aspects of an adaptive finite element method for the fractional laplacian a priori and a posteriori error estimates efficient implementation and multigrid solver abstract we develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations on non trivial domains in d 1 dimensions our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and where necessary modifying them so that they can be applied to the fractional pde case improved a priori error estimates are derived for the case of quasi uniform meshes which are seen to deliver sub optimal rates of convergence owing to the presence of singularities attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure we assume that the resulting refined meshes are locally quasi uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods including the multigrid method the storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered the performance and efficiency of the resulting algorithm is illustrated for a variety of examples

influence in principal components analysis summary in linear regression the theoretical influence function and the various sample versions of it have an established place as diagnostic tools these same functions are developed here to provide methods for the detection of influential observations in principal components analysis the perturbation theory of real symmetric matrices unifies this development some interesting points of contrast with the regression case are noted and explained theoretically

a riemannian approach to graph embedding in this paper we make use of the relationship between the laplace beltrami operator and the graph laplacian for the purposes of embedding a graph onto a riemannian manifold to embark on this study we review some of the basics of riemannian geometry and explain the relationship between the laplace beltrami operator and the graph laplacian using the properties of jacobi fields we show how to compute an edge weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths on the manifold between nodes for the particular case of a constant sectional curvature surface we use the kruskal coordinates to compute edge weights that are proportional to the geodesic distance between points we use the resulting edge weight matrix to embed the nodes of the graph onto a riemannian manifold to do this we develop a method that can be used to perform double centring on the laplacian matrix computed from the edge weights the embedding coordinates are given by the eigenvectors of the centred laplacian with the set of embedding coordinates at hand a number of graph manipulation tasks can be performed in this paper we are primarily interested in graph matching we recast the graph matching problem as that of aligning pairs of manifolds subject to a geometric transformation we show that this transformation is pro crustean in nature we illustrate the utility of the method on image matching using the coil database

a numerical scheme for impact problems i the one dimensional case we consider a mechanical system with impact and one degree of freedom the system is not necessarily lagrangian the representative point is subject to the constraint u t in er for all t we assume that at impact the velocity is reversed and multiplied by a given coefficient of restitution e in 0 1 we define a numerical scheme which enables us to approximate the solutions of the cauchy problem this is an ad hoc scheme which does not require a systematic search for the times of impact we prove the convergence of this numerical scheme to a solution many of the features of this proof will be reused in the nonconvex multidimensional case written in generalized coordinates given in the companion paper l paoli and m schatzman siam j numer anal 40 2002 pp 734 768 we present some numerical results obtained with the scheme for a spring dashpot system and we compare them to the results obtained by impact detection and penalization

a minimal solution to radial distortion autocalibration simultaneous estimation of radial distortion epipolar geometry and relative camera pose can be formulated as a minimal problem and solved from a minimal number of image points finding the solution to this problem leads to solving a system of algebraic equations in this paper we provide two different solutions to the problem of estimating radial distortion and epipolar geometry from eight point correspondences in two images unlike previous algorithms which were able to solve the problem from nine correspondences only we enforce the determinant of the fundamental matrix be zero this leads to a system of eight quadratic and one cubic equation in nine variables we first simplify this system by eliminating six of these variables and then solve the system by two alternative techniques the first one is based on the grobner basis method and the second one on the polynomial eigenvalue computation we demonstrate that our solutions are efficient robust and practical by experiments on synthetic and real data

representations of the q deformed algebra uq so4 we study the nonstandard q deformation uq so4 of the universal enveloping algebra u so4 obtained by deforming the defining relations for skew symmetric generators of u so4 this algebra is used in quantum gravity and algebraic topology we construct a homomorphism of uq so4 to the certain nontrivial extension of the drinfeld jimbo quantum algebra uq sl2 2 and show that this homomorphism is an isomorphism by using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra uq so4 it is proved that for q not a root of unity each irreducible finite dimensional representation of uq so4 is equivalent to one of these representations we prove that every finite dimensional representation of uq so4 for q not a root of unity is completely reducible it is shown how to construct by using the homomorphism tensor products of irreducible representations of uq so4 note that no hopf algebra structure is known fo

time like minimal submanifolds as singular limits of nonlinear wave equations abstract we consider the sharp interface limit 0 of the semilinear wave equation u w u 2 0 in r 1 n where u takes values in r k k 1 2 and w is a double well potential if k 1 and vanishes on the unit circle and is positive elsewhere if k 2 for fixed 0 we find some special solutions constructed around minimal surfaces in r n in the general case under some additional assumptions we show that the solutions converge to a radon measure supported on a time like k codimensional minimal submanifold of the minkowski space time this result holds also after the appearance of singularities and enforces the observation made by j neu that this semilinear equation can be regarded as an approximation of the born infeld equation

