generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

information theoretic subspace clustering this paper addresses the problem of grouping the data points sampled from a union of multiple subspaces in the presence of outliers information theoretic objective functions are proposed to combine structured low rank representations lrrs to capture the global structure of data and information theoretic measures to handle outliers in theoretical part we point out that group sparsity induced measures ell 2 1 norm ell alpha norm and correntropy can be justified from the viewpoint of half quadratic hq optimization which facilitates both convergence study and algorithmic development in particular a general formulation is accordingly proposed to unify hq based group sparsity methods into a common framework in algorithmic part we develop information theoretic subspace clustering methods via correntropy with the help of parzen window estimation correntropy is used to handle either outliers under any distributions or sample specific errors in data pairwise link constraints are further treated as a prior structure of lrrs based on the hq framework iterative algorithms are developed to solve the nonconvex information theoretic loss functions experimental results on three benchmark databases show that our methods can further improve the robustness of lrr subspace clustering and outperform other state of the art subspace clustering methods

learning graphs with monotone topology properties and multiple connected components recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph laplacian constraints while such problems are convex existing methods cannot guarantee that solutions will have specific graph topology properties e g being a tree which are desirable for some applications the problem of learning a graph with topology properties is in general non convex in this paper we propose an approach to solve these problems by decomposing them into two sub problems for which efficient solutions are known specifically a graph topology inference gti step is employed to select a feasible graph topology then a graph weight estimation gwe step is performed by solving a generalized graph laplacian estimation problem where edges are constrained by the topology found in the gti step our main result is a bound on the error of the gwe step as a function of the error in the gti step this error bound indicates that the gti step should be solved using an algorithm that approximates the data similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology the gti stage can leverage existing methods which are typically based on minimizing the total weight of removed edges since the gwe stage is an inverse covariance estimation problem with linear constraints it can be solved using existing convex optimization methods we demonstrate that our approach can achieve good results for both synthetic and texture image data

multibody grouping from motion images we want to deduce from a sequence of noisy two dimensional images of a scene of several rigid bodies moving independently in three dimensions the number of bodies and the grouping of given feature points in the images to the bodies prior processing is assumed to have identified features or points common to all frames and the images are assumed to be created by orthographic projection i e perspective effects are minimal we describe a computationally inexpensive algorithm that can determine which points or features belong to which rigid body using the fact that with exact observations in orthographic projection points on a single body lie in a three or less dimensional linear manifold of frame space if there are enough observations and independent motions these manifolds can be viewed as a set linearly independent four or less dimensional subspaces we show that the row echelon canonical form provides direct information on the grouping of points to these subspaces treatment of the noise is the most difficult part of the problem this paper uses a statistical approach to estimate the grouping of points to subspaces in the presence of noise by computing which partition has the maximum likelihood the input data is assumed to be contaminated with independent gaussian noise the algorithm can base its estimates on a user supplied standard deviation of the noise or it can estimate the noise from the data the algorithm can also be used to estimate the probability of a user specified partition so that the hypothesis can be combined with others using bayesian statistics

an interpretive model of line continuation in human visual perception abstract continuation is a fundamental intuitive property of perceptual grouping in line segregation to explain this property a hypothetical model from the standpoint of computer engineering is presented in this paper this model includes four mechanisms i e signal beaming mechanism feedback transmitting mechanism image partitioning mechanism and image revising mechanism and a temporary 3 d space called rho space or quotient space the map of excitatory and inhibitory signals from an input image is first constructed by applying some functions of the so called simple cortical cells which are sensitive to the position and orientation of the stimulus the signal beaming mechanism and feedback transmitting mechanism interact in human behaviours such as walking to resolve problems such as continuity transition fuzziness and ambiguity by using these two mechanisms the constructed map is then converted into a three dimensional quotient space in which all the members voxels satisfy the equivalence relationships so that the image partitioning mechanism can accomplish the work of line segregation the image revising mechanism is further used to eliminate the jaggedness caused by the signal transmission so that the final result of line segregation is consistent with the one from human observation this interpretive model was built on a 32 bits workstation sun 3 160 the results of several image processing simulations suggest that the hypothetical model is feasible and reasonable

translated poisson mixture model for stratification learning a framework for the regularized and robust estimation of non uniform dimensionality and density in high dimensional noisy data is introduced in this work this leads to learning stratifications that is mixture of manifolds representing different characteristics and complexities in the data set the basic idea relies on modeling the high dimensional sample points as a process of translated poisson mixtures with regularizing restrictions leading to a model which includes the presence of noise the translated poisson distribution is useful to model a noisy counting process and it is derived from the noise induced translation of a regular poisson distribution by maximizing the log likelihood of the process counting the points falling into a local ball we estimate the local dimension and density we show that the sequence of all possible local countings in a point cloud formed by samples of a stratification can be modeled by a mixture of different translated poisson distributions thus allowing the presence of mixed dimensionality and densities in the same data set with this statistical model the parameters which best describe the data estimated via expectation maximization divide the points in different classes according to both dimensionality and density together with an estimation of these quantities for each class theoretical asymptotic results for the model are presented as well the presentation of the theoretical framework is complemented with artificial and real examples showing the importance of regularized stratification learning in high dimensional data analysis in general and computer vision and image analysis in particular

