map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

linear programming for the 0 1 quadratic knapsack problem abstract in this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we propose a new method for computing an upper bound this method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0 1 variables but nonredundant in continuous variables the obtained upper bound is better than the bounds given by other known methods we also propose an algorithm for computing a good feasible solution this algorithm is an elaboration of the heuristic methods proposed by chaillou hansen and mahieu and by gallo hammer and simeone the relative error between this feasible solution and the optimum solution is generally less than 1 we show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum finally we propose a branch and bound algorithm for solving the quadratic knapsack problem and report extensive computational tests

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

optimal design of a distributed network with a two level hierarchical structure abstract this paper deals with the topological design of a network with a two level hierarchical structure where the embedded backbone network is full meshed and the local networks attached to it are of star type the whole problem unlike the conventional approaches partitioning it into two subproblems is directly handled in such a general setting that both a backbone network and local access networks are to be simultaneously determined we formulate this problem as a quadratic 0 1 programming model and also propose an equivalent linearized version of the model based on the observation that the linearized model is a variant of the well established uncapacitated facility location problem a dual based solution procedure is developed and shown to be efficient from computing experiments which were done on a wide variety of problems with up to 50 backbone nodes and 50 user nodes

inference methods for crfs with co occurrence statistics the markov and conditional random fields crfs used in computer vision typically model only local interactions between variables as this is generally thought to be the only case that is computationally tractable in this paper we consider a class of global potentials defined over all variables in the crf we show how they can be readily optimised using standard graph cut algorithms at little extra expense compared to a standard pairwise field this result can be directly used for the problem of class based image segmentation which has seen increasing recent interest within computer vision here the aim is to assign a label to each pixel of a given image from a set of possible object classes typically these methods use random fields to model local interactions between pixels or super pixels one of the cues that helps recognition is global object co occurrence statistics a measure of which classes such as chair or motorbike are likely to occur in the same image together there have been several approaches proposed to exploit this property but all of them suffer from different limitations and typically carry a high computational cost preventing their application on large images we find that the new model we propose produces a significant improvement in the labelling compared to just using a pairwise model and that this improvement increases as the number of labels increases

contour detection and hierarchical image segmentation this paper investigates two fundamental problems in computer vision contour detection and image segmentation we present state of the art algorithms for both of these tasks our contour detector combines multiple local cues into a globalization framework based on spectral clustering our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree in this manner we reduce the problem of image segmentation to that of contour detection extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms the automatically generated hierarchical segmentations can be interactively refined by user specified annotations computation at multiple image resolutions provides a means of coupling our system to recognition applications

thermostatistical persistency a powerful improving concept for simulated annealing algorithms abstract we propose a new heuristic method to solve 0 1 optimisation problems basically the method is an extension of the simulated annealing algorithm at a given temperature approximations of the variable value expectations are computed this information allows to fix the variables as the temperature decreases so that the size of the problem is progressively reduced it follows a significant improvement compared with pure simulated annealing we obtain better solutions in less cpu time variable value expectations can be estimated by simulation procedures in the special case of unconstrained 0 1 optimisation problems estimates can be obtained by solving an analytic system two problems are examined the travelling salesman problem and the minimisation of an unconstrained 0 1 quadratic function

neural networks error correcting codes and polynomials over the binary n cube several ways of relating the concept of error correcting codes to the concept of neural networks are presented performing maximum likelihood decoding in a linear block error correcting code is shown to be equivalent to finding a global maximum of the energy function of a certain neural network given a linear block code a neural network can be constructed in such a way that every codeword corresponds to a local maximum the connection between maximization of polynomials over the n cube and error correcting codes is also investigated the results suggest that decoding techniques can be a useful tool for solving such maximization problems the results are generalized to both nonbinary and nonlinear codes

concave extensions for nonlinear 0 1 maximization problems a well known linearization technique for nonlinear 0 1 maximization problems can be viewed as extending any polynomial in 0 1 variables to a concave function defined on 0 1 n some properties of this standard concave extension are investigated polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix the standard extension is proved to be identical to another type of concave extension defined as the lower envelope of a class of affine functions majorizing the given polynomial

