nonlocal discrete regularization on weighted graphs a framework for image and manifold processing we introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing the approach considers the problem as a variational one which consists of minimizing a weighted sum of two energy terms a regularization one that uses a discrete weighted dirichlet energy and an approximation one this is the discrete analogue of recent continuous euclidean nonlocal regularization functionals the proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted laplace operator parameterized by the degree of regularity the graph structure and the graph weight function these discrete processing methods provide a graph based version of recently proposed semi local or nonlocal processing methods used in image and mesh processing such as the bilateral filter the tv digital filter or the nonlocal means filter it works with equal ease on regular 2 d and 3 d images manifolds or any data we illustrate the abilities of the approach by applying it to various types of images meshes manifolds and data represented as graphs

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

tests for changing mean with monotonic power abstract several widely used tests for a changing mean exhibit nonmonotonic power in finite samples due to incorrect estimation of nuisance parameters under the alternative in this paper we study the issue of nonmonotonic power in testing for changing mean we investigate the asymptotic power properties of the tests using a new framework where alternatives are characterized as having large changes the asymptotic analysis provides a theoretical explanation to the power problem modified tests that have monotonic power against a wide range of alternatives of structural change are proposed instead of estimating the nuisance parameters based on ordinary least squares residuals the proposed tests use modified estimators based on nonparametric regression residuals it is shown that tests based on the modified long run variance estimator provide an improved rate of divergence of the tests under the alternative of a change in mean tests for structural breaks based on such an estimator are able to remain consistent while still retaining the same asymptotic distribution under the null hypothesis of constant mean

kernels with prescribed surface tension mobility for threshold dynamics schemes abstract we show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces including networks of them in both two and three dimensions moreover we discuss necessary and sufficient conditions for the positivity of the kernel which in the case of two phase flow ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow in particular we show in a barrier type statement that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions and give a complete characterization among other results is a threshold dynamics scheme that is guaranteed to dissipate a non local approximation to the interfacial energy in the fully anisotropic multiphase setting using the new kernel construction

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

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

regularization on graphs with function adapted diffusion processes harmonic analysis and diffusion on discrete data has been shown to lead to state of the art algorithms for machine learning tasks especially in the context of semi supervised and transductive learning the success of these algorithms rests on the assumption that the function s to be studied learned interpolated etc are smooth with respect to the geometry of the data in this paper we present a method for modifying the given geometry so the function s to be studied are smoother with respect to the modified geometry and thus more amenable to treatment using harmonic analysis methods among the many possible applications we consider the problems of image denoising and transductive classification in both settings our approach improves on standard diffusion based methods

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

classification by pairwise coupling we discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes and then coupling the estimates together the coupling model is similar to the bradley terry method for paired comparisons we study the nature of the class probability estimates that arise and examine the performance of the procedure in real and simulated data sets classifiers used include linear discriminants nearest neighbors adaptive nonlinear methods and the support vector machine

a wavelet laplace variational technique for image deconvolution and inpainting we construct a new variational method for blind deconvolution of images and inpainting motivated by recent pde based techniques involving the ginzburg landau functional but using more localized wavelet based methods we present results for both binary and grayscale images comparable speeds are achieved with better sharpness of edges in the reconstruction

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

consistent model specification tests for time series econometric models in this paper we consider general hypothesis testing problems for nonparametric and semiparametric time series econometric models we apply the general methodology to construct a consistent test for omitted variables and a consistent test for a partially linear model the proposed tests are shown to have asymptotic normal distributions under their respective null hypotheses we also discuss the problems of testing portfolio conditional mean variance efficiency and testing a semiparametric single index model monte carlo simulations are conducted to examine the finite sample performances of the nonparametric omitted variable test and the test for a partially linear specification

powerful tests for structural changes in volatility abstract detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence we propose modified cusum and lm tests that are built on a robust estimator of the long run variance of squared series we establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative the theory allows smooth and abrupt structural changes that can be small the smoothing parameter is automatically selected such that the proposed test has good finite sample size and meanwhile achieves decent power gain