flexible spatial configuration of local image features local image features have been designed to be informative and repeatable under rigid transformations and illumination deformations even though current state of the art local image features present a high degree of repeatability their local appearance alone usually does not bring enough discriminative power to support a reliable matching resulting in a relatively high number of mismatches in the correspondence set formed during the data association procedure as a result geometric filters commonly based on global spatial configuration have been used to reduce this number of mismatches however this approach presents a trade off between the effectiveness to reject mismatches and the robustness to nonrigid deformations in this paper we propose two geometric filters based on a semilocal spatial configuration of local features that are designed to be robust to nonrigid deformations and to rigid transformations without compromising its efficacy to reject mismatches we compare our methods to the hough transform which is an efficient and effective mismatch rejection step based on the global spatial configuration of features in these comparisons our methods are shown to be more effective in the task of rejecting mismatches for rigid transformations and nonrigid deformations at comparable time complexity figures finally we demonstrate how we can integrate these methods in a probabilistic recognition system such that the final verification step uses not only the similarity between features but also their semilocal configuration

nonlinear wave equations as limits of convex minimization problems proof of a conjecture by de giorgi we prove a conjecture by de giorgi which states that global weak solutions of nonlinear wave equations such as w jwj p 2 w 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations these functionals which are integrals in space time of a convex lagrangian contain an exponential weight with a parameter and the initial data of the wave equation serve as boundary conditions as tends to zero the minimizers v converge up to subsequences to a solution of the nonlinear wave equation there is no restriction on the nonlinearity exponent and the method is easily extended to more general equations

on the fitting of surfaces to data with covariances we consider the problem of estimating parameters of a model described by an equation of special form specific models arise in the analysis of a wide class of computer vision problems including conic fitting and estimation of the fundamental matrix we assume that noisy data are accompanied by known covariance matrices characterizing the uncertainty of the measurements a cost function is first obtained by considering a maximum likelihood formulation and applying certain necessary approximations that render the problem tractable a newton like iterative scheme is then generated for determining a minimizer of the cost function unlike alternative approaches such as sampson s method or the renormalization technique the new scheme has as its theoretical limit the minimizer of the cost function furthermore the scheme is simply expressed efficient and unsurpassed as a general technique in our testing an important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches

robust structure and motion from outlines of smooth curved surfaces this paper addresses the problem of estimating the motion of a camera as it observes the outline or apparent contour of a solid bounded by a smooth surface in successive image frames in this context the surface points that project onto the outline of an object depend on the viewpoint and the only true correspondences between two outlines of the same object are the projections of frontier points where the viewing rays intersect in the tangent plane of the surface in turn the epipolar geometry is easily estimated once these correspondences have been identified given the apparent contours detected in an image sequence a robust procedure based on ransac and a voting strategy is proposed to simultaneously estimate the camera configurations and a consistent set of frontier point projections by enforcing the redundancy of multiview epipolar geometry the proposed approach is in principle applicable to orthographic weak perspective and affine projection models experiments with nine real image sequences are presented for the orthographic projection case including a quantitative comparison with the ground truth data for the six data sets for which the latter information is available sample visual hulls have been computed from all image sequences for qualitative evaluation

representations of the q deformed algebra uq iso2 an algebra homomorphism from the q deformed algebra uq iso2 with generating elements i t1 t2 and defining relations i t2 q t1 t1 i q t2 t2 t1 q 0 where a b q q1 2ab q 1 2ba to the extension uq m2 of the hopf algebra uq m2 is constructed the algebra uq iso2 at q 1 leads to the lie algebra iso2 m2 of the group iso 2 of motions of the euclidean plane the hopf algebra uq m2 which is not isomorphic to uq iso2 is treated as a hopf q deformation of the universal enveloping algebra of iso2 and is well known in the literature not all irreducible representations of uq m2 can be extended to representations of the extension uq m2 composing the homomorphism with irreducible representations of uq m2 we obtain representations of uq iso2 not all of these representations of uq iso2 are irreducible the reducible representations of uq iso2 are decomposed into irreducible components in this way we obtain all irreducible representations of uq iso2 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra iso2 when q1 representations of the other part have no classical analogue

