learning layered motion segmentations of video we present an unsupervised approach for learning a layered representation of a scene from a video for motion segmentation our method is applicable to any video containing piecewise parametric motion the learnt model is a composition of layers which consist of one or more segments the shape of each segment is represented using a binary matte and its appearance is given by the rgb value for each point belonging to the matte included in the model are the effects of image projection lighting and motion blur furthermore spatial continuity is explicitly modeled resulting in contiguous segments unlike previous approaches our method does not use reference frame s for initialization the two main contributions of our method are i a novel algorithm for obtaining the initial estimate of the model by dividing the scene into rigidly moving components using efficient loopy belief propagation and ii refining the initial estimate using swap and expansion algorithms which guarantee a strong local minima results are presented on several classes of objects with different types of camera motion e g videos of a human walking shot with static or translating cameras we compare our method with the state of the art and demonstrate significant improvements

on ɛ spectra and stability radii techniques of krylov subspace iterations play an important role in computing e spectra of large matrices to obtain results about the reliability of this kind of approximations we propose to compare the position of the e spectrum of a with those of its diagonal submatrices we give theoretical results which are valid for any block decomposition in four blocks a11 a12 a21 a22 we then illustrate our results by numerical experiments the same kind of problem arises when we compute the stability radius of a large matrix in that context we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

infinite toeplitz and laurent matrices with localized impurities this paper is concerned with the change of the spectra of infinite toeplitz and laurent matrices under perturbations in a prescribed finite set of sites the main result says that the spectrum of a toeplitz matrix with a non constant rational symbol is not affected by small localized impurities while such impurities can nevertheless enlarge the spectrum of the corresponding laurent matrix we also study the spectra that may emerge when randomly perturbing toeplitz or laurent matrices in a randomly chosen single site

motion competition a variational approach to piecewise parametric motion segmentation we present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence our model is based on a conditional probability for the spatio temporal image gradient given a particular velocity model and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length exploiting the bayesian framework we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions the resulting functional can be interpreted as an extension of the mumford shah functional from intensity segmentation to motion segmentation in contrast to most alternative approaches the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set we propose two different representations of this motion boundary an explicit spline based implementation which can be applied to the motion based tracking of a single moving object and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects numerical results both for simulated ground truth experiments and for real world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion

spectral value sets of closed linear operators we study how the spectrum of a closed linear operator on a complex banach space changes under affine perturbations of the form a a a d e here a d and e are given linear operators whereas is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation d e the union of the spectra of the perturbed operators a with the norm of smaller than a given 0 is called the spectral value set of a at level in this paper we extend a known characterization of these sets for the matrix case to infinite dimensions and in so doing present a framework that allows for unbounded perturbations of closed linear operators on banach spaces the results will be illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

structured pseudospectra in structural engineering this paper presents a new method for computing the pseudospectra of a matrix that respects a prescribed sparsity structure the pseudospectrum is defined as the set of points in the complex plane to which an eigenvalue of the matrix can be shifted by a perturbation of a certain size a canonical form for sparsity preserving perturbations is given and a computable formula for the corresponding structured pseudospectra is derived this formula relates the computation of structured pseudospectra to the computation of the structured singular value ssv of an associated matrix although the computation of the ssv in general is an np hard problem algorithms for its approximation are available and demonstrate good performance when applied to the computation of structured pseudospectra of medium sized or highly sparse matrices the method is applied to a wing vibration problem where it is compared with the matrix polynomial approach and to the stability analysis of truss structures new measures for the vulnerability of a truss structure are proposed which are related to the distance to singularity of the associated stiffness matrix copyright 2005 john wiley sons ltd

isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space the classical isoperimetric inequality for a domain sigma subset mathbb r k with smooth boundary partial sigma is displaystyle k k omega k mathrm vol sigma k 1 leq mathrm vol partial sigma k where equality holds if and only if sigma is a ball in mathbb r k

stability of eigenvalues and spectral decompositions under linear perturbation abstract given an n by n matrix a and a fixed perturbation matrix e the effect of the linear perturbations a te t c on certain spectral properties of a is analyzed specially the effect of the structure of e on these properties is investigated a geometric framework for spectral analysis of a te is developed to achieve this goal which also leads to a better understanding of the sensitivity of eigenvalues and spectral decompositions of a

on the largest principal angle between random subspaces formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p dirnensional subspaces of r n chosen from the uniform distribution on the grassmann manifold which is the unique distribution invariant by orthogonal transformations of r n the formulas involve the gamma function and the hypergeometric function of a matrix argument c 2005 elsevier inc all rights reserved

spectral value sets a graphical tool for robustness analysis in this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix a can be shifted by real or complex perturbations of the form a a a d e where d and e are fixed matrices and the unknown disturbance matrix satisfies 0 both real and complex perturbations are studied the graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations the results are illustrated by various examples

