universal manifold embedding for geometrically deformed functions assume we have a set of observations for example images of different objects each undergoing a different geometric deformation yet all the deformations belong to the same family as a result of the action of these deformations the set of different observations on each object is generally a manifold in the ambient space of observations in this paper we show that in those cases where the set of deformations admits a finite dimensional representation there is a mapping from the space of observations to a low dimensional linear space the manifold corresponding to each object is mapped to a distinct linear subspace of euclidean space the dimension of the subspace is the same as that of the manifold this mapping which we call universal manifold embedding enables the estimation of geometric deformations using the classical linear theory the universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework the embedding of the space of observations depends on the deformation model and is independent of the specific observed object hence it is universal we study two cases of this embedding that of elastic deformations of 1 d signals and the case of affine deformations of n dimensional signals

on the a posteriori error analysis for equations of prescribed mean curvature we present two approaches to the a posteriori error analysis for prescribed mean curvature equations the main difference between them concerns the estimation of the residual without or with computable weights in the second case the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning we analyze the two approaches from a theoretical viewpoint moreover we investigate and compare the performance of the derived indicators in an adaptive procedure our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way

pseudo transient continuation for nonsmooth nonlinear equations pseudo transient continuation is a newton like iterative method for computing steady state solutions of differential equations in cases where the initial data are far from a steady state the iteration mimics a temporal integration scheme with the time step being increased as steady state is approached the iteration is an inexact newton iteration in the terminal phase in this paper we show how steady state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo transient continuation an example of such a case is a discretized pde with a lipschitz continuous but nondifferentiable constitutive relation as part of the nonlinearity in this case we can approximate a generalized derivative with a difference quotient the existing theory for pseudo transient continuation requires lipschitz continuity of the jacobian newton like methods for nonsmooth equations have been globalized by trust region methods smooth approximations and splitting methods in the past but these approaches are not designed to find steady state solutions of time dependent problems the method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo transient continuation

on the deterministic computation of functional integrals in application to quantum mechanical problems abstract new approximate formulae for functional integrals with gaussian measure in separable frechet spaces are derived as a special case the integration with respect to the conditional wiener measure is investigated for conditional wiener integrals a family of approximate formulae with weight is constructed the quantum mechanical models namely the linear and the inharmonic oscillators are described the efficiency of the formulas is demonstrated in the numerical comparison with the monte carlo method on lattice

learning smooth pattern transformation manifolds manifold models provide low dimensional representations that are useful for processing and analyzing data in a transformation invariant way in this paper we study the problem of learning smooth pattern transformation manifolds from image sets that represent observations of geometrically transformed signals to construct a manifold we build a representative pattern whose transformations accurately fit various input images we examine two objectives of the manifold building problem namely approximation and classification for the approximation problem we propose a greedy method that constructs a representative pattern by selecting analytic atoms from a continuous dictionary manifold we present a dc optimization scheme that is applicable to a wide range of transformation and dictionary models and demonstrate its application to the transformation manifolds generated by the rotation translation and anisotropic scaling of a reference pattern then we generalize this approach to a setting with multiple transformation manifolds where each manifold represents a different class of signals we present an iterative multiple manifold building algorithm such that the classification accuracy is promoted in the learning of the representative patterns the experimental results suggest that the proposed methods yield high accuracy in the approximation and classification of data compared with some reference methods while the invariance to geometric transformations is achieved because of the transformation manifold model

new mathematical and algorithmic schemes for pattern classification with application to the identification of writers of important ancient documents abstract in this paper a novel approach is introduced for classifying curves into proper families according to their similarity first a mathematical quantity we call plane curvature is introduced and a number of propositions are stated and proved proper similarity measures of two curves are introduced and a subsequent statistical analysis is applied first the efficiency of the curve fitting process has been tested on 2 shapes datasets of reference next the methodology has been applied to the very important problem of classifying 23 byzantine codices and 46 ancient inscriptions to their writers thus achieving correct dating of their content the inscriptions have been attributed to ten individual hands and the byzantine codices to four writers