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

an algorithm for projective point matching in the presence of spurious points point matching is the task of finding correspondences between two sets of points such that the two sets of points are aligned with each other pure point matching uses only the location of the points to constrain the problem this is a problem with broad practical applications but it has only been well studied when the geometric transformation relating the two point sets is of a relatively low order here we present a heuristic local search algorithm that can find correspondences between point sets in two dimensions that are related by a projective transform point matching is a harder problem when spurious points appear in the sets to be matched we present a heuristic algorithm which minimizes the effects of spurious points

sequential updating of projective and affine structure from motion a structure from motion algorithm is described which recovers structure and camera position modulo a projective ambiguity camera calibration is not required and camera parameters such as focal length can be altered freely during motion the structure is updated sequentially over an image sequence in contrast to schemes which employ a batch process a specialisation of the algorithm to recover structure and camera position modulo an affine transformation is described together with a method to periodically update the affine coordinate frame to prevent drift over time we describe the constraint used to obtain this specialisation structure is recovered from image corners detected and matched automatically and reliably in real image sequences results are shown for reference objects and indoor environments and accuracy of recovered structure is fully evaluated and compared for a number of reconstruction schemes a specific application of the work is demonstrated affine structure is used to compute free space maps enabling navigation through unstructured environments and avoidance of obstacles the path planning involves only affine constructions

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

global propagation of affine invariant features for robust matching local invariant features have been successfully used in image matching to cope with viewpoint change partial occlusion and clutters however when these factors become too strong there will be a lot of mismatches due to the limited repeatability and discriminative power of features in this paper we present an efficient approach to remove the false matches and propagate the correct ones for the affine invariant features which represent the state of the art local invariance first a pair wise affine consistency measure is proposed to evaluate the consensus of the matches of affine invariant regions the measure takes into account both the keypoint location and the region shape size and orientation based on this measure a geometric filter is then presented which can efficiently remove the outliers from the initial matches and is robust to severe clutters and non rigid deformation to increase the correct matches we propose a global match refinement and propagation method that simultaneously finds a optimal group of local affine transforms to relate the features in two images the global method is capable of producing a quasi dense set of matches even for the weakly textured surfaces that suffer strong rigid transformation or non rigid deformation the strong capability of the proposed method in dealing with significant viewpoint change non rigid deformation and low texture objects is demonstrated in experiments of image matching object recognition and image based rendering

estimation of 2d displacement field based on affine geometric invariance and scene constraints in the paper a novel approach using affine transfer and scene constraint for estimation of 2d displacement field was developed in this approach we derived a system of 5 linear equations for computing corresponding point of any image point via the utilisation of affine invariant subsequently the characteristics of such a linear system was thoroughly studied and the way to obtain a reliable solution of the system via robust least squares rls approach in conjunction with total least squares technique was proposed in addition through the interpretation of the approach from both geometric point of view and numerical point of view we gave the limitation of the algorithm the limitation was then relaxed to a certain extent by using full fundamental matrix these findings were further verified through the experimental results

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

n dimensional tensor voting and application to epipolar geometry estimation we address the problem of epipolar geometry estimation by formulating it as one of hyperplane inference from a sparse and noisy point set in an 8d space given a set of noisy point correspondences in two images of a static scene without correspondences even in the presence of moving objects our method extracts good matches and rejects outliers the methodology is novel and unconventional since unlike most other methods optimizing certain scalar objective functions our approach does not involve initialization or any iterative search in the parameter space therefore it is free of the problem of local optima or poor convergence further since no search is involved it is unnecessary to impose simplifying assumption to the scene being analyzed for reducing the search complexity subject to the general epipolar constraint only we detect wrong matches by a computation scheme 8d tensor voting which is an instance of the more general n dimensional tensor voting framework in essence the input set of matches is first transformed into a sparse 8d point set dense 8d tensor kernels are then used to vote for the most salient hyperplane that captures all inliers inherent in the input with this filtered set of matches the normalized eight point algorithm can be used to estimate the fundamental matrix accurately by making use of efficient data structure and locality our method is both time and space efficient despite the higher dimensionality we demonstrate the general usefulness of our method using example image pairs for aerial image analysis with widely different views and from nonstatic 3d scenes each example contains a considerable number of wrong matches

mathematical modelling of surface vibration with volume constraint and its analysis abstract a hyperbolic problem with volume constraint is investigated by means of a minimizing method called the discrete morse flow a weak solution is introduced in a constructive way and results of numerical computations are shown

the dual bootstrap iterative closest point algorithm with application to retinal image registration motivated by the problem of retinal image registration this paper introduces and analyzes a new registration algorithm called dual bootstrap iterative closest point dual bootstrap icp the approach is to start from one or more initial low order estimates that are only accurate in small image regions called bootstrap regions in each bootstrap region the algorithm iteratively 1 refines the transformation estimate using constraints only from within the bootstrap region 2 expands the bootstrap region and 3 tests to see if a higher order transformation model can be used stopping when the region expands to cover the overlap between images steps 1 and 3 the bootstrap steps are governed by the covariance matrix of the estimated transformation estimation refinement step 2 uses a novel robust version of the icp algorithm in registering retinal image pairs dual bootstrap icp is initialized by automatically matching individual vascular landmarks and it aligns images based on detected blood vessel centerlines the resulting quadratic transformations are accurate to less than a pixel on tests involving approximately 6000 image pairs it successfully registered 99 5 of the pairs containing at least one common landmark and 100 of the pairs containing at least one common landmark and at least 35 image overlap

