ambiguity in structure from motion sphere versus plane if 3d rigid motion can be correctly estimated from image sequences the structure of the scene can be correctly derived using the equations for image formation however an error in the estimation of 3d motion will result in the computation of a distorted version of the scene structure of computational interest are these regions in space where the distortions are such that the depths become negative because in order for the scene to be visible it has to lie in front of the image and thus the corresponding depth estimates have to be positive the stability analysis for the structure from motion problem presented in this paper investigates the optimal relationship between the errors in the estimated translational and rotational parameters of a rigid motion that results in the estimation of a minimum number of negative depth values the input used is the value of the flow along some direction which is more general than optic flow or correspondence for a planar retina it is shown that the optimal configuration is achieved when the projections of the translational and rotational errors on the image plane are perpendicular furthermore the projection of the actual and the estimated translation lie on a line through the center for a spherical retina given a rotational error the optimal translation is the correct one given a translational error the optimal rotational negative deptherror depends both in direction and value on the actual and estimated translation as well as the scene in view the proofs besides illuminating the confounding of translation and rotation in structure from motion have an important application to ecological optics the same analysis provides a computational explanation of why it is easier to estimate self motion in the case of a spherical retina and why shape can be estimated easily in the case of a planar retina thus suggesting that nature s design of compound eyes or panoramic vision for flying systems and camera type eyes for primates and other systems that perform manipulation is optimal

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

universal gaussian approximations under random censorship universal gaussian approximations are established for empirical cumulative hazard and product limit processes under random censorship they hold uniformly up to some large order statistics in the sample with the approximation rates depending on the order of these statistics and require no assumptions on the censoring mechanism weak convergence results and laws of the iterated logarithm follow on the whole line if the respective processes are stopped at certain large order statistics depending on the type of result some new consequences and negative results for confidence band construction are discussed some new uniform consistency rates up to large order statistics are also derived and shown to be universally best possible for a wide range of tail order statistics

gaussian approximation of local empirical processes indexed by functions an extended notion of a local empirical process indexed by functions is introduced which includes kernel density and regression function estimators and the conditional empirical process as special cases under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of gaussian processes are established for such processes a compact law of the iterated logarithm lil is then inferred from the corresponding lil for the approximating sequence of gaussian processes a number of statistical applications of our results are indicated

extracting structure from optical flow using the fast error search technique in this paper we present a globally optimal and computationally efficient technique for estimating the focus of expansion foe of an optical flow field using fast partial search for each candidate location on a discrete sampling of the image area we generate a linear system of equations for determining the remaining unknowns viz rotation and inverse depth we compute the least squares error of the system without actually solving the equations to generate an error surface that describes the goodness of fit across the hypotheses using fourier techniques we prove that given an n n flow field the foe and subsequently rotation and structure can be estimated in cal o n 2 log n operations since the resulting system is linear bounded perturbations in the data lead to bounded errors we support the theoretical development and proof of our technique with experiments on synthetic and real data through a series of experiments on synthetic data we prove the correctness robustness and operating envelope of our algorithm we demonstrate the utility of our technique by applying it for detecting obstacles from a monocular sequence of images

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

model based brightness constraints on direct estimation of structure and motion we describe a direct method for estimating structure and motion from image intensities of multiple views we extend the direct methods of horn and weldon 1988 to three views adding the third view enables us to solve for motion and compute a dense depth map of the scene directly from image spatio temporal derivatives in a linear manner without first having to find point correspondences or compute optical flow we describe the advantages and limitations of this method which are then verified with experiments using real images

functional laws of the iterated logarithm for local empirical processes indexed by sets we introduce the notion of a multivariate local and tail empirical process indexed by sets and establish a number of functional laws of the iterated logarithm for such processes this leads to a unified methodology to study the almost sure behavior of various statistics which are local functionals of the empirical distribution such statistics include density estimators and the bahadur kiefer representation