robust algebraic segmentation of mixed rigid body and planar motions from two views this paper studies segmentation of multiple rigid body motions in a 3 d dynamic scene under perspective camera projection we consider dynamic scenes that contain both 3 d rigid body structures and 2 d planar structures based on the well known epipolar and homography constraints between two views we propose a hybrid perspective constraint hpc to unify the representation of rigid body and planar motions given a mixture of k hybrid perspective constraints we propose an algebraic process to partition image correspondences to the individual 3 d motions called robust algebraic segmentation ras particularly we prove that the joint distribution of image correspondences is uniquely determined by a set of 2k th degree polynomials a global signature for the union of k motions of possibly mixed type the first and second derivatives of these polynomials provide a means to recover the association of the individual image samples to their respective motions finally using robust statistics we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers we conduct extensive simulations and real experiments to validate the performance of the new algorithm the results demonstrate that ras achieves notably higher accuracy than most existing robust motion segmentation methods including random sample consensus ransac and its variations the implementation of the algorithm is also two to three times faster than the existing methods the implementation of the algorithm and the benchmark scripts are available at http perception csl illinois edu ras

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

two view multibody structure from motion we present an algebraic geometric approach to 3 d motion estimation and segmentation of multiple rigid body motions from noise free point correspondences in two perspective views our approach exploits the algebraic and geometric properties of the so called multibody epipolar constraint and its associated multibody fundamental matrix which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions we derive a rank constraint on a polynomial embedding of the correspondences from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix we then show how to compute the epipolar lines from the first order derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using generalized pca gpca given the epipoles and epipolar lines the estimation of individual fundamental matrices becomes a linear problem the clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices although our approach is mostly designed for noise free correspondences we also test its performance on synthetic and real data with moderate levels of noise

spatio temporal union of subspaces for multi body non rigid structure from motion abstract non rigid structure from motion nrsfm has so far been mostly studied for recovering 3d structure of a single non rigid deforming object to handle the real world challenging multiple deforming objects scenarios existing methods either pre segment different objects in the scene or treat multiple non rigid objects as a whole to obtain the 3d non rigid reconstruction however these methods fail to exploit the inherent structure in the problem as the solution of segmentation and the solution of reconstruction could not benefit each other in this paper we propose a unified framework to jointly segment and reconstruct multiple non rigid objects to compactly represent complex multi body non rigid scenes we propose to exploit the structure of the scenes along both temporal direction and spatial direction thus achieving a spatio temporal representation specifically we represent the 3d non rigid deformations as lying in a union of subspaces along the temporal direction and represent the 3d trajectories as lying in the union of subspaces along the spatial direction this spatio temporal representation not only provides competitive 3d reconstruction but also outputs robust segmentation of multiple non rigid objects the resultant optimization problem is solved efficiently using the alternating direction method of multipliers admm extensive experimental results on both synthetic and real multi body nrsfm datasets demonstrate the superior performance of our proposed framework compared with the state of the art methods

motion recovery by integrating over the joint image manifold recovery of epipolar geometry is a fundamental problem in computer vision the introduction of the joint image manifold jim allows to treat the recovery of camera motion and epipolar geometry as the problem of fitting a manifold to the data measured in a stereo pair the manifold has a singularity and boundary therefore special care must be taken when fitting it four fitting methods are discussed direct algebraic geometric and the integrated maximum likelihood iml based method the first three methods are the exact analogues of three common methods for recovering epipolar geometry the more recently introduced iml method seeks the manifold which has the highest support in the sense that the largest measure of its points are close to the data while computationally more intensive than the other methods its results are better in some scenarios both simulations and experiments suggest that the advantages of iml manifold fitting carry over to the task of recovering epipolar geometry especially when the extent of the data and or the motion are small

a regularized approach for geodesic based semisupervised multimanifold learning geodesic distance as an essential measurement for data dissimilarity has been successfully used in manifold learning however most geodesic distance based manifold learning algorithms have two limitations when applied to classification 1 class information is rarely used in computing the geodesic distances between data points on manifolds and 2 little attention has been paid to building an explicit dimension reduction mapping for extracting the discriminative information hidden in the geodesic distances in this paper we regard geodesic distance as a kind of kernel which maps data from linearly inseparable space to linear separable distance space in doing this a new semisupervised manifold learning algorithm namely regularized geodesic feature learning algorithm is proposed the method consists of three techniques a semisupervised graph construction method replacement of original data points with feature vectors which are built by geodesic distances and a new semisupervised dimension reduction method for feature vectors experiments on the mnist usps handwritten digit data sets mit cbcl face versus nonface data set and an intelligent traffic data set show the effectiveness of the proposed algorithm

mdpe a very robust estimator for model fitting and range image segmentation in this paper we propose a novel and highly robust estimator called mdpe1 maximum density power estimator this estimator applies nonparametric density estimation and density gradient estimation techniques in parametric estimation model fitting mdpe optimizes an objective function that measures more than just the size of the residuals both the density distribution of data points in residual space and the size of the residual corresponding to the local maximum of the density distribution are considered as important characteristics in our objective function mdpe can tolerate more than 85 outliers compared with several other recently proposed similar estimators mdpe has a higher robustness to outliers and less error variance we also present a new range image segmentation algorithm based on a modified version of the mdpe quick mdpe and its performance is compared to several other segmentation methods segmentation requires more than a simple minded application of an estimator no matter how good that estimator is our segmentation algorithm overcomes several difficulties faced with applying a statistical estimator to this task

pricing vulnerable path dependent options using integral transforms abstract in the over the counter otc markets the holders of many contracts are vulnerable to counterparty credit risk because of this issue vulnerable options must be considered in addition in a financial environment the pricing of path dependent options yields many interesting mathematical challenges in this paper we study the pricing of vulnerable path dependent options using double mellin transforms to investigate an explicit closed form pricing formula or semi analytic formula in each path dependent option