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

completion energies and scale the detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping we examine the notion of completion energy of curve elements showing and exploiting its intrinsic dependence on length and width scales we introduce a fast method for computing the most likely completion between two elements by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy we then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields this is done through multiscale procedures i e separate processing at different scales with some interscale interactions such procedures allow the summation of all induction fields to be done in a total of only o n log n operations where n is the number of pixels in the image more important such procedures yield a more realistic dependence of the induction field on the length and width scales the field of a long element is very different from the sum of the fields of its composing short segments

using a mixed integer quadratic programming solver for the unconstrained quadratic 0 1 problem in this paper we consider problem p of minimizing a quadratic function q x x tqx ctx of binary variables our main idea is to use the recent mixed integer quadratic programming miqp solvers but for this we have to first convexify the objective function q x a classical trick is to raise up the diagonal entries of q by a vector u until q diag u is positive semidefinite then using the fact that xi2 xi we can obtain an equivalent convex objective function which can then be handled by an miqp solver hence computing a suitable vector u constitutes a preprocessing phase in this exact solution method we devise two different preprocessing methods the first one is straightforward and consists in computing the smallest eigenvalue of q in the second method vector u is obtained once a classical sdp relaxation of p is solved we carry out computational tests using the generator of pardalos and rodgers 1990 and we compare our two solution methods to several other exact solution methods furthermore we report computational results for the max cut problem

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

segmentation of multiple salient closed contours from real images using a saliency measure based on the global property of contour closure we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images the saliency measure incorporates the gestalt principles of proximity and good continuity that previous methods have also exploited unlike previous methods we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a markov process where edges from the image serve as states element i j of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i we show how the saliency measure defined for individual edges can be used to derive a saliency relation defined for pairs of edges and further show that strongly connected components of the graph representing the saliency relation correspond to smooth closed contours in the image finally we report for the first time results on large real images for which segmentation takes an average of about 10 seconds per object on a general purpose workstation

testing optimality for quadratic 0 1 problems abstract the issue tackled is testing whether a given solution of a quadratic 0 1 problem is optimal the paper presents an algorithm based on the necessary and sufficient optimality condition introduced by hirriart urruty for general convex problems a measure of the quality of the solution is provided computational results show the applicability of the method the method is extended to constrained quadratic 0 1 problems such as quadratic assignment and quadratic knapsack

graph cut based discrete valued image reconstruction efficient graph cut methods have been used with great success for labeling and denoising problems occurring in computer vision unfortunately the presence of linear image mappings has prevented the use of these techniques in most discrete amplitude image reconstruction problems in this paper we develop a graph cut based framework for the direct solution of discrete amplitude linear image reconstruction problems cast as regularized energy function minimizations we first analyze the structure of discrete linear inverse problem cost functions to show that the obstacle to the application of graph cut methods to their solution is the variable mixing caused by the presence of the linear sensing operator we then propose to use a surrogate energy functional that overcomes the challenges imposed by the sensing operator yet can be utilized efficiently in existing graph cut frameworks we use this surrogate energy functional to devise a monotonic iterative algorithm for the solution of discrete valued inverse problems we first provide experiments using local convolutional operators and show the robustness of the proposed technique to noise and stability to changes in regularization parameter then we focus on nonlocal tomographic examples where we consider limited angle data problems we compare our technique with state of the art discrete and continuous image reconstruction techniques experiments show that the proposed method outperforms state of the art techniques in challenging scenarios involving discrete valued unknowns

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

data dependent higher order clique selection for artery vein segmentation by energy minimization we propose a novel segmentation method based on energy minimization of higher order potentials we introduce higher order terms into the energy to incorporate prior knowledge on the shape of the segments the terms encourage certain sets of pixels to be entirely in one segment or the other the sets can for instance be smooth curves in order to help delineate pulmonary vessels which are known to run in almost straight lines the higher order terms can be converted to submodular first order terms by adding auxiliary variables which can then be globally minimized using graph cuts we also determine the weight of these terms or the degree of the aforementioned encouragement in a principled way by learning from training data with the ground truth we demonstrate the effectiveness of the method in a real world application in fully automatic pulmonary artery vein segmentation in ct images