comutations underlying the measuremnt of visual motion abstract the organization of movement in a changing image provides a valuable source of information for analyzing the environment in terms of objects their motion in space and their three dimensional structure a description of this movement is not provided to our visual system directly however it must be inferred from the pattern of changing intensity that reaches the eye this paper examines the problem of motion measurement which we formulate as the computation of an instantaneous two dimensional velocity field initial motion measurements occur at the location of significant intensity changes as suggested by marr and ullman 1 these measurements provide only one component of velocity and must be integrated to compute the two dimensional velocity field a fundamental problem for this integration is that motion is not determined uniquely from the changing image an additional constraint of smoothness of the velocity field is formulated based on the physical assumption that surfaces are generally smooth allowing the computation of a unique solution a theoretical analysis of the conditions under which this computation yields the correct velocity field suggests that the solution is physically plausible empirical studies show the results to be consistent with human motion perception

strong laws for local quantile processes we show that increments of size h n from the uniform quantile and uniform empirical processes in the neighborhood of a fixed point t 0 0 1 may have different rates of almost sure convergence to 0 in the range where h n 0 and nh n log n in particular when h n n with 0 0 the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed kiefer process is not better than o n 1 4 log n e

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

derivatives of maximum entropy basis functions on the boundary theory and computations summary in this paper we obtain explicit expressions to evaluate the derivatives of maximum entropy max ent basis function on the boundary of a convex domain in the max ent formulation the basis functions are obtained by maximizing a concave functional subjected to linear constraints reproducing conditions in doing so it is found that the lagrange multipliers blow up when x and the expressions for the derivatives of the max ent basis functions in are of an indeterminate form for points on we appeal to l hopital s rule to derive expressions to determine the derivatives of the basis functions we consider the shannon entropy functional and the relative entropy functional with different choices of the prior weight function the first order derivatives of all basis functions are bounded in contrast on an irregular grid with a certain nodal spacing some of the second derivatives of the basis functions are unbounded on the boundary necessary and sufficient conditions on the priors to obtain bounded lagrange multipliers are established optimal convergence rates for fourth order problems are demonstrated for a galerkin approach with a quadratically complete partition of unity enriched max ent approximation copyright 2013 john wiley sons ltd

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

integration constraint in diffuse element method abstract the patch test verifies whether a linear solution is reproduced exactly in an elasticity problem this approach to test the numerical formulation and the code itself is standard in the finite element method the mls shape functions do not have a polynomial form therefore the integration is not well performed by the classical gauss legendre scheme and the patch test is only satisfied asymptotically at convergence in this paper we propose a custom quadrature scheme for mls shape functions in order to ensure the properties needed for an exact verification of the patch test

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

strong embedding of the estimator of the distribution function under random censorship on etudie le comportement asymptotique de l estimateur limite produit f n d une distribution inconnue on donne une approximation de la difference f n x f x par un processus gaussien

motion from flow along contours a note on robustness and ambiguous cases the projected contours of a rigid body in motion define a velocity field a flow in the image plane where the tangential components are invisible we investigate robustness when estimating motion parameters from the normal components of velocity along contours ambiguous curves are contours for which no unique motion can be recovered from normal flow we show that they are field lines of some extra velocity field which causes an invisible motion when added on top of the actual velocity field these field lines coincide with the contour this is an analytical description of what waxman and wohn 1984 called the aperture problem in the large which actually was studied by lie 1893 under the concept of self projective curves if we specialize to a planar surface patch we make a catalogue of the class of ambiguous curves we study the robustness of motion parameters obtained by least squares estimation by calculating the a mount by which noise in the normal flow is magnified for simple curves and a motion model with four unknowns two translation components rotation and expansion there are explicit formulas relating the robustness of motion parameters to how large a portion of the contour we use sometimes it helps to use more global information a larger portion of e g a closed contour and sometimes there is no way out because we are working with contour segments that too closely resemble ambiguous curves neither is there any hope to recover the true velocity field from normal flow locally since any contour falls into one of the ill conditioned contours