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

a new approach to two view motion segmentation using global dimension minimization we present a new approach to rigid body motion segmentation from two views we use a previously developed nonlinear embedding of two view point correspondences into a 9 dimensional space and identify the different motions by segmenting lower dimensional subspaces in order to overcome nonuniform distributions along the subspaces whose dimensions are unknown we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation we propose a fast projected gradient algorithm for minimizing global dimension and thus segmenting motions from 2 views we develop an outlier detection framework around the proposed method and we present state of the art results on outlier free and outlier corrupted two view data for segmenting motion

on the approximability of minimizing nonzero variables or unsatisfied relations in linear systems we investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning operations research and pattern recognition in the first class min ulr one wishes given a possibly infeasible system of linear relations to find a solution that violates as few relations as possible while satisfying all the others in the second class min rvls the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible for both min ulr and min rvls the four basic types of relational operators and are considered while min rvls with equations was mentioned to be np hard in garey and johnson 1979 we established in amaldi 1992 amaldi and kann 1995 that min ulr with equalities and inequalities are np hard even when restricted to homogeneous systems with bipolar coefficients the latter problems have been shown hard to approximate in arora et al 1993 in this paper we determine strong bounds on the approximability of various variants of min rvls and min ulr including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional the various np hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints but none of them can be approximated within any constant factor unless p np particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning in particular we disprove a conjecture of van horn and martinez 1992 regarding the existence of a polynomial time algorithm to design linear classifiers or perceptrons that involve a close to minimum number of features

self regularized fixed rank representation for subspace segmentation abstract consider a set of data points generated from various linear subspaces subspace segmentation tasks which are important in fields such as computer vision and image processing aim to partition the set of data points so as to recover these subspaces the subspace segmentation method fixed rank representation frr was introduced to remedy the problem of insufficient sampling in classical low rank representation lrr in many subspace segmentation applications frr has achieved much better results than those of lrr in this paper a new frr related algorithm called self regularized fixed rank representation srfrr is proposed in srfrr a laplacian regularizer is constructed using the coefficient matrix obtained by srfrr itself further by proving that the laplacian regularizer can be transformed into a structure constraint on the coefficient matrix we show that another existing method sparse frr sfrr is a special case of srfrr to implement the srfrr method we present two optimization algorithms experiments on both synthetic and real databases show that srfrr outperforms some existing frr and lrr related algorithms

subspace learning by l0 induced sparsity subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure such methods with sparsity prior such as sparse subspace clustering ssc elhamifar and vidal in ieee trans pattern anal mach intell 35 11 2765 2781 2013 with the sparsity induced by the ell 1 norm are demonstrated to be effective in subspace clustering most of those methods require certain assumptions e g independence or disjointness on the subspaces however these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods in this paper we propose ell 0 induced sparse subspace clustering ell 0 ssc in contrast to the required assumptions such as independence or disjointness on subspaces for most existing sparse subspace clustering methods we prove that ell 0 ssc guarantees the subspace sparse representation a key element in subspace clustering for arbitrary distinct underlying subspaces almost surely under the mild i i d assumption on the data generation we also present the no free lunch theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding ell 0 sparse representation problem of ell 0 ssc a novel approximate algorithm named approximate ell 0 ssc a ell 0 ssc is developed which employs proximal gradient descent to obtain a sub optimal solution to the optimization problem of ell 0 ssc with theoretical guarantee the sub optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results extensive experimental results on various data sets demonstrate the superiority of a ell 0 ssc compared to other competing clustering methods furthermore we extend ell 0 ssc to semi supervised learning by performing label propagation on the sparse similarity matrix learnt by a ell 0 ssc and demonstrate the effectiveness of the resultant semi supervised learning method termed ell 0 sparse subspace label propagation ell 0 sslp

minimum error entropy based sparse representation for robust subspace clustering this paper addresses the problem of clustering data points that are approximately drawn from multiple subspaces recently a large family of spectral clustering based methods such as sparse subspace clustering ssc and low rank representation lrr have been proposed in this paper we present a general formulation to unify many of them within a common framework based on atomic representation since mean square error mse relies heavily on the gaussianity assumption the previous mse based subspace clustering methods have the limitation of being sensitive to non gaussian noise in this paper we develop a novel subspace clustering method termed meessc by specifying the minimum error entropy mee as the loss function and the sparsity inducing atomic set we show that meessc can well overcome the above limitation the experimental results on both synthetic and real data verify the effectiveness of the proposed method

