empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

shape recognition with spectral distances recent works have shown the use of diffusion geometry for various pattern recognition applications including nonrigid shape analysis in this paper we introduce spectral shape distance as a general framework for distribution based shape similarity and show that two recent methods for shape similarity due to rustamov and mahmoudi and sapiro are particular cases thereof

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

algorithms to automatically quantify the geometric similarity of anatomical surfaces we describe approaches for distances between pairs of two dimensional surfaces embedded in three dimensional space that use local structures and global information contained in interstructure geometric relationships we present algorithms to automatically determine these distances as well as geometric correspondences this approach is motivated by the aspiration of students of natural science to understand the continuity of form that unites the diversity of life at present scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points landmarks identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists this necessity renders these studies inaccessible to nonmorphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns unlike other algorithms presented for morphological correspondences our approach does not require any preliminary marking of special features or landmarks by the user it also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster making pairwise comparisons feasible for significantly larger numbers of digitized surfaces we illustrate our approach using three datasets representing teeth and different bones of primates and humans and show that it leads to highly accurate results

applying tensor based morphometry to parametric surfaces can improve mri based disease diagnosis many methods have been proposed for computer assisted diagnostic classification full tensor information and machine learning with 3d maps derived from brain images may help detect subtle differences or classify subjects into different groups here we develop a new approach to apply tensor based morphometry to parametric surface models for diagnostic classification we use this approach to identify cortical surface features for use in diagnostic classifiers first with holomorphic 1 forms we compute an efficient and accurate conformal mapping from a multiply connected mesh to the so called slit domain next the surface parameterization approach provides a natural way to register anatomical surfaces across subjects using a constrained harmonic map to analyze anatomical differences we then analyze the full riemannian surface metric tensors which retain multivariate information on local surface geometry as the number of voxels in a 3d image is large sparse learning is a promising method to select a subset of imaging features and to improve classification accuracy focusing on vertices with greatest effect sizes we train a diagnostic classifier using the surface features selected by an l1 norm based sparse learning method stability selection is applied to validate the selected feature sets we tested the algorithm on mri derived cortical surfaces from 42 subjects with genetically confirmed williams syndrome and 40 age matched controls multivariate statistics on the local tensors gave greater effect sizes for detecting group differences relative to other tbm based statistics including analysis of the jacobian determinant and the largest eigenvalue of the surface metric our method also gave reasonable classification results relative to the jacobian determinant the pair of eigenvalues of the jacobian matrix and volume features this analysis pipeline may boost the power of morphometry studies and may assist with image based classification

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

adaptive hypothesis testing using wavelets let a function f be observed with a noise we wish to test the null hypothesis that the function is identically zero against a composite nonparametric alternative functions from the alternative set are separated away from zero in an integral e g l 2 norm and also possess some smoothness properties the minimax rate of testing for this problem was evaluated in earlier papers by ingster and by lepski and spokoiny under different kinds of smoothness assumptions it was shown that both the optimal rate of testing and the structure of optimal in rate tests depend on smoothness parameters which are usually unknown in practical applications in this paper the problem of adaptive assumption free testing is considered it is shown that adaptive testing without loss of efficiency is impossible an extra log log factor is inessential but unavoidable payment for the adaptation a simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of besov classes

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

multivariate tensor based morphometry on surfaces application to mapping ventricular abnormalities in hiv aids here we developed a new method called multivariate tensor based surface morphometry tbm and applied it to study lateral ventricular surface differences associated with hiv aids using concepts from differential geometry and the theory of differential forms we created mathematical structures known as holomorphic one forms to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain the new meshing approach also provides a natural way to register anatomical surfaces across subjects and improves on prior methods as it handles surfaces that branch and join at complex 3d junctions to analyze anatomical differences we computed new statistics from the riemannian surface metrics these retain multivariate information on local surface geometry we applied this framework to analyze lateral ventricular surface morphometry in 3d mri data from 11 subjects with hiv aids and 8 healthy controls our method detected a 3d profile of surface abnormalities even in this small sample multivariate statistics on the local tensors gave better effect sizes for detecting group differences relative to other tbm based methods including analysis of the jacobian determinant the largest and smallest eigenvalues of the surface metric and the pair of eigenvalues of the jacobian matrix the resulting analysis pipeline may improve the power of surface based morphometry studies of the brain

