e the flow closest to the identity in some sense in this paper we describe a new particle mesh discretization for the evolution of the geodesic flow and the embedded shape particle mesh algorithms are very natural for this problem because lagrangian particles particles moving with the flow can represent the movement of the shape whereas the vector field is eulerian and hence best represented on a static mesh we explain the derivation of the method and prove the conservation properties the discrete method has a set of conserved momenta corresponding to the particle relabelling symmetry which converge to conserved quantities in the continuous problem we also introduce a new discretization for the geometric current matching condition of vaillant and glaunes 2005 surface matching via currents proc conf ipmi pp 381 92 we illustrate the method and the derived properties with numerical examples

on the shape of plane images this paper studies set patterns in mathbb r 2 and gives a definition of shape in terms of transformation groups where the transformations need not be rigid and linear but more structured than arbitrary homeomorphisms the resulting concept of shape classes and shape groups is studied analytically on such shape classes prior measures intended for bayesian image processing are introduced the priors are given as markov processes of the gibbs type on certain graphs and where the generators are simple geometric objects in mathbb r 2 for example line segments or other arcs these continuum based models differ from earlier lattice based ones in that they incorporate more shape information in the prior measures this leads to algorithms for pattern synthesis and image processing which have been implemented by apl code and applied in an extensive series of computer experiments since the algorithms are computer intensive a limit theorem for the prior measures is presented that is intended to sp

an image morphing technique based on optimal mass preserving mapping image morphing or image interpolation in the time domain deals with the metamorphosis of one image into another in this paper a new class of image morphing algorithms is proposed based on the theory of optimal mass transport the l2 mass moving energy functional is modified by adding an intensity penalizing term in order to reduce the undesired double exposure effect it is an intensity based approach and thus is parameter free the optimal warping function is computed using an iterative gradient descent approach this proposed morphing method is also extended to doubly connected domains using a harmonic parameterization technique along with finite element methods

a pattern theoretic characterization of biological growth mathematical and statistical modeling of biological growth is an important problem in medical diagnostics here we seek tools to analyze changes in anatomical parts using images collected over time we introduce a structured model called growth by random iterated diffeomorphisms grid that treats a cumulative growth deformation as a composition of several elementary deformations each elementary deformation applies to a small region by capturing deformation local to that region and is characterized by a seed and a radial deformation pattern around that seed these grid variables seed locations and radial deformation patterns are estimated from observed images in two steps 1 estimate a cumulative deformation over an observation interval 2 estimate grid variables using maximum likelihood criterion from this estimated cumulative deformation we demonstrate this framework using an mri image data of a rat s brain growth for future statistical analysis we propose a time varying poisson process for the seed placements and a random drawing from a predetermined catalog of deformations for the radial deformation patterns

sparse adaptive parameterization of variability in image ensembles this paper introduces a new parameterization of diffeomorphic deformations for the characterization of the variability in image ensembles dense diffeomorphic deformations are built by interpolating the motion of a finite set of control points that forms a hamiltonian flow of self interacting particles the proposed approach estimates a template image representative of a given image set an optimal set of control points that focuses on the most variable parts of the image and template to image registrations that quantify the variability within the image set the method automatically selects the most relevant control points for the characterization of the image variability and estimates their optimal positions in the template domain the optimization in position is done during the estimation of the deformations without adding any computational cost at each step of the gradient descent the selection of the control points is done by adding a l 1 prior to the objective function which is optimized using the fista algorithm

automated sulcal segmentation using watersheds on the cortical surface the human cortical surface is a highly complex folded structure sulci the spaces between the folds define location on the cortex and provide a parcellation into anatomically distinct areas a topic that has recently received increased attention is the segmentation of these sulci from magnetic resonance images with most work focusing on extracting either the sulcal spaces between the folds or curve representations of sulci unlike these methods we propose a technique that extracts actual regions of the cortical surface that surround sulci which we call sulcal regions the method is based on a watershed algorithm applied to a geodesic depth measure on the cortical surface a well known problem with the watershed algorithm is a tendency toward oversegmentation meaning that a single region is segmented as several pieces to address this problem we propose a postprocessing algorithm that merges appropriate segments from the watershed algorithm the sulcal regions are then manually labeled by simply selecting the appropriate regions with a mouse click and a preliminary study of sulcal depth is reported finally a scheme is presented for computing a complete parcellation of the cortical surface