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

a rayleigh chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse hermitian matrices a procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse hermitian matrices the procedure a modification of an inverse subspace iteration procedure uses adaptively determined chebyshev polynomials to approximate the required application of the inverse operator on the subspace the method is robust converges with acceptable rapidity and can easily handle operators with eigenvalues of multiplicity greater than one numerical results are shown that demonstrate the utility of the procedure

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

power watershed a unifying graph based optimization framework in this work we extend a common framework for graph based image segmentation that includes the graph cuts random walker and shortest path optimization algorithms viewing an image as a weighted graph these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework we then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm which we term the power watershed in particular when q 2 the power watershed leads to a multilabel scale and contrast invariant unique global optimum obtained in practice in quasi linear time placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation

the cusum test with ols residuals we show that the cusum test of the stability over time of the coefficients of a linear regression model which is usually based on recursive residuals can also be applied to ordinary least squares residuals we derive the limiting null distribution of the resulting test and compare its local power to that of the standard procedure it turns out that neither version is uniformly superior to the other henceforth bde on the cusum and cusum of squares tests for the constancy over time of the coefficients of a linear regression model both tests are based on recursive residuals which are independent n o o 2 under ho and therefore ideal ingredients for all types of tests while the cusum of squares test has been generalized to ordinary least squares ols residuals mccabe and harrison 1980 a generalization of the cusum test to ols residuals which are dependent and heteroskedastic even under ho has not yet been made except for a very special case macneill 1978 in fact both bde 1975 p 151 and mccabe and harrison 1980 p 142 argue that a least squares variant of the cusum test poses intractable problems due to an alleged difficulty in assessing the significance of the departure of the cumulated ols residuals from their mean value zero below we show that it is no more difficult to derive the limiting distribution for a cusum test based on ols residuals than for a cusum test based on recursive residuals while the cusums of the recursive residuals properly standardized tend in distribution to a standard wiener process sen 1982 krimer ploberger and alt 1988 henceforth kpa we show that the ols based cusums tend in distribution to a brownian bridge or tied down brownian motion see billingsley 1968 p 64 this generalizes macneill s 1978 polynomial regression result to arbitrary regressor sequences we also demonstrate that the resulting cusum test has higher local power for certain types of structural change than the one based on recursive residuals though neither variant is uniformly superior to the other

auction dynamics a volume constrained mbo scheme abstract we show how auction algorithms originally developed for the assignment problem can be utilized in merriman bence and osher s threshold dynamics scheme to simulate multi phase motion by mean curvature in the presence of equality and inequality volume constraints on the individual phases the resulting algorithms are highly efficient and robust and can be used in simulations ranging from minimal partition problems in euclidean space to semi supervised machine learning via clustering on graphs in the case of the latter application numerous experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state of the art methods while requiring a fraction of the computation time

a non local perspective on the power properties of the cusum and cusum of squares tests for structural change abstract we consider the power properties of the cusum and cusum of squares cusq tests in the presence of a one time change in the parameters of a linear regression model a result due to ploberger and kramer 1990 the local power of the cusum and cusum of squares tests econometric theory 6 335 347 is that the cusq test has only trivial asymptotic local power in this case while the cusum test has non trivial local asymptotic power unless the change is orthogonal to the mean regressor the main theme of the paper is that such conclusions obtained from a local asymptotic framework are not reliable guides to what happens in finite samples the approach we take is to derive expansions of the test statistics that retain terms related to the magnitude of the change under the alternative hypothesis this enables us to analyze what happens for non local to zero breaks our theoretical results are able to explain how the power function of the tests can be drastically different depending on whether one deals with a static regression with uncorrelated errors a static regression with correlated errors a dynamic regression with lagged dependent variables or whether a correction for non normality is applied in the case of the cusq we discuss in which cases the tests are subject to a non monotonic power function that goes to zero as the magnitude of the change increases and uncover some curious properties all theoretical results are verified to yield good guides to the finite sample power through simulation experiments we finally highlight the practical importance of our results