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

the moments of ols and 2sls when the disturbances are non normal abstract this paper examines the exact moments of ols and 2sls structural estimators under the assumption that the reduced form disturbances follow a non normal distribution of the edgeworth type with specified skewness 3 and kurtosis 4 the exact results are given as functions of hypergeometric functions and 3 and 4 the technique used is that developed by davis 1976 and it is shown that when 3 4 0 the results correspond to sawa s 1972

surface fluid registration of conformal representation application to detect disease burden and genetic influence on hippocampus in this paper we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1 forms inverse consistent surface fluid registration and multivariate tensor based morphometry mtbm first we conformally map a surface onto a planar rectangle space with holomorphic 1 forms second we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface third we align the feature image with a chosen template image via the fluid image registration algorithm which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization the inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images this alignment induces a corresponding deformation on the surface we tested the system on alzheimer s disease neuroimaging initiative adni baseline dataset to study ad symptoms on hippocampus in our system by modeling a hippocampus as a 3d parametric surface we nonlinearly registered each surface with a selected template surface then we used mtbm to analyze the morphometry difference between diagnostic groups experimental results show that the new system has better performance than two publicly available subcortical surface registration tools first and spharm we also analyzed the genetic influence of the apolipoprotein e 4 allele apoe4 which is considered as the most prevalent risk factor for ad our work successfully detected statistically significant difference between apoe4 carriers and non carriers in both patients of mild cognitive impairment mci and healthy control subjects the results show evidence that the apoe genotype may be associated with accelerated brain atrophy so that our work provides a new mri analysis tool that may help presymptomatic ad research

on the use of nonparametric regression for checking linear relationships summary the problem of checking the linearity of a regression relationship is addressed through the idea of smoothing of a residual plot a pseudolikelihood ratio test statistic which measures the distance between the nonparametric and the parametric models is derived as a ratio of quadratic forms the distribution of this statistic under the null hypothesis of linearity is calculated numerically by using johnson curves a power study shows the new statistic to be more sensitive to non linearity than the durbin watson statistic

towards a holistic cortical thickness descriptor heat kernel based grey matter morphology signatures in this paper we propose a heat kernel based regional shape descriptor that may be capable of better exploiting volumetric morphological information than other available methods thereby improving statistical power on brain magnetic resonance imaging mri analysis the mechanism of our analysis is driven by the graph spectrum and the heat kernel theory to capture the volumetric geometry information in the constructed tetrahedral meshes in order to capture profound brain grey matter shape changes we first use the volumetric laplace beltrami operator to determine the point pair correspondence between white grey matter and csf grey matter boundary surfaces by computing the streamlines in a tetrahedral mesh secondly we propose multi scale grey matter morphology signatures to describe the transition probability by random walk between the point pairs which reflects the inherent geometric characteristics thirdly a point distribution model is applied to reduce the dimensionality of the grey matter morphology signatures and generate the internal structure features with the sparse linear discriminant analysis we select a concise morphology feature set with improved classification accuracies in our experiments the proposed work outperformed the cortical thickness features computed by freesurfer software in the classification of alzheimer s disease and its prodromal stage i e mild cognitive impairment on publicly available data from the alzheimer s disease neuroimaging initiative the multi scale and physics based volumetric structure feature may bring stronger statistical power than some traditional methods for mri based grey matter morphology analysis

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

quadrilateral and hexahedral mesh generation based on surface foliation theory ii abstract automatic hexahedral mesh generation plays a fundamental role in cad cae fields especially for iga isogeometric analysis recently an automatic hex mesh generation method has been proposed which is based on the equivalence among three key concepts colorable quadrilateral meshes finite measured foliations and strebel differentials this work focuses on the computational aspect of strebel differentials on high genus surfaces which is based on graph valued harmonic mapping the algorithmic pipeline is as follows first the user inputs an admissible curve system which induces a cylindric decomposition graph then the user specifies the lengths of the edges of the graph third the algorithm finds the unique harmonic map from the surface to the metric graph by a non linear heat flow finally the harmonic map induces a strebel differential which can be further utilized to generate the hex mesh the method has solid theoretic foundation which guarantees the existence and the uniqueness of the graph valued harmonic map the algorithm is capable of handling surfaces with complicated topologies and producing all possible strebel differentials on the surface the user has full control of the combinatorial type and the geometry of the strebel differential the computational pipeline is automatic the experimental results show the efficiency and efficacy of the algorithm and demonstrate the great potential for automating the structured hexahedral mesh generation

a parameterization based numerical method for isotropic and anisotropic diffusion smoothing on non flat surfaces neuroimaging data such as 3d maps of cortical thickness or neural activation can often be analyzed more informatively with respect to the cortical surface rather than the entire volume of the brain any cortical surface based analysis should be carried out using computations in the intrinsic geometry of the surface rather than using the metric of the ambient 3d space we present parameterization based numerical methods for performing isotropic and anisotropic filtering on triangulated surface geometries in contrast to existing fem based methods for triangulated geometries our approach accounts for the metric of the surface in order to discretize and numerically compute the isotropic and anisotropic geometric operators we first parameterize the surface using a p harmonic mapping we then use this parameterization as our computational domain and account for the surface metric while carrying out isotropic and anisotropic filtering to validate our method we compare our numerical results to the analytical expression for isotropic diffusion on a spherical surface we apply these methods to smoothing of mean curvature maps on the cortical surface a step commonly required for analysis of gyrification or for registering surface based maps across subjects

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