motion segmentation in the presence of outlying incomplete or corrupted trajectories in this paper we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence using the affine camera model this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces in practice due to limitations of the tracker occlusions and the presence of nonrigid objects in the scene the obtained motion trajectories may contain grossly mistracked features missing entries or corrupted entries in this paper we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework our methods draw strong connections between lossy compression rank minimization and sparse representation we test our methods extensively on the hopkins155 motion segmentation database and other motion sequences with outliers and missing data we compare the performance of our methods to state of the art motion segmentation methods based on expectation maximization and spectral clustering for data without outliers or missing information the results of our methods are on par with the state of the art results and in many cases exceed them in addition our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above all code and results are publicly available at http perception csl uiuc edu coding motion

recovery guarantees for exemplar based clustering for a certain class of distributions we prove that the linear programming relaxation of k medoids clustering a variant of k means clustering where means are replaced by exemplars from within the dataset distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball in this case clustering by thresholding pairwise distances between points can fail we also exhibit numerical evidence of high probability recovery in a substantially more permissive regime

adaptive graph construction using data self representativeness for pattern classification graph construction from data constitutes a pre stage in many machine learning and computer vision tasks like semi supervised learning manifold learning and spectral clustering the influence of graph construction procedures on learning tasks and their related applications has only received limited study despite its critical impact on accuracy state of the art graphs are built via sparse coding adopting 1 regularization those graphs exhibit good performance in many computer vision applications however the locality and similarity among instances are not explicitly used in the coding scheme furthermore due to the use of 1 regularization these construction approaches can be computationally expensive in this paper we investigate graph construction using the data self representativeness property by incorporating a variant of locality constrained linear coding llc we introduce and derive four variants for graph construction these variants adopt a two phase llc tpllc compared with the recent 1 graphs our proposed objective function associated with three variants has an analytical solution and thus is more efficient a key element of the proposed methods is the second phase of coding that allows data closeness or locality to be naturally incorporated it performs a coding over some selected relevant samples and reinforces the individual regularization terms by exploiting the coefficients estimated in the first phase comprehensive experimental results using several benchmark datasets show that it can achieve or outperform existing state of the art results furthermore it is shown to be more efficient than the robust 1 graph construction schemes

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

robust subspace clustering via thresholding the problem of clustering noisy and incompletely observed high dimensional data points into a union of low dimensional subspaces and a set of outliers is considered the number of subspaces their dimensions and their orientations are assumed unknown we propose a simple low complexity subspace clustering algorithm which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points in other words the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance a statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect specifically our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level we furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be up to a log factor linear in the ambient dimension we also propose a simple scheme that provably detects outliers and we present numerical results on real and synthetic data

coherence pursuit fast simple and robust principal component analysis this paper presents a remarkably simple yet powerful algorithm termed coherence pursuit cop to robust principal component analysis pca as inliers lie in a low dimensional subspace and are mostly correlated an inlier is likely to have strong mutual coherence with a large number of data points by contrast outliers either do not admit low dimensional structures or form small clusters in either case an outlier is unlikely to bear strong resemblance to a large number of data points given that cop sets an outlier apart from an inlier by comparing their coherence with the rest of the data points the mutual coherences are computed by forming the gram matrix of the normalized data points subsequently the sought subspace is recovered from the span of the subset of the data points that exhibit strong coherence with the rest of the data as cop only involves one simple matrix multiplication it is significantly faster than the state of the art robust pca algorithms we derive analytical performance guarantees for cop under different models for the distributions of inliers and outliers in both noise free and noisy settings cop is the first robust pca algorithm that is simultaneously non iterative provably robust to both unstructured and structured outliers and can tolerate a large number of unstructured outliers

a geometric analysis of subspace clustering with outliers this paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes as is common in computer vision or unsupervised learning applications we do not know in advance how many subspaces there are nor do we have any information about their dimensions we develop a novel geometric analysis of an algorithm named sparse subspace clustering ssc 11 which signicantly broadens the range of problems where it is provably eective for instance we show that ssc can recover multiple subspaces each of dimension comparable to the ambient dimension we also prove that ssc can correctly cluster data points even when the subspaces of interest intersect further we develop an extension of ssc that succeeds when the data set is corrupted with possibly overwhelmingly many outliers underlying our analysis are clear geometric insights which may bear on other sparse recovery problems a numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods

robust recovery of multiple subspaces by geometric lp minimization we assume i i d data sampled from a mixture distribution with k components along fixed d dimensional linear subspaces and an additional outlier component for p 0 we study the simultaneous recovery of the k fixed subspaces by minimizing the lp averaged distances of the sampled data points from any k subspaces under some conditions we show that if 0 1 and p 1 then the underlying subspaces cannot be recovered or even nearly recovered by lp minimization the results of this paper partially explain the successes and failures of the basic approach of lp energy minimization for modeling data by multiple subspaces