optimal image alignment with random projections of manifolds algorithm and geometric analysis this paper addresses the problem of image alignment based on random measurements image alignment consists of estimating the relative transformation between a query image and a reference image we consider the specific problem where the query image is provided in compressed form in terms of linear measurements captured by a vision sensor we cast the alignment problem as a manifold distance minimization problem in the linear subspace defined by the measurements the transformation manifold that represents synthesis of shift rotation and isotropic scaling of the reference image can be given in closed form when the reference pattern is sparsely represented over a parametric dictionary we show that the objective function can then be decomposed as the difference of two convex functions dc in the particular case where the dictionary is built on gaussian functions thus the optimization problem becomes a dc program which in turn can be solved globally by a cutting plane method the quality of the solution is typically affected by the number of random measurements and the condition number of the manifold that describes the transformations of the reference image we show that the curvature which is closely related to the condition number remains bounded in our image alignment problem which means that the relative transformation between two images can be determined optimally in a reduced subspace

parameterized principal component analysis abstract when modeling multivariate data one might have an extra parameter of contextual information that could be used to treat some observations as more similar to others for example images of faces can vary by age and one would expect the face of a 40 year old to be more similar to the face of a 30 year old than to a baby face we introduce a novel manifold approximation method parameterized principal component analysis ppca that models data with linear subspaces that change continuously according to the extra parameter of contextual information e g age instead of ad hoc atlases special care has been taken in the loss function and the optimization method to encourage smoothly changing subspaces across the parameter values the approach ensures that each observation s projection will share information with observations that have similar parameter values but not with observations that have large parameter differences we tested ppca on artificial data based on known smooth functions of an added parameter as well as on two real datasets with different types of parameters we compared ppca to pca sparse pca and to independent principal component analysis ipca an atlas based method that groups observations by their parameter values and projects each group using pca with no sharing of information between groups ppca recovers the known functions with less error and projects the datasets test set observations with consistently less reconstruction error than ipca does in some cases where the manifold is truly nonlinear pca outperforms all the other manifold approximation methods compared

an improved method for solving quasi linear convection diffusion problems on a coarse mesh a method is developed for solving quasi linear convection diffusion problems starting on a coarse mesh where the data and solution dependent coefficients are unresolved the problem is unstable and approximation properties do not hold the newton like iterations of the solver are based on the framework of regularized pseudotransient continuation where the proposed time integrator is a variation on the newmark strategy designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite an algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning the method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization derivation and q linear loc

image registration using adaptive polar transform image registration is an essential step in many image processing applications that need visual information from multiple images for comparison integration or analysis recently researchers have introduced image registration techniques using the log polar transform lpt for its rotation and scale invariant properties however it suffers from nonuniform sampling which makes it not suitable for applications in which the registered images are altered or occluded inspired by lpt this paper presents a new registration algorithm that addresses the problems of the conventional lpt while maintaining the robustness to scale and rotation we introduce a novel adaptive polar transform apt technique that evenly and effectively samples the image in the cartesian coordinates combining apt with an innovative projection transform along with a matching mechanism the proposed method yields less computational load and more accurate registration than that of the conventional lpt translation between the registered images is recovered with the new search scheme using gabor feature extraction to accelerate the localization procedure moreover an image comparison scheme is proposed for locating the area where the image pairs differ experiments on real images demonstrate the effectiveness and robustness of the proposed approach for registering images that are subjected to occlusion and alteration in addition to scale rotation and translation

stabilized and inexact adaptive methods for capturing internal layers in quasilinear pde a method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh the solution process is assumed to start on a mesh where the problem is badly resolved and approximation properties of the exact problem and its corresponding finite element solution do not hold a sequence of stabilized and inexact partial solves allows the mesh to be refined to capture internal layers while an approximate solution is built eventually leading to an accurate approximation of both the problem and its solution the innovations in the current work include a closed form definition for the numerical dissipation and inexact scaling parameters on each mesh refinement as well as a convergence result for the residual of the discrete problem numerical experiments demonstrate the method on a range of problems featuring steep internal layers and high solution dependent frequencies of the diffusion coefficients