model fitting with sufficient random sample coverage it has been observed previously that the number of iterations required to derive good model parameter values used by ransac like model estimators is too optimistic we present the derivation of an analytical formula that allows the calculation of the sufficient limit of iterations needed to obtain good parameter values with the prescribed probability for any number of model parameters it explains the values that had been found experimentally for certain numbers of model parameters by others very well furthermore the improvement that our approach of sufficient random sample coverage sursac offers in comparison to the original ransac algorithm as well as to its adaptive modification by hartley and zisserman is demonstrated with synthetic data for the case of a non linear model function over a wide range of outlier fractions and different ratios of inlier and outlier densities

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

optimal motion and structure estimation the causes of existing linear algorithms exhibiting various high sensitivities to noise are analyzed it is shown that even a small pixel level perturbation may override the epipolar information that is essential for the linear algorithms to distinguish different motions this analysis indicates the need for optimal estimation in the presence of noise methods are introduced for optimal motion and structure estimation under two situations of noise distribution known and unknown computationally the optimal estimation amounts to minimizing a nonlinear function for the correct convergence of this nonlinear minimization a two step approach is used the first step is using a linear algorithm to give a preliminary estimate for the parameters the second step is minimizing the optimal objective function starting from that preliminary estimate as an initial guess a remarkable accuracy improvement has been achieved by this two step approach over using the linear algorithm alone

impsac synthesis of importance sampling and random sample consensus this paper proposes a new method for recovery of epipolar geometry and feature correspondence between images which have undergone a significant deformation either due to large rotation or wide baseline of the cameras the method also encodes the uncertainty by providing an arbitrarily close approximation to the posterior distribution of the two view relation the method operates on a pyramid from coarse to fine resolution thus raising the problem of how to propagate information from one level to another in a statistically consistent way the distribution of the parameters at each resolution is encoded nonparametrically as a set of particles at the coarsest level a random sample consensus monte carlo markov chain ransac mcmc estimator is used to initialize this set of particles the posterior can then be approximated as a mixture of gaussians fitted to these particles the distribution at a coarser level influences the distribution at a finer level using the technique of sampling importance resampling sir and mcmc which allows for asymptotically correct approximations of the posterior distribution the estimate of the posterior distribution at the level above is being used as the importance sampling function to generate a new set of particles which can be further improved by mcmc it is shown that the method is superior to previous single resolution ransac style feature matchers

statistical bias of conic fitting and renormalization introducing a statistical model of noise in terms of the covariance matrix of the n vector we point out that the least squares conic fitting is statistically biased we present a new fitting scheme called renormalization for computing an unbiased estimate by automatically adjusting to noise relationships to existing methods are discussed and our method is tested using real and synthetic data

an efficient solution to the five point relative pose problem an efficient algorithmic solution to the classical five point relative pose problem is presented the problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points the algorithm consists of computing the coefficients of a tenth degree polynomial in closed form and subsequently finding its roots it is the first algorithm well suited for numerical implementation that also corresponds to the inherent complexity of the problem we investigate the numerical precision of the algorithm we also study its performance under noise in minimal as well as overdetermined cases the performance is compared to that of the well known 8 and 7 point methods and a 6 point scheme the algorithm is used in a robust hypothesize and test framework to estimate structure and motion in real time with low delay the real time system uses solely visual input and has been demonstrated at major conferences

representations of the q deformed algebras uq so2 1 and uq so3 1 representations of the algebra uq so2 1 are studied this algebra is a q deformation of the universal enveloping algebra u so2 1 of the lie algebra of the group so0 2 1 and differs from the quantum algebra uq su1 1 classifications of irreducible representations and of infinitesimally unitary irreducible representations of uq so2 1 with simple and discrete spectrum of one of generators are derived the set of infinitesimally unitary representations of uq so2 1 does not coincide with that for the algebra uq su1 1 the sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra uq so3 1 are given we also consider representations of uq son 1 which are of class 1 with respect to the subalgebra uq son