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

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

three view multibody structure from motion we propose a geometric approach to 3d motion segmentation from point correspondences in three perspective views we demonstrate that after applying a polynomial embedding to the point correspondences they become related by the so called multibody trilinear constraint and its associated multibody trifocal tensor which are natural generalizations of the trilinear constraint and the trifocal tensor to multiple motions we derive a rank constraint on the embedded correspondences from which one can estimate the number of independent motions as well as linearly solve for the multibody trifocal tensor we then show how to compute the epipolar lines associated with each image point from the common root of a set of univariate polynomials and the epipoles by solving a pair of plane clustering problems using generalized principal component analysis gpca the individual trifocal tensors are then obtained from the second order derivatives of the multibody trilinear constraint given epipolar lines and epipoles or trifocal tensors one can immediately obtain an initial clustering of the correspondences we use this clustering to initialize an iterative algorithm that alternates between the computation of the trifocal tensors and the segmentation of the correspondences we test our algorithm on various synthetic and real scenes and compare it with other algebraic and iterative algorithms

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

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

enhancing low rank subspace clustering by manifold regularization recently low rank representation lrr method has achieved great success in subspace clustering sc which aims to cluster the data points that lie in a union of low dimensional subspace given a set of data points lrr seeks the lowest rank representation among the many possible linear combinations of the bases in a given dictionary or in terms of the data itself however lrr only considers the global euclidean structure while the local manifold structure which is often important for many real applications is ignored in this paper to exploit the local manifold structure of the data a manifold regularization characterized by a laplacian graph has been incorporated into lrr leading to our proposed laplacian regularized lrr laplrr an efficient optimization procedure which is based on alternating direction method of multipliers admm is developed for laplrr experimental results on synthetic and real data sets are presented to demonstrate that the performance of lrr has been enhanced by using the manifold regularization

transformation invariant subspace clustering subspace clustering has achieved great success in many computer vision applications however most subspace clustering algorithms require well aligned data samples which is often not straightforward to achieve this paper proposes a transformation invariant subspace clustering framework by jointly aligning data samples and learning subspace representation by alignment the transformed data samples become highly correlated and a better affinity matrix can be obtained the joint problem can be reduced to a sequence of least squares regression problems which can be efficiently solved we verify the effectiveness of the proposed method with extensive experiments on unaligned real data demonstrating its higher clustering accuracy than the state of the art subspace clustering and transformation invariant clustering algorithms highlightsa transformation invariant subspace clustering algorithm is proposed alignment and subspace clustering are mutually dependent state of the art clustering results on the unconstrained face dataset

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

an investigation into the closure process in human visual perception for line patterns closure property can provide a key to the image understanding in human visual perception it is a fundamental intuitive property of pattern grouping as advocated by the gestalt school of psychology the closure possesses a property to organize the world by filling in the gaps in the stimulation so that we perceive a whole object and not disjointed parts organization principle to investigate the organization principle in human visual perception for line patterns closure process and the function of visual receptive field are considered in this paper two corresponding methods from the viewpoint of computer science are presented and applied to the removal of the closure noise appearing in the binary dpc degraded paint character image one is called de disk expansion method based on the boundary information of stimulation to simulate the closure process the other is called vf visual receptive field method based on the lateral spread of excitation in neurophysiology to provide the effective vf for closure process the combination of de and vf methods is also developed a series of detailed experiments are given to illustrate the proposed algorithms and to compare the performance of the proposed algorithms with that of related existing methods morphology method and relaxation method experiments show that the proposed algorithms can obtain the better smoothing results and can reasonably simulate the closure process in human visual perception for line patterns

convex sparse spectral clustering single view to multi view spectral clustering sc is one of the most widely used methods for data clustering it first finds a low dimensional embedding u of data by computing the eigenvectors of the normalized laplacian matrix and then performs k means on text u top to get the final clustering result in this paper we observe that in the ideal case text u text u top should be block diagonal and thus sparse therefore we propose the sparse sc ssc method that extends the sc with sparse regularization on text u text u top to address the computational issue of the nonconvex ssc model we propose a novel convex relaxation of ssc based on the convex hull of the fixed rank projection matrices then the convex ssc model can be efficiently solved by the alternating direction method of multipliers furthermore we propose the pairwise ssc that extends ssc to boost the clustering performance by using the multi view information of data experimental comparisons with several baselines on real world datasets testify to the efficacy of our proposed methods

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

block sparse recovery via convex optimization given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks we study the problem of finding a block sparse representation of the signal i e a representation that uses the minimum number of blocks motivated by signal image processing and computer vision applications such as face recognition we consider the block sparse recovery problem in the case where the number of atoms in each block is arbitrary possibly much larger than the dimension of the underlying subspace to find a block sparse representation of a signal we propose two classes of nonconvex optimization programs which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks respectively since both classes of problems are np hard we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original nonconvex formulation our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary which are natural generalizations of existing notions of mutual and cumulative coherence we evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition we show that treating the face recognition problem as a block sparse recovery problem improves the state of the art results by 10 with only 25 of the training data