multi target tracking by online learning a crf model of appearance and motion patterns we introduce an online learning approach for multi target tracking detection responses are gradually associated into tracklets in multiple levels to produce final tracks unlike most previous approaches which only focus on producing discriminative motion and appearance models for all targets we further consider discriminative features for distinguishing difficult pairs of targets the tracking problem is formulated using an online learned crf model and is transformed into an energy minimization problem the energy functions include a set of unary functions that are based on motion and appearance models for discriminating all targets as well as a set of pairwise functions that are based on models for differentiating corresponding pairs of tracklets the online crf approach is more powerful at distinguishing spatially close targets with similar appearances as well as in tracking targets in presence of camera motions an efficient algorithm is introduced for finding an association with low energy cost we present results on four public data sets and show significant improvements compared with several state of art methods

weighted local intensity fusion method for variational optical flow estimation estimating a dense motion field of successive video frames is a fundamental problem in image processing the multi scale variational optical flow method is a critical technique that addresses this issue despite the considerable progress over the past decades there are still some challenges such as dealing with large displacements and estimating the smoothness parameter we present a local intensity fusion lif method to tackle these difficulties by evaluating the local interpolation error in terms of l1 block match on the corresponding set of images we fuse flow proposals which are obtained from different methods and from different parameter settings integrally under a unified lif this approach has two benefits 1 the incorporated matching information is helpful to recover large displacements and 2 the obtained optimal fusion solution gives a tradeoff between the data term and the smoothness term in addition a selective gradient based weight is introduced to improve the performance of the lif finally we propose a corrected weighted median filter cwmf which applies the motion information to correct errors of the color distance weight to denoise the intermediate flow fields during optimization experiments demonstrate the effectiveness of our method highlightsa lif is proposed to handle both large and small motion and to estimate smoothness parameter a selective gradient is introduced to the lif to reduce errors that are caused by outliers a cwmf is designed to overcome the defect of traditional wmf

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

learning probabilistic models for contour completion in natural images using a large set of human segmented natural images we study the statistics of region boundaries we observe several power law distributions which likely arise from both multi scale structure within individual objects and from arbitrary viewing distance accordingly we develop a scale invariant representation of images from the bottom up using a piecewise linear approximation of contours and constrained delaunay triangulation to complete gaps we model curvilinear grouping on top of this graphical geometric structure using a conditional random field to capture the statistics of continuity and different junction types quantitative evaluations on several large datasets show that our contour grouping algorithm consistently dominates and significantly improves on local edge detection

filter based mean field inference for random fields with higher order terms and product label spaces recently a number of cross bilateral filtering methods have been proposed for solving multi label problems in computer vision such as stereo optical flow and object class segmentation that show an order of magnitude improvement in speed over previous methods these methods have achieved good results despite using models with only unary and or pairwise terms however previous work has shown the value of using models with higher order terms e g to represent label consistency over large regions or global co occurrence relations we show how these higher order terms can be formulated such that filter based inference remains possible we demonstrate our techniques on joint stereo and object labelling problems as well as object class segmentation showing in addition for joint object stereo labelling how our method provides an efficient approach to inference in product label spaces we show that we are able to speed up inference in these models around 10 30 times with respect to competing graph cut move making methods as well as maintaining or improving accuracy in all cases we show results on pascalvoc 10 for object class segmentation and leuven for joint object stereo labelling

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

robust estimation of adaptive tensors of curvature by tensor voting although curvature estimation from a given mesh or regularly sampled point set is a well studied problem it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise such input is ubiquitous in computer vision in this paper we propose a three pass tensor voting algorithm to robustly estimate curvature tensors from which accurate principal curvatures and directions can be calculated our quantitative estimation is an improvement over the previous two pass algorithm where only qualitative curvature estimation sign of gaussian curvature is performed to overcome misalignment errors our improved method automatically corrects input point locations at subvoxel precision which also rejects outliers that are uncorrectable to adapt to different scales locally we define the radiushit of a curvature tensor to quantify estimation accuracy and applicability our curvature estimation algorithm has been proven with detailed quantitative experiments performing better in a variety of standard error metrics percentage error in curvature magnitudes absolute angle difference in curvature direction in the presence of a large amount of misalignment noise