multimodal brain warping using the demons algorithm and adaptative intensity corrections this paper presents an original method for three dimensional elastic registrat ion of multimodal images we propose to make use of a scheme that iterates between correcting for intensity differences between images and performing standard monomodal registration the core of our contribution resides in providing a method that finds the transformation that maps the intensities of one image to those of another it makes the assumption that there are at most two functional dependences between the intensities of structures present in the images to register and relies on robust estimation techniques to evaluate these functions we provide results showing successful registratio n between several imaging modalities involving segmentations t1 magnetic resonance mr t2 mr proton density pd mr and computed tomography ct we also argue that our intensity modeling may be more appropriate than mutual information mi in the context of evaluating high dimensional deformations as it puts more constraints on the parameters to be estimated and thus permits a better search of the parameter space

automatic construction of parts geometry models for initializing groupwise registration groupwise nonrigid image registration is a powerful tool to automatically establish correspondences across sets of images such correspondences are widely used for constructing statistical models of shape and appearance as existing techniques usually treat registration as an optimization problem a good initialization is required although the standard initialization affine transformation generally works well it is often inadequate when registering images of complex structures in this paper we present a more sophisticated method that uses the sparse matches of a parts geometry model as the initialization we show that both the model and its matches can be automatically obtained and that the matches are able to effectively initialize a groupwise nonrigid registration algorithm leading to accurate dense correspondences we also show that the dense mesh models constructed during the groupwise registration process can be used to accurately annotate new images we demonstrate the efficacy of the approach on three datasets of increasing difficulty and report on a detailed quantitative evaluation of its performance

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

sharpmean groupwise registration guided by sharp mean image and tree based registration groupwise registration has become more and more popular due to its attractiveness for unbiased analysis of population data one of the most popular approaches for groupwise registration is to iteratively calculate the group mean image and then register all subject images towards the latest estimated group mean image however its performance might be undermined by the fuzzy mean image estimated in the very beginning of groupwise registration procedure because all subject images are far from being well aligned at that moment in this paper we first point out the significance of always keeping the group mean image sharp and clear throughout the entire groupwise registration procedure which is intuitively important but has not been explored in the literature yet to achieve this we resort to developing the robust mean image estimator by the adaptive weighting strategy where the weights are adaptive across not only the individual subject images but also all spatial locations in the image domain on the other hand we notice that some subjects might have large anatomical variations from the group mean image which challenges most of the state of the art registration algorithms to ensure good registration results in each iteration we explore the manifold of subject images and build a minimal spanning tree mst with the group mean image as the root of the mst therefore each subject image is only registered to its parent node often with similar shapes and its overall transformation to the group mean image space is obtained by concatenating all deformations along the paths connecting itself to the root of the mst the group mean image as a result all the subjects will be well aligned to the group mean image adaptively our method has been evaluated in both real and simulated datasets in all experiments our method outperforms the conventional algorithm which generally produces a fuzzy group mean image throughout the entire groupwise registration

magnetic resonance image tissue classification using a partial volume model abstract we describe a sequence of low level operations to isolate and classify brain tissue within t1 weighted magnetic resonance images mri our method first removes nonbrain tissue using a combination of anisotropic diffusion filtering edge detection and mathematical morphology we compensate for image nonuniformities due to magnetic field inhomogeneities by fitting a tricubic b spline gain field to local estimates of the image nonuniformity spaced throughout the mri volume the local estimates are computed by fitting a partial volume tissue measurement model to histograms of neighborhoods about each estimate point the measurement model uses mean tissue intensity and noise variance values computed from the global image and a multiplicative bias parameter that is estimated for each region during the histogram fit voxels in the intensity normalized image are then classified into six tissue types using a maximum a posteriori classifier this classifier combines the partial volume tissue measurement model with a gibbs prior that models the spatial properties of the brain we validate each stage of our algorithm on real and phantom data using data from the 20 normal mri brain data sets of the internet brain segmentation repository our method achieved average indices of 0 746 0 114 for gray matter gm and 0 798 0 089 for white matter wm compared to expert labeled data our method achieved average indices 0 893 0 041 for gm and 0 928 0 039 for wm compared to the ground truth labeling on 12 volumes from the montreal neurological institute s brainweb phantom

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

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility

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