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

blessing of dimensionality recovering mixture data via dictionary pursuit this paper studies the problem of recovering the authentic samples that lie on a union of multiple subspaces from their corrupted observations due to the high dimensional and massive nature of today s data driven community it is arguable that the target matrix i e authentic sample matrix to recover is often low rank in this case the recently established robust principal component analysis rpca method already provides us a convenient way to solve the problem of recovering mixture data however in general rpca is not good enough because the incoherent condition assumed by rpca is not so consistent with the mixture structure of multiple subspaces namely when the subspace number grows the row coherence of data keeps heightening and accordingly rpca degrades to overcome the challenges arising from mixture data we suggest to consider lrr in this paper we elucidate that lrr can well handle mixture data as long as its dictionary is configured appropriately more precisely we mathematically prove that lrr can weaken the dependence on the row coherence provided that the dictionary is well conditioned and has a rank of not too high in particular if the dictionary itself is sufficiently low rank then the dependence on the row coherence can be completely removed these provide some elementary principles for dictionary learning and naturally lead to a practical algorithm for recovering mixture data our experiments on randomly generated matrices and real motion sequences show promising results

segmentation and association among lines and junctions for a line image abstract the relationships among line primitives and junctions for a line image are of great importance in pattern recognition with a conventional thinning method the thinning result may yield several short branches and multiple fork points at a junction which is the intersection of line primitives from the view of an original line image this is called the lability of the shape of the skeleton and will become an impediment in the construction of relationships among line primitives and junctions in this paper an effective approach to performing the segmentation and association among line primitives and junctions for a line image is presented the proposed approach cannot only segment the line primitives and junctions but also construct their relationships for the further interpretation of the given line image the segmentation is first performed by the analysis of line continuation to construct a map of uni olls orientation of the longest line in sight and a map of multi olls for an input line image the former is used to find the line primitives the latter to find the junctions next the two orientational maps are processed individually via a 2d median filter 2d labeling and partitioning and symmetry measuring then the line primitive containing the orientation and symmetry information and the junction containing the reasonable central point information are obtained finally the relationships among the found features are constructed by the transition of the orientation from the partitioned line primitives to the junctions the experimental results show that the proposed approach is feasible for the line type of characters under consideration

simultaneous bayesian sparse approximation with structured sparse models sparse approximation is key to many signal processing image processing and machine learning applications if multiple signals maintain some degree of dependency for example the support sets are statistically related then it will generally be advantageous to jointly estimate the sparse representation vectors from the measurement vectors as opposed to solving for each signal individually in this paper we propose simultaneous sparse bayesian learning sbl for joint sparse approximation with two structured sparse models ssms where one is row sparse with embedded element sparse and the other one is row sparse plus element sparse while sbl has attracted much attention as a means to deal with a single sparse approximation problem it is not obvious how to extend sbl to ssms by capitalizing on a dual space view of existing convex methods for sms we showcase the precision component model and covariance component model for ssms where both models involve a common hyperparameter and an innovation hyperparameter that together control the prior variance for each coefficient the statistical perspective of precision component versus covariance component models unfolds the intrinsic mechanism in ssms and also leads to our development of sbl inspired cost functions for ssms centralized algorithms that include ell 1 and ell 2 reweighting algorithms and consensus based decentralized algorithms are developed for simultaneous sparse approximation with ssms in addition theoretical analysis is conducted to provide valuable insights into the proposed approach which includes global minima analysis of the sbl inspired nonconvex cost functions and convergence analysis of the proposed ell 1 reweighting algorithms for ssms superior performance of the proposed algorithms is demonstrated by numerical experiments

valuing vulnerable geometric asian options an option on a defaultable instrument is prone to default risk arising from the probability that the instrument s issuer will not honor its contractual obligations under this financial circumstance we consider vulnerable asian options in this paper we obtain parabolic partial differential equations with time dependent coefficients on the vulnerable asian option model and derive a closed formula of the pricing of the vulnerable geometric asian option using the pricing formula of the vulnerable european option with time dependent coefficients

robust kernel low rank representation recently low rank representation lrr has shown promising performance in many real world applications such as face clustering however lrr may not achieve satisfactory results when dealing with the data from nonlinear subspaces since it is originally designed to handle the data from linear subspaces in the input space meanwhile the kernel based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel induced mapping to effectively cope with the nonlinear data we first propose the kernelized version of lrr in the clean data case we also present a closed form solution for the resultant optimization problem moreover to handle corrupted data we propose the robust kernel lrr rklrr approach and develop an efficient optimization algorithm to solve it based on the alternating direction method in particular we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved and it is guaranteed to obtain a globally optimal solution besides our proposed algorithm can also solve the original lrr problem which is a special case of our rklrr when using the linear kernel in addition based on our new optimization technique the kernelization of some variants of lrr can be similarly achieved comprehensive experiments on synthetic data sets and real world data sets clearly demonstrate the efficiency of our algorithm as well as the effectiveness of rklrr and the kernelization of two variants of lrr

stroke segmentation by bernstein be zier curve fitting abstract stroke segmentation is essential to the recognition of handwritten chinese characters here some new and reliable techniques are proposed first the thinning process is applied to each character to obtain the skeleton then a maximum circle technique is used to remove the spurious branches and merge the split 4 fork points all possible pairs of stroke segments connected at the same fork point are considered and the bernstein bezier curve is used to fit each pair to smooth the data and find its trend then from the result of this curve fitting we can decide which pair belongs to the same stroke finally the inflection points with very small radius of curvature are found and the stroke segmentation is carried out based on these inflection points experiments on some chinese characters show that the proposed new techniques are reliable and time saving compared with the direction method