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

geometry and radiometry invariant matched manifold detection consider a set of deformable objects undergoing geometric and radiometric transformations as a result of the action of these transformations the set of different realizations of each object is generally a manifold in the space of observations assuming the geometric deformations an object undergoes belong to some finite dimensional family it has been shown that the universal manifold embedding ume provides a set of nonlinear operators that universally maps each of the different manifolds where each manifold is generated by the set all of possible appearances of a single object into a distinct linear subspace of an euclidean space in this paper we generalize this framework to the case where the observed object undergoes both an affine geometric transformation and a monotonic radiometric transformation and present a novel framework for the detection and recognition of the deformable objects applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations but is geometry covariant with the affine transformation the set of all possible observations on that object is mapped by the ume into a single linear subspace invariant with respect to both the geometric and radiometric transformations the embedding of the space of observations is independent of the specific observed object hence it is universal the invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations the observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities since each tile is mapped by the radiometry invariant ume to a distinct linear subspace the detection and tracking problems are solved by evaluating distances between linear subspaces classification in this context becomes a problem of determining which labeled subspace in a grassmannian is closest to a subspace in the same grassmannian where the latter has been generated by radiometry invariant ume from an unlabeled observation

3d face recognition with sparse spherical representations this paper addresses the problem of 3d face recognition using simultaneous sparse approximations on the sphere the 3d face point clouds are first aligned with a fully automated registration process they are then represented as signals on the 2 sphere in order to preserve depth and geometry information next we implement a dimensionality reduction process with simultaneous sparse approximations and subspace projection it permits to represent each 3d face by only a few spherical functions that are able to capture the salient facial characteristics and hence to preserve the discriminant facial information we eventually perform recognition by effective matching in the reduced space where linear discriminant analysis can be further activated for improved recognition performance the 3d face recognition algorithm is evaluated on the frgc v 1 0 data set where it is shown to outperform classical state of the art solutions that work with depth images

asymptotic mesh independence of newton galerkin methods via a refined mysovskii theorem the paper presents a theoretical characterization of the often observed asymptotic mesh independence of newton s method which means that newton s method applied to discretized operator equations behaves essentially the same for all sufficiently fine discretizations unlike previous theoretical approaches the theory presented here does not need any uniform lipschitz assumptions which may be hard to verify in realistic problems or in some cases may not hold at all moreover this approach leads to a significantly simpler presentation the new results are obtained by means of a new refined newton mysovskii theorem in affine invariant formulation this theorem will be of interest in a wider context because it gives both existence and uniqueness of the solution and quadratic convergence for sufficiently good starting points attention is restricted to galerkin approximations even though similar results should hold for finite difference methods but corresponding proofs would certainly be more technical

discretization of parametrizable signal manifolds transformation invariant analysis of signals often requires the computation of the distance from a test pattern to a transformation manifold in particular the estimation of the distances between a transformed query signal and several transformation manifolds representing different classes provides essential information for the classification of the signal in many applications the computation of the exact distance to the manifold is costly whereas an efficient practical solution is the approximation of the manifold distance with the aid of a manifold grid in this paper we consider a setting with transformation manifolds of known parameterization we first present an algorithm for the selection of samples from a single manifold that permits to minimize the average error in the manifold distance estimation then we propose a method for the joint discretization of multiple manifolds that represent different signal classes where we optimize the transformation invariant classification accuracy yielded by the discrete manifold representation experimental results show that sampling each manifold individually by minimizing the manifold distance estimation error outperforms baseline sampling solutions with respect to registration and classification accuracy performing an additional joint optimization on all samples improves the classification performance further moreover given a fixed total number of samples to be selected from all manifolds an asymmetric distribution of samples to different manifolds depending on their geometric structures may also increase the classification accuracy in comparison with the equal distribution of samples