directions of motion fields are hardly ever ambiguous if instead of the full motion field we consider only the direction of the motion field due to a rigid motion what can we say about the three dimensional motion information contained in it this paper provides a geometric analysis of this question based solely on the constraint that the depth of the surfaces in view is positive the motivation behind this analysis is to provide a theoretical foundation for image constraints employing only the sign of flow in various directions and justify their utilization for addressing 3d dynamic vision problems it is shown that considering as the imaging surface the whole sphere independently of the scene in view two different rigid motions cannot give rise to the same directional motion field if we restrict the image to half of a sphere or an infinitely large image plane two different rigid motions with instantaneous translational and rotational velocities t 1 1 and t 2 2 cannot give rise to the same directional motion field unless the plane through t 1 and t 2 is perpendicular to the plane through 1 and 2 i e t 1 t 2 1 2 e 0 in addition in order to give practical significance to these uniqueness results for the case of a limited field of view we also characterize the locations on the image where the motion vectors due to the different motions must have different directions if 1 2 t 1 t 2 e 0 and certain additional constraints are met then the two rigid motions could produce motion fields with the same direction for this to happen the depth of each corresponding surface has to be within a certain range defined by a second and a third order surface similar more restrictive constraints are obtained for the case of multiple motions consequently directions of motion fields are hardly ever ambiguous a byproduct of the analysis is that full motion fields are never ambiguous with a half sphere as the imaging surface

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

functional limit laws for the increments of kaplan meier product limit processes and applications we prove functional limit laws for the increment functions of empirical processes based upon randomly right censored data the increment sizes we consider are classified into different classes covering the whole possible spectrum we apply these results to obtain a description of the strong limiting behavior of a series of estimators of local functionals of lifetime distributions in particular we treat the case of kernel density and hazard rate estimators

the rate of strong uniform consistency for the product limit estimator the maximal deviation of the product limit estimate from the estimated distribution function is investigated as a consequence of a functional law of the iterated logarithm the log log law is proved for this deviation on appropriate half lines with the precise constants this result implies that the log log law need not hold in general for the maximal deviation on the whole line then a general asymptotic order of magnitude is determined for the latter deviation this order depends on the joint behaviour of the censoring and censored distributions in a well defined way corresponding to specific joint behaviours several limsup results are deduced as consequence including all the previously known log log type laws in improved form the results are also extended to the variable censoring model

improved robustness for nearly incompressible large deformation meshfree simulations on delaunay tessellations abstract a displacement based galerkin meshfree method for large deformation analysis of nearly incompressible elastic solids is presented nodal discretization of the domain is defined by a delaunay tessellation three node triangles and four node tetrahedra which is used to form the meshfree basis functions and to numerically integrate the weak form integrals in the proposed approach for nearly incompressible solids a volume averaged nodal projection operator is constructed to average the dilatational constraint at a node from the displacement field of surrounding nodes the nodal dilatational constraint is then projected onto the linear approximation space the displacement field is constructed on the linear space and enriched with bubble like meshfree basis functions for stability the new procedure leads to a displacement based formulation that is similar to f bar methodologies in finite elements and isogeometric analysis we adopt maximum entropy meshfree basis functions and the performance of the meshfree method is demonstrated on benchmark problems using structured and unstructured background meshes in two and three dimensions the nonlinear simulations reveal that the proposed methodology provides improved robustness for nearly incompressible large deformation analysis on delaunay meshes

hazard rate estimation under random censoring with varying kernels and bandwidths we discuss the estimation of hazard rates under random censoring with the kernel method two practically relevant problems that occur when applying unmodified kernel estimators are boundary effects near the endpoints of the support of the hazard rate and a substantial increase in the variance from left to right over the range of abscissae where the hazard rate is estimated a new class of boundary kernels is proposed for the first problem explicit formulas for these kernels are developed and it is shown that this boundary correction works well in practice a data adaptive varying bandwidth selection procedure is proposed for the second problem this procedure generally will lead to increasing bandwidths near the left endpoint and toward the right endpoint and will lead to smaller integrated mean squared error of the hazard rate estimator as compared to a fixed bandwidth method a practically feasible method incorporating the new boundary kernels and local bandwidth choices is implemented and illustrated with survival data from a leukemia study