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

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

hybrid linear modeling via local best fit flats we present a simple and fast geometric method for modeling data by a union of affine subspaces the method begins by forming a collection of local best fit affine subspaces i e subspaces approximating the data in local neighborhoods the correct sizes of the local neighborhoods are determined automatically by the jones 2 numbers we prove under certain geometric conditions that our method finds the optimal local neighborhoods the collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model we discuss applications to tracking based motion segmentation and clustering of faces under different illuminating conditions we give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the mnist handwritten digits data and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces

thinning approach for noisy digital patterns abstract many problems difficult to solve by traditional computer algorithms can be explored by the principles of human visual perception from the hint of human visual symmetry a thinning approach for noisy digital patterns is presented in this paper it includes three main algorithms algorithm 1 is designed to find the effective circular range for each pixel containing the maximum effective information algorithm 2 is designed to compute the symmetry score of the pixel distribution in the circular range and convert the symmetry information into a gray scale image algorithm 3 provides a modified gray scale thinning algorithm for thinning the gray scale image to obtain a connected thin line experiments show that the thinning results obtained by the proposed method are properly acceptable whensnr 2 2 db and the connectedness of the thin line can be preserved

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

hypergraph modelling for geometric model fitting in this paper we propose a novel hypergraph based method called hf to fit and segment multi structural data the proposed hf formulates the geometric model fitting problem as a hypergraph partition problem based on a novel hypergraph model in the hypergraph model vertices represent data points and hyperedges denote model hypotheses the hypergraph with large and data determined degrees of hyperedges can express the complex relationships between model hypotheses and data points in addition we develop a robust hypergraph partition algorithm to detect sub hypergraphs for model fitting hf can effectively and efficiently estimate the number of and the parameters of model instances in multi structural data heavily corrupted with outliers simultaneously experimental results show the advantages of the proposed method over previous methods on both synthetic data and real images highlightswe propose a novel hypergraph based method to fit and segment multi structural data the proposed method includes a hypergraph model with large degrees of hyperedges the proposed method includes a robust hypergraph partition algorithm experimental results show that the proposed method is superior to some state of the art fitting methods

pricing vulnerable power exchange options in an intensity based framework abstract in this article we study the valuation of a european vulnerable power exchange option in an intensity based framework we assume that the default by the counter party is the time of the first jump of a doubly stochastic poisson process whose intensity is modeled by a jump diffusion process the dynamics of the two assets are assumed to be driven by correlated jump diffusion processes all the three processes are assumed to be correlated in continuous part as well as in the jump part in the proposed framework employing the measure change technique we obtain the explicit formula for the price of the power exchange option with counter party risk furthermore sensitivity analysis is given to illustrate the effects of counterparty risk on the price of the option and effect of various parameters on the option prices

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

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

sparse subspace clustering algorithm theory and applications many real world problems deal with collections of high dimensional data such as images videos text and web documents dna microarray data and more often such high dimensional data lie close to low dimensional structures corresponding to several classes or categories to which the data belong in this paper we propose and study an algorithm called sparse subspace clustering to cluster data points that lie in a union of low dimensional subspaces the key idea is that among the infinitely many possible representations of a data point in terms of other points a sparse representation corresponds to selecting a few points from the same subspace this motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces since solving the sparse optimization program is in general np hard we consider a convex relaxation and show that under appropriate conditions on the arrangement of the subspaces and the distribution of the data the proposed minimization program succeeds in recovering the desired sparse representations the proposed algorithm is efficient and can handle data points near the intersections of subspaces another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances such as noise sparse outlying entries and missing entries by incorporating the model of the data into the sparse optimization program we demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real world problems of motion segmentation and face clustering

the pricing of vulnerable options with double mellin transforms abstract many options traded in the over the counter markets are subject to default risks resulting from the probability that the option writer could not honor its contractual obligations there have been growing concerns about financial derivatives subject to default risks in particular since the global financial crisis and eurozone crisis this paper uses double mellin transforms to study european vulnerable options under constant as well as stochastic the hull white interest rates we obtain explicitly an analytic closed form pricing formula in each interest rate case so that the pricing of the options can be computed both accurately and efficiently

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

algebraic clustering of affine subspaces subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition prior work has studied this problem using algebraic iterative statistical low rank and sparse representation techniques while these methods have been applied to both linear and affine subspaces theoretical results have only been established in the case of linear subspaces for example algebraic subspace clustering asc is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal in this paper we study in a rigorous fashion the properties of asc in the case of affine subspaces using notions from algebraic geometry we prove that the homogenization trick which embeds points in a union of affine subspaces into points in a union of linear subspaces preserves the general position of the points and the transversality of the union of subspaces in the embedded space thus establishing the correctness of asc for affine subspaces

joint sparse representation and embedding propagation learning a framework for graph based semisupervised learning in this paper we propose a novel graph based semisupervised learning framework called joint sparse representation and embedding propagation learning jsrepl the idea of jsrepl is to join epl with sparse representation to perform label propagation like most of graph based semisupervised propagation learning algorithms jsrepl also constructs weights graph matrix from given data different from classical approaches which build weights graph matrix and estimate the labels of unlabeled data in sequence jsrepl simultaneously builds weights graph matrix and estimates the labels of unlabeled data we also propose an efficient algorithm to solve the proposed problem the proposed method is applied to the problem of semisupervised image clustering using the orl yale pie and yaleb data sets our experiments demonstrate the effectiveness of our proposed algorithm