a pyramid approach to subpixel registration based on intensity we present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set which can be either images two dimensional or volumes three dimensional it uses an explicit spline representation of the images in conjunction with spline processing and is based on a coarse to fine iterative strategy pyramid approach the minimization is performed according to a new variation ml of the marquardt levenberg algorithm for nonlinear least square optimization the geometric deformation model is a global three dimensional 3 d affine transformation that can be optionally restricted to rigid body motion rotation and translation combined with isometric scaling it also includes an optional adjustment of image contrast differences we obtain excellent results for the registration of intramodality positron emission tomography pet and functional magnetic resonance imaging fmri data we conclude that the multiresolution refinement strategy is more robust than a comparable single stage method being less likely to be trapped into a false local optimum in addition our improved version of the marquardt levenberg algorithm is faster

computation of conditional wiener integrals by the composite approximation formulas with weight abstract new approximation formulas with weight for the functional integrals with conditional wiener measure are derived the formulas are exact on a class of polynomial functionals of a given degree the convergence of approximations to the exact value of the integral is proved the estimate of the remainder is obtained the results are illustrated with numerical examples the advantages of the formulas over lattice monte carlo method are demonstrated in computation of some quantities in euclidean quantum mechanics

geodesic remeshing using front propagation in this paper we propose a complete framework for 3d geometry modeling and processing that uses only fast geodesic computations the basic building block for these techniques is a novel greedy algorithm to perform a uniform or adaptive remeshing of a triangulated surface our other contributions include a parameterization scheme based on barycentric coordinates an intrinsic algorithm for computing geodesic centroidal tessellations and a fast and robust method to flatten a genus 0 surface patch on large meshes more than 500 000 vertices our techniques speed up computation by over one order of magnitude in comparison to classical remeshing and parameterization methods our methods are easy to implement and do not need multilevel solvers to handle complex models that may contain poorly shaped triangles

a spherical representation for recognition of free form surfaces introduces a new surface representation for recognizing curved objects the authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object the mesh is computed from the data by deforming a standard shaped mesh for example an ellipsoid until it fits the surface of the object the authors define local regularity constraints that the mesh must satisfy the authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh a surface curvature index that is pose invariant is stored at every node of the mesh the authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene the authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation the authors present results on real range images which show that this approach to modelling and recognizing 3d objects has three main advantages 1 it is applicable to complex curved surfaces that cannot be handled by conventional techniques 2 it reduces the recognition problem to the computation of similarity between spherical distributions in particular the recognition algorithm does not require any combinatorial search and 3 even though it is based on a spherical mapping the approach can handle occlusions and partial views

out of sample generalizations for supervised manifold learning for classification supervised manifold learning methods for data classification map high dimensional data samples to a lower dimensional domain in a structure preserving way while increasing the separation between different classes most manifold learning methods compute the embedding only of the initially available data however the generalization of the embedding to novel points i e the out of sample extension problem becomes especially important in classification applications in this paper we propose a semi supervised method for building an interpolation function that provides an out of sample extension for general supervised manifold learning algorithms studied in the context of classification the proposed algorithm computes a radial basis function interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples defined as their distance to the embeddings of the manifolds of their own class as well as a regularization term that controls the smoothness of the interpolation function in a direction dependent way the class labels of test data and the interpolation function parameters are estimated jointly with an iterative process experimental results on face and object images demonstrate the potential of the proposed out of sample extension algorithm for the classification of manifold modeled data sets

functional integrals for nuclear many particle systems the new method for computation of the physical characteristics of quantum systems with many degrees of freedom is described this method is based on the representation of the matrix element of the evolution operator in euclidean metrics in a form of the functional integral with a certain measure in the corresponding space and on the use of approximation formulae which we constructed for this kind of integral the method does not require preliminary discretization of space and time and allows us to use the deterministic algorithms this approach proved to have important advantages over the other known methods including the higher efficiency of computations examples of application of the method to the numerical study of some potential nuclear models as well as comparison of results with the experimental data and with the values obtained by the other authors are presented

studying open quantum systems by means of a deterministic approach to approximate functional integration representation of propagator for open quantum systems in the form of a double functional integral with respect to conditional wiener measure is proposed it allows one to apply the approximate formulas exact for functional polynomials of a certain power to calculation of such integrals within this deterministic approach the problem is reduced to evaluation of usual riemann integrals of low multiplicity the formulas are in fact the basis of a numerical method of studying time evolution of the systems the features of the method are discussed and some examples of calculations are given