roof duality for polynomial 0 1 optimization the purpose of this note is to generalize the roof duality theory of hammer hansen and simeone to the case of polynomial 0 1 optimization 0 1pp by reformulating 0 1pp and expanding some of their definitions we show that most of the results for quadratic 0 1 problem 0 1qp can be extended to the general polynomial case

a comparison of measures for detecting natural shapes in cluttered backgrounds we propose a new measure of perceptual saliency and quantitatively compare its ability to detect natural shapes in cluttered backgrounds to five previously proposed measures as defined in the new measure the saliency of an edge is the fraction of closed random walks which contain that edge the transition probability matrix defining the random walk between edges is based on a distribution of natural shapes modeled by a stochastic motion each of the saliency measures in our comparison is a function of a set of affinity values assigned to pairs of edges although the authors of each measure define the affinity between a pair of edges somewhat differently all incorporate the gestalt principles of good continuation and proximity in some form in order to make the comparison meaningful we use a single definition of affinity and focus instead on the performance of the different functions for combining affinity values the primary performance criterion is accuracy we compute false positive rates in classifying edges as signal or noise for a large set of test figures in almost every case the new measure significantly outperforms previous measures

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

an efficient optimization framework for multi region segmentation based on lagrangian duality we introduce a multi region model for simultaneous segmentation of medical images in contrast to many other models geometric constraints such as inclusion and exclusion between the regions are enforced which makes it possible to correctly segment different regions even if the intensity distributions are identical we efficiently optimize the model using a combination of graph cuts and lagrangian duality which is faster and more memory efficient than current state of the art as the method is based on global optimization techniques the resulting segmentations are independent of initialization we apply our framework to the segmentation of the left and right ventricles myocardium and the left ventricular papillary muscles in magnetic resonance imaging and to lung segmentation in full body x ray computed tomography we evaluate our approach on a publicly available benchmark with competitive results

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

combinatorial optimization of the discretized multiphase mumford shah functional the mumford shah model has been one of the most influential models in image segmentation and denoising the optimization of the multiphase mumford shah energy functional has been performed using level sets methods that optimize the mumford shah energy by evolving the level sets via the gradient descent these methods are very slow and prone to getting stuck in local optima due to the use of gradient descent after the reformulation of the 2 phase mumford shah functional on a graph several groups investigated the hierarchical extension of the graph representation to multi class the discrete hierarchical approaches are more effective than hierarchical or direct multiphase formulation using level sets however they provide approximate solutions and can diverge away from the optimal solution in this paper we present a discrete alternating optimization for the discretized vese chan approximation of the piecewise constant multiphase mumford shah functional that directly minimizes the multiphase functional without recursive bisection on the labels our approach handles the nonsubmodularity of the multiphase energy function and provides a global optimum if the image estimation data term is known apriori

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

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

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

an extension of the ko nig egerva ry property to node weighted bidirected graphs given a bidirected graphg and a vectorb of positive integral node weights an integer linear program ip is defined on g b ip generalizes the node packing problem on a node weighted undirected graph in the sense that it reduces to the latter wheng is undirected a polynomial time algorithm is given that recognizes whether cp the linear program obtained by relaxing the integrality constraints of ip has an integral optimal solution also an efficient method for solving the linear programming dual of cp is described

persistency in quadratic 0 1 optimization this paper is concerned with persistency properties which allow the evaluation of some variables at all maximizing points of a quadratic pseudo boolean function hammer hansen and simeone 1984 have proposed to determine these variables using a procedure described by balinski for computing a strongly complementary pair of optimal primal and dual solutions for arbitrary linear programs we propose a linear time algorithm for determining these variables from a best roof off i e from a lowest upper linear bound off

higher order energies for image segmentation a novel energy minimization method for general higher order binary energy functions is proposed in this paper we first relax a discrete higher order function to a continuous one and use the taylor expansion to obtain an approximate lower order function which is optimized by the quadratic pseudo boolean optimization or other discrete optimizers the minimum solution of this lower order function is then used as a new local point where we expand the original higher order energy function again our algorithm does not restrict to any specific form of the higher order binary function or bring in extra auxiliary variables for concreteness we show an application of segmentation with the appearance entropy which is efficiently solved by our method experimental results demonstrate that our method outperforms the state of the art methods