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

motion field and optical flow qualitative properties it is shown that the motion field the 2 d vector field which is the perspective projection on the image plane of the 3 d velocity field of a moving scene and the optical flow defined as the estimate of the motion field which can be derived from the first order variation of the image brightness pattern are in general different unless special conditions are satisfied therefore dense optical flow is often ill suited for computing structure from motion and for reconstructing the 3 d velocity field by algorithms which require a locally accurate estimate of the motion field a different use of the optical flow is suggested it is shown that the smoothed optical flow and the motion field can be interpreted as vector fields tangent to flows of planar dynamical systems stable qualitative properties of the motion field which give useful informations about the 3 d velocity field and the 3 d structure of the scene usually can be obtained from the optical flow the idea is supported by results from the theory of structural stability of dynamical systems

a four point integration scheme with quadratic exactness for three dimensional element free galerkin method based on variationally consistent formulation abstract the formulation of three dimensional element free galerkin efg method based on the hu washizu three field variational principle is described the orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions this leads to a variationally consistent formulation which has a similar form as the formulation of standard galerkin weak form based on this formulation an integration scheme which employs only four cubature points in each background tetrahedral element cell is rationally developed for three dimensional efg with quadratic approximation the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved extension of the proposed method to small strain elastoplasticity is also presented the proposed method can exactly pass quadratic patch test that is quadratic exactness is achieved and thus it is named as quadratically consistent 4 point qc4 integration method in contrast efg with standard tetrahedral cubature and the existing linearly consistent 1 point lc1 integration fail to exactly pass quadratic patch test numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy convergence efficiency and stability the capability of the proposed qc4 scheme in solving elastoplastic problems is also demonstrated by numerical examples

assumed strain nodally integrated hexahedral finite element formulation for elastoplastic applications summary in this work a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems the element stems from the nodally integrated continuum element nice formulation and its extensions assumed gradient operators are derived via nodal integration from the kinematic weighted residual the degrees of freedom are only the displacements at the nodes the adopted constitutive model is the classical associative von mises plasticity model with isotropic and kinematic hardening in particular a double step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions copyright 2014 john wiley sons ltd

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

the jackknife estimate of variance of a kaplan meier integral let f n be the kaplan meier estimator of a distribution function f computed from randomly censored data it is known that under certain integrability assumptions on a function the kaplan meier integral df n when properly standardized is asymptotically normal in this paper it is shown that with probability 1 the jackknife estimate of variance consistently estimates the limit variance

effect of numerical integration on meshless methods abstract in this paper we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k 1 the meshless method was used on a second order neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration this estimate was obtained under the assumption that the numerical integration scheme satisfied a form of green s formula we also indicated how to obtain numerical integration schemes satisfying this property

consistent and stable meshfree galerkin methods using the virtual element decomposition summary over the past two decades meshfree methods have undergone significant development as a numerical tool to solve partial differential equations pdes in contrast to finite elements the basis functions in meshfree methods are smooth nonpolynomial functions and they do not rely on an underlying mesh structure for their construction these features render meshfree methods to be particularly appealing for higher order pdes and for large deformation simulations of solid continua however a deficiency that still persists in meshfree galerkin methods is the inaccuracies in numerical integration which affects the consistency and stability of the method several previous contributions have tackled the issue of integration errors with an eye on consistency but without explicitly ensuring stability in this paper we draw on the recently proposed virtual element method to present a formulation that guarantees both the consistency and stability of the approximate bilinear form we adopt maximum entropy meshfree basis functions but other meshfree basis functions can also be used within this framework numerical results for several two dimensional and three dimensional elliptic poisson and linear elastostatic boundary value problems that demonstrate the effectiveness of the proposed formulation are presented copyright 2017 john wiley sons ltd

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

the central limit theorem under random censorship let f n be the kaplan meier estimator of a distribution function f computed from randomly censored data we show that under optimal integrability assumptions on a function the kaplan meier integral df n when properly standardized is asymptotically normal

consistent element free galerkin method an element free galerkin efg method with linear quadratic and cubic approximations which can exactly in a numerical sense pass the corresponding patch tests is proposed and is named as consistent efg cefg method the development of this method is based on the hu washizu three field variational principle numerical integration schemes with corrected nodal derivatives at quadrature points are proposed according to the satisfaction of the orthogonality condition between stress and strain difference thus the method is variationally consistent the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions are theoretically proved and also numerically validated numerical results show that the proposed cefg method greatly improves the numerical performance of the efg method in terms of accuracy convergence efficiency and stability especially the proposed cubic cefg method which shows exceptional accuracy and convergence copyright 2014 john wiley sons ltd