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

spectral matting we present spectral matting a new approach to natural image matting that automatically computes a basis set of fuzzy matting components from the smallest eigenvectors of a suitably defined laplacian matrix thus our approach extends spectral segmentation techniques whose goal is to extract hard segments to the extraction of soft matting components these components may then be used as building blocks to easily construct semantically meaningful foreground mattes either in an unsupervised fashion or based on a small amount of user input

sources of nonmonotonic power when testing for a shift in mean of a dynamic time series sources of nonmonotonic power are uncovered for a wide variety of tests for a shift in the mean of a dynamic time series two main sources of nonmonotonic power are found the first source is the behavior of the estimate of error variance under the alternative hypothesis of a shift in mean in particular if the error variance is estimated under the null hypothesis nonmonotonic power can result the second source is the presence of a lagged dependent variable in the estimated regression

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

a robust approach for object matching and classification using partial dominant orientation descriptor this paper introduces a novel approach to measure the correspondence between objects and exploit it for object and image classification tasks using the proposed partial dominant orientation descriptor pdod in particular the object is represented by a set of stable and informative key locations sampled using difference of gaussian the proposed pdod at each extracted key location takes into account the position and partially computes the dominant orientation of other key locations relative to it thus offering a global distinctive and discriminative characterization this allows us to learn features that are largely invariant to common image transformations including changes in object colors and textures the correspondence in between two objects is performed by finding for each key location in one object the key location in the other object that has the most similar descriptor object classification proceeds by assigning the most relevant category that has maximally similar stored prototype objects to the query object using k nearest neighbors algorithm with adaptive object distance for efficiency we further investigate pdod for image classification by developing powerful image representations based on the popular bag of words model the extensive experiments demonstrate that the proposed approach greatly improves the matching and classification results while achieving the state of the art performances on several challenging benchmark datasets the obtained results suggest also broader applicability to other classification modalities we propose a novel descriptor for complex object matching and classification the proposed descriptor combines both local and global properties of interest points the proposed descriptor is highly distinctive and insensitive to common image transformations adaptive object distance is proposed to measure the similarity between objects we compare our approach with many existing methods and we achieve the state of the art performances on several benchmarks

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

testing for structural change in conditional models in the past decade we have seen the development of a new set of tests for structural change of unknown timing in regression models most notably the supf statistic of andrews 1993 econometrica 61 825 856 the expf and avef statistics of andrewsploberger 1994 econometrica 62 1383 1414 and the statistic of nyblom 1989 journal of american statistical association 84 223 230 the distribution theory used for these tests is primarily asymptotic and has been derived under the maintained assumption that the regressors are stationary this excludes structural change in the marginal distribution of the regressors as a result these tests technically cannot discriminate between structural change in the conditional and marginal distributions this paper attempts to remedy this de ciency by deriving the large sample distributions of the test statistics allowing for structural change in the marginal distribution of the regressors we nd that the asymptotic distributions of the supf expf avef and statistics are not invariant to structural change in the regressors to solve the size problem we introduce a xed regressor bootstrap which achieves the rst order asymptotic distribution and appears to possess reasonable size properties in small samples our bootstrap theory allows for arbitrary structural change in the regressors including structural shifts polynomial trends and exogenous stochastic trends it allows for lagged dependent variables and heteroskedastic error processes 2000 elsevier science s a all rights reserved

multiclass data segmentation using diffuse interface methods on graphs we present two graph based algorithms for multiclass segmentation of high dimensional data on graphs the algorithms use a diffuse interface model based on the ginzburg landau functional related to total variation and graph cuts a multiclass extension is introduced using the gibbs simplex with the functional s double well potential modified to handle the multiclass case the first algorithm minimizes the functional using a convex splitting numerical scheme the second algorithm uses a graph adaptation of the classical numerical merriman bence osher mbo scheme which alternates between diffusion and thresholding we demonstrate the performance of both algorithms experimentally on synthetic data image labeling and several benchmark data sets such as mnist coil and webkb we also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph laplacian and take advantage of the sparsity of the matrix experiments indicate that the results are competitive with or better than the current state of the art in multiclass graph based segmentation algorithms for high dimensional data