robust parameter estimation in computer vision estimation techniques in computer vision applications must estimate accurate model parameters despite small scale noise in the data occasional large scale measurement errors outliers and measurements from multiple populations in the same data set increasingly robust estimation techniques some borrowed from the statistics literature and others described in the computer vision literature have been used in solving these parameter estimation problems ideally these techniques should effectively ignore the outliers and measurements from other populations treating them as outliers when estimating the parameters of a single population two frequently used techniques are least median of squares lms p j rousseeuw j amer statist assoc 79 1984 pp 871 880 and m estimators robust statistics the approach based on influence functions f r hampel et al john wiley 1986 robust statistics p j huber john wiley 1981 lms handles large fractions of outliers up to the theoretical limit of 50 for estimators invariant to affine changes to the data but has low statistical efficiency m estimators have higher statistical efficiency but tolerate much lower percentages of outliers unless properly initialized while robust estimators have been used in a variety of computer vision applications three are considered here in analysis of range images images containing depth or x y z measurements at each pixel instead of intensity measurements robust estimators have been used successfully to estimate surface model parameters in small image regions in stereo and motion analysis they have been used to estimate parameters of what is called the fundamental matrix which characterizes the relative imaging geometry of two cameras imaging the same scene recently robust estimators have been applied to estimating a quadratic image to image transformation model necessary to create a composite mosaic image from a series of images of the human retina in each case a straightforward application of standard robust estimators is insufficient and carefully developed extensions are used to solve the problem

the geometry and matching of lines and curves over multiple views this paper describes the geometry of imaged curves in two and three views multi view relationships are developed for lines conics and non algebraic curves the new relationships focus on determining the plane of the curve in a projective reconstruction and in particular using the homography induced by this plane for transfer from one image to another it is shown that given the fundamental matrix between two views and images of the curve in each view then the plane of a conic may be determined up to a two fold ambiguity but local curvature of a curve uniquely determines the plane it is then shown that given the trifocal tensor between three views this plane defines a homography map which may be used to transfer a conic or the curvature from two views to a third simple expressions are developed for the plane and homography in each case a set of algorithms are then described for automatically matching individual line segments and curves between images the algorithms use both photometric information and the multiple view geometric relationships for image pairs the homography facilitates the computation of a neighbourhood cross correlation based matching score for putative line curve correspondences for image triplets cross correlation matching scores are used in conjunction with line curve transfer based on the trifocal geometry to disambiguate matches algorithms are developed for both short and wide baselines the algorithms are robust to deficiencies in the segment extraction and partial occlusion experimental results are given for image pairs and triplets for varying motions between views and for different scene types the methods are applicable to line curve matching in stereo and trinocular rigs and as a starting point for line curve matching through monocular image sequences

a voting based computational framework for visual motion analysis and interpretation most approaches for motion analysis and interpretation rely on restrictive parametric models and involve iterative methods which depend heavily on initial conditions and are subject to instability further difficulties are encountered in image regions where motion is not smooth typically around motion boundaries this work addresses the problem of visual motion analysis and interpretation by formulating it as an inference of motion layers from a noisy and possibly sparse point set in a 4d space the core of the method is based on a layered 4d representation of data and a voting scheme for affinity propagation the inherent problem caused by the ambiguity of 2d to 3d interpretation is usually handled by adding additional constraints such as rigidity however enforcing such a global constraint has been problematic in the combined presence of noise and multiple independent motions by decoupling the processes of matching outlier rejection segmentation and interpretation we extract accurate motion layers based on the smoothness of image motion and then locally enforce rigidity for each layer in order to infer its 3d structure and motion the proposed framework is noniterative and consistently handles both smooth moving regions and motion discontinuities without using any prior knowledge of the motion model

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

a variational approach to a constrained hyperbolic free boundary problem abstract this paper concerns the construction of a weak solution to a constrained hyperbolic equation in one space dimension the problem includes a nonlocal term corresponding to a volume constraint and the presence of a free boundary we approach our problem by means of a variational method known as the discrete morse flow

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

a probabilistic criterion to detect rigid point matches between two images and estimate the fundamental matrix the perspective projections of n physical points on two views stereovision are constrained as soon as n 8 however to prove in practice the existence of a rigid motion between two images more than 8 point matches are desirable in order to compensate for the limited accuracy of the matches in this paper we propose a computational definition of rigidity and a probabilistic criterion to rate the meaningfulness of a rigid set as a function of both the number of pairs of points n and the accuracy of the matches this criterion yields an objective way to compare say precise matches of a few points and approximate matches of a lot of points it gives a yes no answer to the question could this rigid points correspondence have occurred by chance since it guarantees that the expected number of meaningful rigid sets found by chance in a random distribution of points is as small as desired it also yields absolute accuracy requirements for rigidity detection in the case of non matched points and optimal values of n depending on the expected accuracy of the matches and on the proportion of outliers we use it to build an optimized random sampling algorithm that is able to detect a rigid motion and estimate the fundamental matrix when the set of point matches contains up to 90 of outliers which outperforms the best currently known methods like m estimators lmeds classical ransac and tensor voting