manifold manifold distance and its application to face recognition with image sets in this paper we address the problem of classifying image sets for face recognition where each set contains images belonging to the same subject and typically covering large variations by modeling each image set as a manifold we formulate the problem as the computation of the distance between two manifolds called manifold manifold distance mmd since an image set can come in three pattern levels point subspace and manifold we systematically study the distance among the three levels and formulate them in a general multilevel mmd framework specifically we express a manifold by a collection of local linear models each depicted by a subspace mmd is then converted to integrate the distances between pairs of subspaces from one of the involved manifolds we theoretically and experimentally study several configurations of the ingredients of mmd the proposed method is applied to the task of face recognition with image sets where identification is achieved by seeking the minimum mmd from the probe to the gallery of image sets our experiments demonstrate that as a general set similarity measure mmd consistently outperforms other competing nondiscriminative methods and is also promisingly comparable to the state of the art discriminative methods

uniqueness of discrete solutions of nonmonotone pdes without a globally fine mesh condition uniqueness of the finite element solution for nonmonotone quasilinear problems of elliptic type is established in one and two dimensions in each case we prove a comparison theorem based on locally bounding the variation of the discrete solution over each element the uniqueness follows from this result and does not require a globally small meshsize

computation of multiple functional integrals in quantum physics the new method of computation of multiple functional integrals of quantum physics is elaborated the method is based on the numerical integration over complete separable metric spaces via approximations exact on a class of polynomial functionals of given degree new approximation formulas for the functional integrals with respect to gaussian measure are constructed the convergence of approximations to an exact value of integral is proved the estimate of the remainder is obtained in the particular case of conditional wiener measure the approximation formulas with the weight are derived the method is applied to the study of the multidimensional calogero model and to computation of the binding of nucleons in the nucleus of tritium

numerical solution of some partial differential equations by means of a deterministic method of approximate functional integration a numerical method of solution of some partial differential equations is presented the method is based on representation of green functions of the equations in the form of functional integrals and subsequent approximate calculation of the integrals with the help of a deterministic approach in this case the solution of the equations is reduced to evaluation of usual riemann integrals of relatively low multiplicity a procedure allowing one to increase accuracy of the solutions is suggested the features of the method are investigated on examples of numerical solution of the schrodinger equation and related diffusion equation

minimum distance between pattern transformation manifolds algorithm and applications transformation invariance is an important property in pattern recognition where different observations of the same object typically receive the same label this paper focuses on a transformation invariant distance measure that represents the minimum distance between the transformation manifolds spanned by patterns of interest since these manifolds are typically nonlinear the computation of the manifold distance md becomes a nonconvex optimization problem we propose representing a pattern of interest as a linear combination of a few geometric functions extracted from a structured and redundant basis transforming the pattern results in the transformation of its constituent parts we show that when the transformation is restricted to a synthesis of translations rotations and isotropic scalings such a pattern representation results in a closed form expression of the manifold equation with respect to the transformation parameters the md computation can then be formulated as a minimization problem whose objective function is expressed as the difference of convex functions dc this interesting property permits optimally solving the optimization problem with dc programming solvers that are globally convergent we present experimental evidence which shows that our method is able to find the globally optimal solution outperforming existing methods that yield suboptimal solutions

maximum likelihood localization of 2 d patterns in the gauss laguerre transform domain theoretic framework and preliminary results usual approaches to localization i e joint estimation of position orientation and scale of a bidimensional pattern employ suboptimum techniques based on invariant signatures which allow for position estimation independent of scale and orientation in this paper a maximum likelihood method for pattern localization working in the gauss laguerre transform glt domain is presented the glt is based on an orthogonal family of circular harmonic functions with specific radial profiles which permits optimum joint estimation of position and scale rotation parameters looking at the maxima of a gauss laguerre likelihood map the fisher information matrix for any given pattern is given and the theoretical asymptotic accuracy of the parameter estimates is calculated through the cramer rao lower bound application of the ml estimation method is discussed and an example is provided