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

volume averaged nodal projection method for nearly incompressible elasticity using meshfree and bubble basis functions we present a displacement based galerkin meshfree method for the analysis of nearly incompressible linear elastic solids where low order simplicial tessellations i e 3 node triangular or 4 node tetrahedral meshes are used as a background structure for numerical integration of the weak form integrals and to get the nodal information for the computation of the meshfree basis functions in this approach a volume averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal or lower order than the approximation space for the displacement field resulting in a locking free method the stability of the method is provided via bubble like basis functions because the notion of an element or cell is not present in the computation of the meshfree basis functions such low order tessellations can be used regardless of the order of the approximation spaces desired first and second order meshfree basis functions are chosen as a particular case in the proposed method numerical examples are provided in two and three dimensions to demonstrate the robustness of the method its ability to avoid volumetric locking in the nearly incompressible regime and its improved performance when compared to the mini finite element scheme on the simplicial mesh

an arbitrary order variationally consistent integration for galerkin meshfree methods author s hillman michael charles advisor s chen jiun shyan abstract the rate of convergence in galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration if insufficiently accurate domain integration is employed the optimal convergence rate cannot be attained for meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature and the lack of accuracy in the domain integration may lead to sub optimal convergence or even solutions that diverge with refinement the difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves this dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature first the conditions for achieving arbitrary order exactness in a boundary value problem using the galerkin approximation with quadrature are derived the conditions are derived in a general form and are applicable to all types of problems the test function gradients in the galerkin approximation should be consistent with the chosen numerical integration and this is termed variational consistency specifically integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature specific problems are then considered with the conditions explicitly stated including elasticity the euler bernoulli beam the kirchhoff love plate and the non linear formulation of solid mechanics treating the type of numerical integration as a given test function gradients are then constructed to satisfy this condition the resulting method is arbitrarily high order exact and applicable to all types of integration the method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems it is demonstrated that the error induced by numerical integration is greatly reduced and optimal convergence is either partially or fully restored further it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error

local convergence of empirical measures in the random censorship situation with application to density and rate estimators on etudie les deviations locales de la mesure empirique definie par l estimateur de kaplan meier 1958 pour la fonction de survie

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

strong limit theorems for oscillation moduli of the uniform empirical process let u n t n tfrac 1 2 gamma n t t 0 leqq t leqq 1 denote the uniform empirical process based on the first n of a sequence 1 2 of iid uniform 0 1 random variables where gamma n t n 1 sum limits i 1 n 1 0 t xi i is the empirical distribution function the oscillation modulus of u n is defined by omega n a sup u n t h u n t 0 leqq t leqq 1 h h leqq a and the lipschitz 1 2 modulus of u n is defined by tilde omega n a sup u n t h u n t h tfrac 1 2 0 leqq t leqq 1 h a leqq h leqq 1

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

large deviations for processes with independent increments abstract the establishment of the large deviation principle ldp has had important implications in various areas in statistics it has been used to obtain the asymptotic efficiencies of tests and estimates and to obtain the asymptotic behavior of functional integrals associated with solutions of stochastic integrals it appears in the evaluation of the free energy in statistical mechanics it is also intimately related to certain types of laws of large numbers this paper studies the ldp for a stochastic process with stationary independent increments with no gaussian component and obtain complete results

quadrature for meshless methods in this paper we discuss quadrature for meshless methods we present results of two analyses the first of these was developed in 1 and the second was developed in 2 using a simple example we show that a natural quadrature scheme that satisfies no special properties leads to severe inaccuracies in the approximate solution we then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature in the second analysis we show that if the quadrature scheme satisfies green s formula then the effect of the quadrature is controlled

the product limit estimator and the bootstrap some asymptotic representations the product limit estimator and its quantile process are represented as i i d mean processes with a remainder of ordern 3 4 logn 3 4 a s corresponding bootstrap versions of these representations are given which can help one visualize how the bootstrap procedure operates in this set up