a generalized kernel consensus based robust estimator in this paper we present a new adaptive scale kernel consensus askc robust estimator as a generalization of the popular and state of the art robust estimators such as random sample consensus ransac adaptive scale sample consensus assc and maximum kernel density estimator mkde the askc framework is grounded on and unifies these robust estimators using nonparametric kernel density estimation theory in particular we show that each of these methods is a special case of askc using a specific kernel like these methods askc can tolerate more than 50 percent outliers but it can also automatically estimate the scale of inliers we apply askc to two important areas in computer vision robust motion estimation and pose estimation and show comparative results on both synthetic and real data

guided mlesac faster image transform estimation by using matching priors mlesac is an established algorithm for maximum likelihood estimation by random sampling consensus devised for computing multiview entities like the fundamental matrix from correspondences between image features a shortcoming of the method is that it assumes that little is known about the prior probabilities of the validities of the correspondences this paper explains the consequences of that omission and describes how the algorithm s theoretical standing and practical performance can be enhanced by deriving estimates of these prior probabilities using the priors in guided mlesac is found to give an order of magnitude speed increase for problems where the correspondences are described by one image transformation and clutter this paper describes two further modifications to guided mlesac the first shows how all putative matches rather than just the best from a particular feature can be taken forward into the sampling stage albeit at the expense of additional computation the second suggests how to propagate the output from one frame forward to successive frames the additional information makes guided mlesac computationally realistic at video rates for correspondence sets modeled by two transformations and clutter

regularity of the obstacle problem for the parabolic biharmonic equation we consider the obstacle problem for the parabolic biharmonic equation we study the problem via an implicit time discretization we prove the existence of a unique solution and discuss its regularity property

simultaneous two view epipolar geometry estimation and motion segmentation by 4d tensor voting we address the problem of simultaneous two view epipolar geometry estimation and motion segmentation from nonstatic scenes given a set of noisy image pairs containing matches of n objects we propose an unconventional efficient and robust method 4d tensor voting for estimating the unknown n epipolar geometries and segmenting the static and motion matching pairs into n independent motions by considering the 4d isotropic and orthogonal joint image space only two tensor voting passes are needed and a very high noise to signal ratio up to five can be tolerated epipolar geometries corresponding to multiple rigid motions are extracted in succession only two uncalibrated frames are needed and no simplifying assumption such as affine camera model or homographic model between images other than the pin hole camera model is made our novel approach consists of propagating a local geometric smoothness constraint in the 4d joint image space followed by global consistency enforcement for extracting the fundamental matrices corresponding to independent motions we have performed extensive experiments to compare our method with some representative algorithms to show that better performance on nonstatic scenes are achieved results on challenging data sets are presented

sampling minimal subsets with large spans for robust estimation when sampling minimal subsets for robust parameter estimation it is commonly known that obtaining an all inlier minimal subset is not sufficient the points therein should also have a large spatial extent this paper investigates a theoretical basis behind this principle based on a little known result which expresses the least squares regression as a weighted linear combination of all possible minimal subset estimates it turns out that the weight of a minimal subset estimate is directly related to the span of the associated points we then derive an analogous result for total least squares which unlike ordinary least squares corrects for errors in both dependent and independent variables we establish the relevance of our result to computer vision by relating total least squares to geometric estimation techniques as practical contributions we elaborate why naive distance based sampling fails as a strategy to maximise the span of all inlier minimal subsets produced in addition we propose a novel method which unlike previous methods can consciously target all inlier minimal subsets with large spans

the development and comparison of robust methodsfor estimating the fundamental matrix this paper has two goals the first is to develop a variety of robust methods for the computation of the fundamental matrix the calibration free representation of camera motion the methods are drawn from the principal categories of robust estimators viz case deletion diagnostics m estimators and random sampling and the paper develops the theory required to apply them to non linear orthogonal regression problems although a considerable amount of interest has focussed on the application of robust estimation in computer vision the relative merits of the many individual methods are unknown leaving the potential practitioner to guess at their value the second goal is therefore to compare and judge the methods comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery in contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth a variety of error measures are examined the experiments allow a statistically satisfying and quasi optimal method to be synthesized which is shown to be stable with up to 50 percent outlier contamination and may still be used if there are more than 50 percent outliers performance bounds are established for the method and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined the results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers

exact two image structure from motion for two image structure from motion we present a simple exact expression for a least squares image reprojection error for finite motion that depends only on the motion optimal estimates of the structure and motion can be computed by minimizing this expression just over the motion parameters also we present a solution to the triangulation problem an exact explicit expression for the optimal structure estimate given the motion we identify a new ambiguity in recovering the structure for known motion we study the exact error s properties experimentally and demonstrate that it often has several local minima for forward or backward motion estimates our experiments also show that the reflected local minimum of oliensis 2001 and soatto et al 1998 occurs for large translational motions our exact results assume that the camera is calibrated and use a least squares image reprojection error that applies most naturally to a spherical imaging surface we approximately extend our approach to the standard least squares error in the image plane and uncalibrated cameras we present an improved version of the sampson error which gives better results experimentally