a linear programming approach to max sum problem a review the max sum labeling problem defined as maximizing a sum of binary i e pairwise functions of discrete variables is a general np hard optimization problem with many applications such as computing the map configuration of a markov random field we review a not widely known approach to the problem developed by ukrainian researchers schlesinger et al in 1976 and show how it contributes to recent results most importantly those on the convex combination of trees and tree reweighted max product in particular we review schlesinger et al s upper bound on the max sum criterion its minimization by equivalent transformations its relation to the constraint satisfaction problem the fact that this minimization is dual to a linear programming relaxation of the original problem and the three kinds of consistency necessary for optimality of the upper bound we revisit problems with boolean variables and supermodular problems we describe two algorithms for decreasing the upper bound we present an example application for structural image analysis

simulated annealing for the unconstrained quadratic pseudo boolean function abstract in this paper we develop a simulated annealing sa based heuristic for the unconstrained quadratic pseudo boolean function an algorithm that solves the problem in o n 2 at each temperature of the cooling schedule is given the performance of sa based heuristic is compared with existing bounding algorithms for this problem computational results and comparisons on several hundred test problems demonstrate the efficiency of our heuristic in terms of solution quality and computational time a new set of hard test problems with their best solution is provided to facilitate future comparison

some thoughts on combinatorial optimisation abstract a group of young researchers from the esi x summer school hec jouy en josas 1994 give their personal views on the current status of and prospects for combinatorial optimisation several issues are considered and discussed with emphasis on a selected number of techniques heuristics and polyhedral approaches and problems knapsack quadratic 0 1 programming machine scheduling routing and network design

learning active basis model for object detection and recognition this article proposes an active basis model a shared sketch algorithm and a computational architecture of sum max maps for representing learning and recognizing deformable templates in our generative model a deformable template is in the form of an active basis which consists of a small number of gabor wavelet elements at selected locations and orientations these elements are allowed to slightly perturb their locations and orientations before they are linearly combined to generate the observed image the active basis model in particular the locations and the orientations of the basis elements can be learned from training images by the shared sketch algorithm the algorithm selects the elements of the active basis sequentially from a dictionary of gabor wavelets when an element is selected at each step the element is shared by all the training images and the element is perturbed to encode or sketch a nearby edge segment in each training image the recognition of the deformable template from an image can be accomplished by a computational architecture that alternates the sum maps and the max maps the computation of the max maps deforms the active basis to match the image data and the computation of the sum maps scores the template matching by the log likelihood of the deformed active basis

a polyhedral study of binary polynomial programs we study the polyhedral convex hull of a mixed integer set x1d4ae defined by a collection of multilinear equations over the unit hypercube such sets appear frequently in the factorable reformulation of mixed integer nonlinear optimization problems in particular the set x1d4ae represents the feasible region of a linearized unconstrained binary polynomial optimization problem we define an equivalent hypergraph representation of the mixed integer set x1d4ae which enables us to derive several families of facet defining inequalities structural properties and lifting operations for its convex hull in the space of the original variables our theoretical developments extend several well known results from the boolean quadric polytope and the cut polytope literature paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub expressions

constrained 0 1 quadratic programming basic approaches and extensions abstract we describe the simplest technique to tackle 0 1 quadratic programs with linear constraints among those that turn out to be successful in practice this method is due to and familiar to the quadratic assignment experts even if it took some time to realize that most approaches to the problem could be interpreted in these terms whereas it does not appear to be widely known outside this community since the technique is completely general and is by far the most successful one in several other cases such as quadratic knapsack we illustrate it in its full generality pointing out its relationship to lagrangian and linear programming relaxations and discussing further extensions we believe that this method should be in the background of every practitioner in combinatorial optimization

a variational model for object segmentation using boundary information and shape prior driven by the mumford shah functional in this paper we propose a new variational model to segment an object belonging to a given shape space using the active contour method a geometric shape prior and the mumford shah functional the core of our model is an energy functional composed by three complementary terms the first one is based on a shape model which constrains the active contour to get a shape of interest the second term detects object boundaries from image gradients and the third term drives globally the shape prior and the active contour towards a homogeneous intensity region the segmentation of the object of interest is given by the minimum of our energy functional this minimum is computed with the calculus of variations and the gradient descent method that provide a system of evolution equations solved with the well known level set method we also prove the existence of this minimum in the space of functions with bounded variation applications of the proposed model are presented on synthetic and medical images