multiframe motion segmentation with missing data using powerfactorization and gpca we consider the problem of segmenting multiple rigid body motions from point correspondences in multiple affine views we cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two three or four our algorithm involves projecting all point trajectories onto a 5 dimensional subspace using the svd the powerfactorization method or ransac and fitting multiple linear subspaces representing different rigid body motions to the points in 5 using gpca unlike previous work our approach does not restrict the motion subspaces to be four dimensional and independent instead it deals gracefully with all the spectrum of possible affine motions from two dimensional and partially dependent to four dimensional and fully independent our algorithm can handle the case of missing data meaning that point tracks do not have to be visible in all images by using the powerfactorization method to project the data in addition our method can handle outlying trajectories by using ransac to perform the projection we compare our approach to other methods on a database of 167 motion sequences with full motions independent motions degenerate motions partially dependent motions missing data outliers etc on motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions

structured pseudospectra and random eigenvalues problems in vibrating systems this paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics different types of pseudospectra of matrices and matrix polynomials are explained particular emphasis is given to structured pseudospectra for matrix polynomials which offer a deterministic way of dealing with uncertainties for structural dynamic systems the pseudospectra analysis is compared with the results from monte carlo simulations of uncertain discrete systems two illustrative example problems one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty are studied in detail excellent agreement is found between the pseudospectra results and monte carlo simulation results

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

exploiting unsupervised and supervised constraints for subspace clustering data in many image and video analysis tasks can be viewed as points drawn from multiple low dimensional subspaces with each subspace corresponding to one category or class one basic task for processing such kind of data is to separate the points according to the underlying subspace referred to as subspace clustering extensive studies have been made on this subject and nearly all of them use unconstrained subspace models meaning the points can be drawn from everywhere of a subspace to represent the data in this paper we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces e g belonging to a submanifold or satisfying the spatial regularity constraint this assumption usually describes the real data better such as differently moving objects in a video scene and face images of different subjects under varying illumination a unified integer linear programming optimization framework is used to approach subspace clustering which can be efficiently solved by a branch and bound bb method we also show that various kinds of supervised information such as subspace number outlier ratio pairwise constraints size prior and etc can be conveniently incorporated into the proposed framework experiments on real data show that the proposed method outperforms the state of the art algorithms significantly in clustering accuracy the effectiveness of the proposed method in exploiting supervised information is also demonstrated

tracking deforming objects using particle filtering for geometric active contours tracking deforming objects involves estimating the global motion of the object and its local deformations as a function of time tracking algorithms using kalman filters or particle filters have been proposed for finite dimensional representations of shape but these are dependent on the chosen parametrization and cannot handle changes in curve topology geometric active contours provide a framework which is parametrization independent and allow for changes in topology in the present work we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects to the best of our knowledge this is the first attempt to implement an approximate particle filtering algorithm for tracking on a theoretically infinite dimensional state space

spectra pseudospectra and localization for random bidiagonal matrices there has been much recent interest initiated by work of the physicists hatano and nelson in the eigenvalues of certain random non hermitian periodic tridiagonal matrices and their bidiagonal limits these eigenvalues cluster along a bubble with wings in the complex plane and the corresponding eigenvectors are localized in the wings delocalized in the bubble here in addition to eigenvalues pseudospectra are analyzed making it possible to treat the nonperiodic analogues of these random matrix problems inside the bubble the resolvent norm grows exponentially with the dimension outside it grows subexponentially in a bounded region that is the spectrum of the infinite dimensional operator localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase respectively with distance from the diagonal this article presents theorems that characterize the spectra pseudospectra and numerical range for the four cases of finite bidiagonal matrices infinite bidiagonal matrices stochastic toeplitz operators finite periodic matrices and doubly infinite bidiagonal matrices stochastic laurent operators c 2001 john wiley sons inc

geometric multiplicity margin for a submatrix let g be a square complex matrix with less than k nonconstant invariant factors we find a complex matrix that gives an optimal approximation to g among all possible matrices that have more than or equal to k invariant factors obtained by varying only the entries of a bottom right submatrix of g

eigenvalues pseudospectrum and structured perturbations we investigate the behavior of eigenvalues under structured perturbations we show that for many common structures such as complex symmetric toeplitz symmetric toeplitz circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations and prove similar results for real perturbations an exception are complex skewsymmetric matrices we also investigate componentwise complex and real perturbations here hermitian and skew hermitian matrices are exceptional for real perturbations furthermore we characterize the structured complex and real pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual unstructured pseudospectrum