variational approach to evolutionary free boundary problems abstract the analysis of evolutionary nonlocal problems represented by volume constrained motion is discussed in the physical sense these problems can describe the motion of droplets on a solid plane the variational approach called the discrete morse flow combined with a fluid dynamics model appears to be a suitable tool

ror rejection of outliers by rotations we address the problem of rejecting false matches of points between two perspective views the two views are taken from two arbitrary unknown positions and orientations we present an algorithm for identification of the false matches between the views the algorithm exploits the possibility of rotating one of the images to achieve some common behavior of the correct matches those matches that deviate from this common behavior turn out to be false matches our algorithm does not in any way use the image characteristics of the matched features in particular it avoids problems that cause the false matches in the first place the algorithm works even in cases where the percentage of false matches is as high as 85 percent the algorithm may be run as a post processing step on output from any point matching algorithm use of the algorithm may significantly improve the ratio of correct matches to incorrect matches we present the algorithm identify the conditions under which it works and present results of the test

so 5 q and contraction chevalley basis representations for q generic and root of unity representations of so 5 q are constructed explicitly on the chevalley basis the classical representations once obtained on this basis are very simply deformed for arbitrary q generic and root of unity this is in sharp contrast to the classical canonical gelfand zetlin gz representations which lead to problems studied elsewhere in q deformations for orthogonal though not for unitary algebras our results are limited to irreducible representations labeled by three variable parameters the maximal number being four within this restriction it is shown how to construct all representations for all q s the contraction of the chevalley generators e2 f2 leads to an easily obtained complete hopf algebra the structure of the contracted algebra is quite different even at the classical level from the familiar e 4 the four dimensional euclidean algebra obtained on contracting the generator j45 in the gz formalism the contracted representations are given the so 5 q representations have their own

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

regularity of a free boundary in parabolic potential theory we study the regularity of the free boundary in a stefan type problem 1u t u in d rn r u u 0 on d with no sign assumptions on u and the time derivative t u

a multi frame approach to visual motion perception one of the main issues in the area of motion estimation given the correspondences of some features in a sequence of images is sensitivity to error in the input the main way to attack the problem as with several other problems in science and engineering is redundancy in the data up to now all the algorithms developed either used two frames or depended on assumptions about the motion or the shape of the scene we present in this paper an algorithm based on multiple frames that employs only the rigidity assumption is simple and mathematically elegant and experimentally proves to be a major improvement over the two frame algorithms the algorithm does minimization of the squared error which we prove equivalent to an eigenvalue minimization problem one of the side effects of this mean square method is that the algorithm can have a very descriptive physical interpretation in terms of the loaded spring model

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

heteroscedastic regression in computer vision problems with bilinear constraint we present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise i e each data point having a different covariance matrix the algorithm is motivated by the recovery of bilinear forms one of the fundamental problems in computer vision which appears whenever the epipolar constraint is imposed or a conic is fit to noisy data points we employ the errors in variables eiv model and show why already at moderate noise levels most available methods fail to provide a satisfactory solution the improved behavior of the new algorithm is due to two factors taking into account the heteroscedastic nature of the errors arising from the linearization of the bilinear form and the use of generalized singular value decomposition gsvd in the computations the performance of the algorithm is compared with several methods proposed in the literature for ellipse fitting and estimation of the fundamental matrix it is shown that the algorithm achieves the accuracy of nonlinear optimization techniques at much less computational cost

adaptive outlier elimination in image registration using genetic programming abstract in feature based methods outlier removal plays an important role in attaining a reasonable accuracy for image registration in this paper we propose a genetic programming gp based adaptive method for outlier removal first features are extracted through the scale invariant feature transform sift from the reference and sensed images which were initially matched using euclidean distance the classification of feature points into inliers and outliers is done in two stages in the first stage feature vectors are computed using various distance and angle information feature points are categorized into three groups inliers outliers and non classified feature ncf points in the second stage a gp based classifier is developed to classify ncf points into inliers and outliers the gp based function takes features as an input feature vector and provides a scalar output by combining features with arithmetic operations finally registration is done by eliminating the outliers the effectiveness of the proposed outlier removal method is analyzed through the classification and positional accuracy the experimental results show a considerable improvement in the registration accuracy