minimization of a quadratic pseudo boolean function abstract we present a branch and bound algorithm for minimizing a quadratic pseudo boolean function f x at each node of the search tree the lower bound is computed in three phases and is equal to b 1 b 2 b 3 computation of b 1 is based upon roof duality b 2 uses the characterization of some positive quadratic posiforms associated with the directed cycles of the implication graph of aspvall plass and tarjan and b 3 is computed by searching in a posiform of degree 4 some subfunctions which cannot be equal to zero these subfunctions are found by using the notion of implication between literals computational results on several hundred test problems with up to 100 variables demonstrate the efficiency of this lower bound

addendum to large deviations asymptotics for spherical integrals abstract we correct an omission in guionnet and zeitouni j funct anal 188 2002 461 and improve the main result there to a full large deviations trajectorial result

efficient reduction of polynomial zero one optimization to the quadratic case we address the problem of optimizing a polynomial with real coefficients over binary variables we show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem the number of variables in the latter linearization is only slightly larger than in the former if polynomial constraints are present in the original problem then their linearized counterparts carry over to the linearized quadratic problem unchanged if the original problem formulation does not contain any constraints we obtain a reduction to unconstrained quadratic zero one optimization which is equivalent to the well studied max cut problem the separation problem for general unconstrained polynomial zero one optimization thus reduces to the separation problem for the cut polytope this allows us to transfer the entire knowledge gained for the latter polytope by intensive research and in particular the sophisticated separation techniques that have been developed we report preliminary experimental results obtained with a straightforward implementation of this approach

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

shape statistics in kernel space for variational image segmentation we present a variational integration of nonlinear shape statistics into a mumford shah based segmentation process the nonlinear statistics are derived from a set of training silhouettes by a novel method of density estimation which can be considered as an extension of kernel pca to a probabilistic framework we assume that the training data forms a gaussian distribution after a nonlinear mapping to a higher dimensional feature space due to the strong nonlinearity the corresponding density estimate in the original space is highly non gaussian applications of the nonlinear shape statistics in segmentation and tracking of 2d and 3d objects demonstrate that the segmentation process can incorporate knowledge on a large variety of complex real world shapes it makes the segmentation process robust against misleading information due to noise clutter and occlusion 2003 pattern recognition society published by elsevier science ltd all rights reserved

quadratic reformulations of nonlinear binary optimization problems very large nonlinear unconstrained binary optimization problems arise in a broad array of applications several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial however no similarly efficient methods are available for the higher degree case since high degree objectives are becoming increasingly important in certain application areas such as computer vision various techniques have been recently developed to reduce the general case to the quadratic one at the cost of increasing the number of variables by introducing additional auxiliary variables in this paper we initiate a systematic study of these quadratization approaches we provide tight lower and upper bounds on the number of auxiliary variables needed in the worst case for general objective functions for bounded degree functions and for a restricted class of quadratizations our upper bounds are constructive thus yielding new quadratization procedures finally we completely characterize all minimal quadratizations of negative monomials

transformation of general binary mrf minimization to the first order case we introduce a transformation of general higher order markov random field with binary labels into a first order one that has the same minima as the original moreover we formalize a framework for approximately minimizing higher order multilabel mrf energies that combines the new reduction with the fusion move and qpbo algorithms while many computer vision problems today are formulated as energy minimization problems they have mostly been limited to using first order energies which consist of unary and pairwise clique potentials with a few exceptions that consider triples this is because of the lack of efficient algorithms to optimize energies with higher order interactions our algorithm challenges this restriction that limits the representational power of the models so that higher order energies can be used to capture the rich statistics of natural scenes we also show that some minimization methods can be considered special cases of the present framework as well as comparing the new method experimentally with other such techniques

an analysis of convex relaxations for map estimation of discrete mrfs the problem of obtaining the maximum a posteriori estimate of a general discrete markov random field i e a markov random field defined using a discrete set of labels is known to be np hard however due to its central importance in many applications several approximation algorithms have been proposed in the literature in this paper we present an analysis of three such algorithms based on convex relaxations i lp s the linear programming lp relaxation proposed by schlesinger 1976 for a special case and independently in chekuri et al 2001 koster et al 1998 and wainwright et al 2005 for the general case ii qp rl the quadratic programming qp relaxation of ravikumar and lafferty 2006 and iii socp ms the second order cone programming socp relaxation first proposed by muramatsu and suzuki 2003 for two label problems and later extended by kumar et al 2006 for a general label set we show that the socp ms and the qp rl relaxations are equivalent furthermore we prove that despite the flexibility in the form of the constraints objective function offered by qp and socp the lp s relaxation strictly dominates i e provides a better approximation than qp rl and socp ms we generalize these results by defining a large class of socp and equivalent qp relaxations which is dominated by the lp s relaxation based on these results we propose some novel socp relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field using some examples we show that the new socp relaxations strictly dominate the previous approaches

persistency in 0 1 polynomial programming an optimal solution to the continuous relaxation of a mixed integer 0 1 linear programming problem is defined to be persistent if the set of 0 1 variables realizing binary values retains those same binary values in at least one integer optimum a mixed integer 0 1 linear program is said to possess the persistency property or equivalently to be persistent if every optimal solution to the continuous relaxation is a persistent solution we study the issue of persistency for a special family of mixed integer 0 1 linear programming problems which derive from the linearization strategy of sherali and adams 1990 in particular we first examine the hierarchy of linearizations which result from applying this strategy to the general class of unconstrained 0 1 polynomial programming problems and then consider a class of specially structured constrained problems our study of persistency differs from previous works in that we use the explicit algebraic representation of the convex hull of feasible binary solutions available at the highest level of this hierarchy as a tool for making inferences relative to optimal integer solutions from optimal continuous solutions each level of this hierarchy consists of a mixed integer 0 1 linear program and we provide for any given optimal solution to the continuous relaxation of any member of this hierarchy a set of sufficient conditions on the optimal dual multipliers for the solution to be persistent these sufficient conditions turn out to be satisfied at every optimal solution of the quadratic level and thus allow for the reformulation of any unconstrained 0 1 quadratic programming problem as a persistent mixed integer linear program in a suitably defined higher dimensional space we subsequently address the fa mily of constrained 0 1 polynomial programs and use a projection operation to prove that a special family of 0 1 linear programs possesses the persistency property included in this family is a generalization of the vertex packing problem that permits quadratic objective function coefficients unlike earlier published works on the vertex packing and related problems our proofs do not rely on balinski s 1965 extreme point characterization

character expansion method for the first order asymptotics of a matrix integral the estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics geometry and free probability questions on a rigorous ground only integrals of one matrix or of several matrices with simple quadratic interaction called ab interaction could be evaluated so far see e g 19 17 or 9 in this article we follow an idea widely developed in the physics literature which is based on character expansion to study more complex interaction in this context we derive a large deviation principle for the empirical measure of young tableaux we then use it to study a matrix model defined in the spirit of the dually weighted graph model introduced in 13 but with a cutoff function such that the matrix integral and its character expansion converge we prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit

a tight linearization and an algorithm for zero one quadratic programming problems this paper is concerned with the solution of linearly constrained zero one quadratic programming problems problems of this kind arise in numerous economic location decision and strategic planning situations including capital budgeting facility location quadratic assignment media selection and dynamic set covering a new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods an implicit enumeration algorithm which uses lagrangian relaxation benders cutting planes and local explorations is designed to exploit the strength of this linearization computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm

salient closed boundary extraction with ratio contour we present ratio contour a novel graph based method for extracting salient closed boundaries from noisy images this method operates on a set of boundary fragments that are produced by edge detection boundary extraction identifies a subset of these fragments and connects them sequentially to form a closed boundary with the largest saliency we encode the gestalt laws of proximity and continuity in a novel boundary saliency measure based on the relative gap length and average curvature when connecting fragments to form a closed boundary this new measure attempts to remove a possible bias toward short boundaries we present a polynomial time algorithm for finding the most salient closed boundary we also present supplementary preprocessing steps that facilitate the application of ratio contour to real images we compare ratio contour to two closely related methods for extracting closed boundaries elder and zucker s method based on the shortest path algorithm and williams and thornber s method based on spectral analysis and a strongly connected components algorithm this comparison involves both theoretic analysis and experimental evaluation on both synthesized data and real images

dense subgraph partition of positive hypergraphs in this paper we present a novel partition framework called dense subgraph partition dsp to automatically precisely and efficiently decompose a positive hypergraph into dense subgraphs a positive hypergraph is a graph or hypergraph whose edges except self loops have positive weights we first define the concepts of core subgraph conditional core subgraph and disjoint partition of a conditional core subgraph then define dsp based on them the result of dsp is an ordered list of dense subgraphs with decreasing densities which uncovers all underlying clusters as well as outliers a divide and conquer algorithm called min partition evolution is proposed to efficiently compute the partition dsp has many appealing properties first it is a nonparametric partition and it reveals all meaningful clusters in a bottom up way second it has an exact and efficient solution called min partition evolution algorithm the min partition evolution algorithm is a divide and conquer algorithm thus time efficient and memory friendly and suitable for parallel processing third it is a unified partition framework for a broad range of graphs and hypergraphs we also establish its relationship with the densest k subgraph problem d k s an np hard but fundamental problem in graph theory and prove that dsp gives precise solutions to d k s for all k in a graph dependent set called critical k set to our best knowledge this is a strong result which has not been reported before moreover as our experimental results show for sparse graphs especially web graphs the size of critical k set is close to the number of vertices in the graph we test the proposed partition framework on various tasks and the experimental results clearly illustrate its advantages

contour completion without region segmentation contour completion plays an important role in visual perception where the goal is to group fragmented low level edge elements into perceptually coherent and salient contours most existing methods for contour completion have focused on pixelwise detection accuracy in contrast fewer methods have addressed the global contour closure effect despite psychological evidences for its importance this paper proposes a purely contour based higher order crf model to achieve contour closure through local connectedness approximation this leads to a simplified problem structure where our higher order inference problem can be transformed into an integer linear program and be solved efficiently compared with the methods based on the same bottom up edge detector our method achieves a superior contour grouping ability measured by rand index a comparable precision recall performance and more visually pleasing results our results suggest that contour closure can be effectively achieved in contour domain in contrast to a popular view that segmentation is essential for this purpose

global stereo reconstruction under second order smoothness priors second order priors on the smoothness of 3d surfaces are a better model of typical scenes than first order priors however stereo reconstruction using global inference algorithms such as graph cuts has not been able to incorporate second order priors because the triple cliques needed to express them yield intractable nonsubmodular optimization problems this paper shows that inference with triple cliques can be effectively performed our optimization strategy is a development of recent extensions to alpha expansion based on the ldquo qpbordquo algorithm the strategy is to repeatedly merge proposal depth maps using a novel extension of qpbo proposal depth maps can come from any source for example frontoparallel planes as in alpha expansion or indeed any existing stereo algorithm with arbitrary parameter settings

an interior point algorithm for minimum sum of squares clustering an exact algorithm is proposed for minimum sum of squares nonhierarchical clustering i e for partitioning a given set of points from a euclidean m space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong this problem is expressed as a constrained hyperbolic program in 0 1 variables the resolution method combines an interior point algorithm i e a weighted analytic center column generation method with branch and bound the auxiliary problem of determining the entering column i e the oracle is an unconstrained hyperbolic program in 0 1 variables with a quadratic numerator and linear denominator it is solved through a sequence of unconstrained quadratic programs in 0 1 variables to accelerate resolution variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region proved minimum sum of squares partitions are determined for the first time for several fairly large data sets from the literature including fisher s 150 iris

partial optimality by pruning for map inference with general graphical models we consider the energy minimization problem for undirected graphical models also known as map inference problem for markov random fields which is np hard in general we propose a novel polynomial time algorithm to obtain a part of its optimal non relaxed integral solution our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the map inference problem and iteratively prunes those which do not satisfy our criterion for partial optimality we show that our pruning strategy is in a certain sense theoretically optimal also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables the method is very general as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem our method s runtime is determined by the runtime of the convex relaxation solver for the map inference problem