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

pseudospectra and delay differential equations in this paper we present a new method for computing the pseudospectra of delay differential equations ddes with fixed finite delay this provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size and hence insight into how stability may change under variation of parameters we also investigate how differently weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectra furthermore we compute pseudospectra of the infinitesimal generator of the dde from which a lower bound on the maximum transient growth can be inferred to illustrate our method we consider a dde modelling a semiconductor laser subject to external feedback

on the dirichlet problem for minimal graphs in hyperbolic space soit r n n 2 un domaine borne on etudie le probleme de dirichlet suivant f f i f j f ij 1 df 2 n f 0 dans f 0 dans f 0 sur on considere le comportement a la frontiere de la solution

spectral curvature clustering scc this paper presents novel techniques for improving the performance of a multi way spectral clustering framework govindu in proceedings of the 2005 ieee computer society conference on computer vision and pattern recognition cvpr 05 vol 1 pp 1150 1157 2005 chen and lerman 2007 preprint in the supplementary webpage for segmenting affine subspaces specifically it suggests an iterative sampling procedure to improve the uniform sampling strategy an automatic scheme of inferring the tuning parameter from the data a precise initialization procedure for k means as well as a simple strategy for isolating outliers the resulting algorithm spectral curvature clustering scc requires only linear storage and takes linear running time in the size of the data it is supported by theory which both justifies its successful performance and guides our practical choices we compare it with other existing methods on a few artificial instances of affine subspaces application of the algorithm to several real world problems is also discussed

enhanced local subspace affinity for feature based motion segmentation we present a new motion segmentation algorithm the enhanced local subspace affinity elsa unlike local subspace affinity elsa is robust in a variety of conditions even without manual tuning of its parameters this result is achieved thanks to two improvements the first is a new model selection technique for the estimation of the trajectory matrix rank the second is an estimation of the number of motions based on the analysis of the eigenvalue spectrum of the symmetric normalized laplacian matrix results using the hopkins155 database and synthetic sequences are presented and compared with state of the art techniques

segmentation and tracking of multiple video objects this paper describes a technique that produces a content based representation of a video shot composed by a background still mosaic and one or more foreground moving objects segmentation of moving objects is based on ego motion compensation and on background modelling using tools from robust statistics region matching is carried out by an algorithm that operates on the mahalanobis distance between region descriptors in two subsequent frames and uses singular value decomposition to compute a set of correspondences satisfying both the principle of proximity and the principle of exclusion the sequence is represented as a layered graph and specific techniques are introduced to cope with crossing and occlusion examples of mpeg 4 main profile encoding are reported

asymptotic behavior of area minimizing currents in hyperbolic space on etudie le comportement asymptotique des courants localement rectifiables minimisant l aire dans un espace hyperbolique on considere l aspect theorie de la mesure geometrique et l aspect equation aux derivees partielles

can spectral value sets of toeplitz band matrices jump abstract the spectral value set speb ca of a bounded linear operator a on l2 is the union of the spectra of all operators of the form a bkc where k

on spectral variations under bounded real matrix perturbations in this paper we investigate the set of eigenvalues of a perturbed matrix ie509 1 wherea is given and n n is arbitrary we determine a lower bound for thisspectral value set which is exact for normal matricesa with well separated eigenvalues we also investigate the behaviour of the spectral value set under similarity transformations the results are then applied tostability radii which measure the distance of a matrixa from the set of matrices having at least one eigenvalue in a given closed instability domain b

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

structured pseudospectra and structured sensitivity of eigenvalues pseudospectra and structured pseudospectra have been investigated widely in this paper for the matrix spectral norm we give a new equivalent definition of structured pseudospectra of a matrix triplet a b c via the so called restricted singular value decomposition rsvd according to this definition we provide a solution of structured wilkinson s problem which concerns structured sensitivity of eigenvalues of square matrices several examples are given to illustrate our results also we extend the new definition of structured pseudospectra to polynomial eigenvalue problems

a note on structured pseudospectra in this note we study the notion of structured pseudospectra we prove that for toeplitz circulant hankel and symmetric structures the structured pseudospectrum equals the unstructured pseudospectrum we show that this is false for hermitian and skew hermitian structures we generalize the result to pseudospectra of matrix polynomials indeed we prove that the structured pseudospectrum equals the unstructured pseudospectrum for matrix polynomials with toeplitz circulant hankel and symmetric structures we conclude by giving a formula for structured pseudospectra of real matrix polynomials the particular type of perturbations used for these pseudospectra arise in control theory

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

erratum on the dirichlet problem for minimal graphs in hyperbolic space r n 0 that can be identified with the asymptotic infinity of hn 1 let nn 1 hn 1 be a closed embedded n 1 dimensional submanifold then there exists a complete absolutely area minimizing integral n current asymptotic to nn 1 at infinity see for example 1 2 it was also noted in 1 2 that in the case n 6 currents obtained are smoothly embedded complete hypersurfaces in the case n 7 as in the euclidean case there can be a singular set of hausdorff dimension at most n 7

regularity at infinity for area minimizing hypersurfaces in hyperbolic space equipped with the hyperbolic metric y z dxz dy2 a standard compactification of ih involves adding the boundary r 0 w so that i i is simply the one point compactification of the euclidean closed half space r 0 suppose 0 7 any interior singularities of m must remain in a bounded region of hyperbolic space near points of f in the euclidean topology m w f may thus be described as the graph of a function this function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to f the second author has recently established l a boundary higher regularity result for this equation which implies in particular that m w f is r if f is cgk for k 2 3 finally in case f bounds a star

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

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have