estimation of planar curves surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation the author addresses the problem of parametric representation and estimation of complex planar curves in 2 d surfaces in 3 d and nonplanar space curves in 3 d curves and surfaces can be defined either parametrically or implicitly with the latter representation used here a planar curve is the set of zeros of a smooth function of two variables x y a surface is the set of zeros of a smooth function of three variables x y z and a space curve is the intersection of two surfaces which are the set of zeros of two linearly independent smooth functions of three variables x y z for example the surface of a complex object in 3 d can be represented as a subset of a single implicit surface with similar results for planar and space curves it is shown how this unified representation can be used for object recognition object position estimation and segmentation of objects into meaningful subobjects that is the detection of interest regions that are more complex than high curvature regions and hence more useful as features for object recognition

constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle this article treats a problem which describs movement of a string that hits to an obstacle in 10 m schatzman solves this problem in a slightly classical way in 5 6 k maruo constructs a solution to this problem by the use of yosida approximation the purpose of this article is to construct a solution to this problem in time semidiscretization method in general approximation by time semidiscretization method is dierent from yosida approximation a simple example is presented in appendix

robust regression methods for computer vision a review regression analysis fitting a model to noisy data is a basic technique in computer vision robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance we review several robust estimation techniques and describe in detail the least median of squares lmeds method the method yields the correct result even when half of the data is severely corrupted its efficiency in the presence of gaussian noise can be improved by complementing it with a weighted least squares based procedure the high time complexity of the lmeds algorithm can be reduced by a monte carlo type speed up technique we discuss the relationship of lmeds with the ransac paradigm and its limitations in the presence of noise corrupting all the data and we compare its performance with the class of robust m estimators references to published applications of robust techniques in computer vision are also given

a review on egomotion by means of differential epipolar geometry applied to the movement of a mobile robot the estimation of camera egomotion is an old problem in computer vision since the 1980s many approaches based on both the discrete and the differential epipolar constraint have been proposed the discrete case is used mainly in self calibrated stereoscopic systems whereas the differential case deals with a single moving camera this article surveys several methods for 3d motion estimation unifying the mathematics convention which are then adapted to the common case of a mobile robot moving on a plane experimental results are given on synthetic data covering more than 0 5 million estimations these surveyed algorithms have been programmed and are available on the internet

inferring segmented dense motion layers using 5d tensor voting we present a novel local spatiotemporal approach to produce motion segmentation and dense temporal trajectories from an image sequence a common representation of image sequences is a 3d spatiotemporal volume x y t and its corresponding mathematical formalism is the fiber bundle however directly enforcing the spatiotemporal smoothness constraint is difficult in the fiber bundle representation thus we convert the representation into a new 5d space x y t vx vy with an additional velocity domain where each moving object produces a separate 3d smooth layer the smoothness constraint is now enforced by extracting 3d layers using the tensor voting framework in a single step that solves both correspondence and segmentation simultaneously motion segmentation is achieved by identifying those layers and the dense temporal trajectories are obtained by converting the layers back into the fiber bundle representation we proceed to address three applications tracking mosaic and 3d reconstruction that are hard to solve from the video stream directly because of the segmentation and dense matching steps but become straightforward with our framework the approach does not make restrictive assumptions about the observed scene or camera motion and is therefore generally applicable we present results on a number of data sets

hop diffusion monte carlo for epipolar geometry estimation between very wide baseline images we present a monte carlo approach for epipolar geometry estimation that efficiently searches for minimal sets of inlier correspondences in the presence of many outliers in the putative correspondence set a condition that is prevalent when we have wide baselines significant scale changes rotations in depth occlusion and repeated patterns the proposed monte carlo algorithm uses balanced local and global search blogs to find the best minimal set of correspondences the local search is a diffusion process using joint feature distributions that captures the dependencies among the correspondences and the global search is a hopping search process across the minimal set space controlled by photometric properties using a novel experimental protocol that involves computing errors for manually marked ground truth points and images with outlier rates as high as 90 percent we find that blogs is better than related approaches such as mapsac 1 napsac 2 and beem 3 blogs results are of similar quality as other approaches but blogs generate them in 10 times fewer iterations the time per iteration for blogs is also the lowest among the ones we studied

implementing a multi model estimation method we revisit the problem of parameter estimation in computer vision reconsidering and implementing what may be called the kanatani s estimation method presented here as a simple optimisation problem so a without any direct reference to a probabilistic framework but b considering i non linear implicit measurement equations and parameter constraints plus ii robust estimation in the presence of outliers and iii multi model comparisons here a a projection algorithm based on generalisations of square root decompositions allows an efficient and numerically stable local resolution of a set of non linear equations on the other hand b a robust estimation module of a hierarchy of non linear models has been designed and validated a step ahead c the software architecture of the estimation module is discussed with the goal of being integrated in reactive software environments or within applications with time constraints while an experimentation considering the parameterisation of retinal displacements between two views is proposed as an illustration of the estimation module