a theory of the motion fields of curves this article reports a study of the motion field generated by moving 3 d curves that are observed by a camera we first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend

laws of the iterated logarithm for nonparametric density estimators we establish a law of the iterated logarithm for a triangular array of independent random variables and apply it to obtain laws for a large class of nonparametric density estimators we consider the case of rosenblatt parzen kernel estimators trigonometric series estimators and orthogonal polynomial estimators in detail and point out that our technique has wider application

triangles and tetrahedra in explicit dynamic codes for solids explicit dynamic codes which are used currently for the study of plastic deformations in impact or with some modification for metal forming suffer two serious limitations first only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing second even with the use of such elements special devices such as reduced integration must be introduced to avoid locking and reduce costs these necessitate complex hour glass control mending type procedures the main difficulties are those due to the need of treatment of almost incompressible deformation modes recently similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem thus allowing an almost unrestricted choice of element interpolations 1998 john wiley sons ltd

functional laws of the iterated logarithm for the product limit estimator of a distribution function under random censorship or truncation functional laws of the iterated logarithm are established for a modified version of the classical product limit estimator of a distribution function when the data are subject to random censorship or truncation making use of stochastic integral representations and empirical process theory strong approximations involving i i d continuous parameter martingales are developed for the product limit estimator and these strong approximations are then applied to derive the functional laws

determining shape and motion from monocular camera determining the spatial motion of a moving camera from a video is a classical problem in computer vision the difficulty of this problem is that the flow pattern directly observable in the video is generally not the complete flow pattern induced by the motion but only the partial information of it which is known as the normal flow in this paper we present a direct method which neither requires the establishment of feature correspondences nor the recovery of optical flow between two image frames but we directly utilize all observable normal flow data to recover the camera motion we propose a two stage iterative algorithm to search the solution in the motion space in a coarse to fine framework the first stage involves the use of the direction part of the normal flow each of these normal flow data can provide a constrained solution space to the direction of motion the intersection of the motion solutions from all the available normal flow data can reduce the motion ambiguity to a certain extent we then use the globality of the rotational magnitude to all image positions to constrain the motion parameters further once the camera motion is determined the depth map of the imaged scene up to an arbitrary scale can be recovered experimental results on synthetic data and real images are provided to reveal the performance of the proposed method highlightswe determine camera motion and shape directly from normal flows our method does not require optical flows or feature correspondences two constraints are developed to determine camera motion experimental results are comparable to the state of the art methods

motion from point matches multiple of solutions in this paper we study the multiplicity of solutions of the motion problem given n point matches between two frames how many solutions are there to the motion problem we show that the maximum number of solutions is 10 when 5 point matches are available this settles a question that has been around in the computer vision community for a while we follow two tracks we show that the two approaches are compatible and yield the same result we then describe a computer implementation of the second approach that uses maple a language for symbolic computation the program allows us to compute exactly the solutions for any configuration of 5 points some experiments are described

on the geometry of visual correspondence image displacement fields optical flow fields stereo disparity fields normal flow fields due to rigid motion possess a global geometric structure which is independent of the scene in view motion vectors of certain lengths and directions are constrained to lie on the imaging surface at particular loci whose location and form depends solely on the 3d motion parameters if optical flow fields or stereo disparity fields are considered then equal vectors are shown to lie on conic sections similarly for normal motion fields equal vectors lie within regions whose boundaries also constitute conics by studying various properties of these curves and regions and their relationships a characterization of the structure of rigid motion fields is given the goal of this paper is to introduce a concept underlying the global structure of image displacement fields this concept gives rise to various constraints that could form the basis of algorithms for the recovery of visual information from multiple views

gradient correction for polygonal and polyhedral finite elements summary previous studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non convergent in this work we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature for finite elements of arbitrary order we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator we compare the proposed approach with the pseudo derivative method developed by belytschko and co workers and for linear elliptic problems with our previous remedy that involves splitting of polynomial and non polynomial of elemental energy bilinear form we present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals copyright 2015 john wiley sons ltd