extrinsic methods for coding and dictionary learning on grassmann manifolds sparsity based representations have recently led to notable results in various visual recognition tasks in a separate line of research riemannian manifolds have been shown useful for dealing with features and models that do not lie in euclidean spaces with the aim of building a bridge between the two realms we address the problem of sparse coding and dictionary learning in grassmann manifolds i e the space of linear subspaces to this end we propose to embed grassmann manifolds into the space of symmetric matrices by an isometric mapping this in turn enables us to extend two sparse coding schemes to grassmann manifolds furthermore we propose an algorithm for learning a grassmann dictionary atom by atom lastly to handle non linearity in data we extend the proposed grassmann sparse coding and dictionary learning algorithms through embedding into higher dimensional hilbert spaces experiments on several classification tasks gender recognition gesture classification scene analysis face recognition action recognition and dynamic texture classification show that the proposed approaches achieve considerable improvements in discrimination accuracy in comparison to state of the art methods such as kernelized affine hull method and graph embedding grassmann discriminant analysis

lucas kanade 20 years on a unifying framework since the lucas kanade algorithm was proposed in 1981 image alignment has become one of the most widely used techniques in computer vision applications range from optical flow and tracking to layered motion mosaic construction and face coding numerous algorithms have been proposed and a wide variety of extensions have been made to the original formulation we present an overview of image alignment describing most of the algorithms and their extensions in a consistent framework we concentrate on the inverse compositional algorithm an efficient algorithm that we recently proposed we examine which of the extensions to lucas kanade can be used with the inverse compositional algorithm without any significant loss of efficiency and which cannot in this paper part 1 in a series of papers we cover the quantity approximated the warp update rule and the gradient descent approximation in future papers we will cover the choice of the error function how to allow linear appearance variation and how to impose priors on the parameters

a grassmann framework for 4d facial shape analysis in this paper we investigate the contribution of dynamic evolution of 3d faces to identity recognition to this end we adopt a subspace representation of the flow of curvature maps computed on 3d facial frames of a sequence after normalizing their pose such representation allows us to embody the shape as well as its temporal evolution within the same subspace representation dictionary learning and sparse coding over the space of fixed dimensional subspaces called grassmann manifold have been used to perform face recognition we have conducted extensive experiments on the bu 4dfe dataset the obtained results of the proposed approach provide promising results highlightsrole of facial shape dynamics in identity recognition effective representation of 3d faces and their dynamics on grassmann manifolds sparse coding and dictionary learning for 4d faces classification

domain class consistency based transfer learning for image classification across domains distribution mismatch between the modeling data and the query data is a known domain adaptation issue in machine learning to this end in this paper we propose a l2 1 norm based discriminative robust kernel transfer learning dktl method for high level recognition tasks the key idea is to realize robust domain transfer by simultaneously integrating domain class consistency dcc metric based discriminative subspace learning kernel learning in reproduced kernel hilbert space and representation learning between source and target domain the dcc metric includes two properties domain consistency used to measure the between domain distribution discrepancy and class consistency used to measure the within domain class separability the essential objective of the proposed transfer learning method is to maximize the dcc metric which is equivalently to minimize the domain class inconsistency dcic such that domain distribution mismatch and class inseparability are well formulated and unified simultaneously the merits of the proposed method include 1 the robust sparse coding selects a few valuable source data with noises outliers removed during knowledge transfer and 2 the proposed dcc metric can pursue more discriminative subspaces of different domains as a result the maximum class separability is also well guaranteed extensive experiments on a number of visual datasets demonstrate the superiority of the proposed method over other state of the art domain adaptation and transfer learning methods

robust subspace clustering subspace clustering refers to the task of nding a multi subspace representation that best ts a collection of points taken from a high dimensional space this paper introduces an algorithm inspired by sparse subspace clustering ssc 25 to cluster noisy data and develops some novel theory demonstrating its correctness in particular the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation and on the number of samples per subspace synthetic as well as real data experiments complement our theoretical study illustrating our approach and demonstrating its eectiveness

robust elastic net subspace representation recently finding the low dimensional structure of high dimensional data has gained much attention given a set of data points sampled from a single subspace or a union of subspaces the goal is to learn or capture the underlying subspace structure of the data set in this paper we propose elastic net subspace representation a new subspace representation framework using elastic net regularization of singular values due to the strong convexity enforced by elastic net the proposed method is more stable and robust in the presence of heavy corruptions compared with existing lasso type rank minimization approaches for discovering a single low dimensional subspace we propose a computationally efficient low rank factorization algorithm called facten using a property of the nuclear norm and the augmented lagrangian method then clusten is proposed to handle the general case in which the data samples are drawn from a union of multiple subspaces for joint subspace clustering and estimation the proposed algorithms are applied to a number of subspace representation problems to evaluate the robustness and efficiency under various noisy conditions and experimental results show the benefits of the proposed method compared with existing methods

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography