Feature-based object matching is a fundamental problem for many applications in computer vision, such as object recognition, 3D reconstruction, tracking, and motion segmentation. In this work, we consider simultaneously matching object instances in a set of images, where both inlier and outlier features are extracted. The task is to identify the inlier features and establish their consistent correspondences across the image set. This is a challenging combinatorial problem, and the problem complexity grows exponentially with the image number. To this end, we propose a novel framework, termed ROML, to address this problem. ROML optimizes simultaneously a partial permutation matrix (PPM) for each image, and feature correspondences are established by the obtained PPMs. Two of our key contributions are summarized as follows. (1) We formulate the problem as rank and sparsity minimization for PPM optimization, and treat simultaneous optimization of multiple PPMs as a regularized consensus problem in the context of distributed optimization. (2) We use the ADMM method to solve the thus formulated ROML problem, in which a subproblem associated with a single PPM optimization appears to be a difficult integer quadratic program (IQP). We prove that under wildly applicable conditions, this IQP is equivalent to a linear sum assignment problem (LSAP), which can be efficiently solved to an exact solution. Extensive experiments on rigid non-rigid object matching, matching instances of a common object category, and common object localization show the efficacy of our proposed method.
Greedy approaches build a matching that has the lowest cost at each iteration, which has the obvious weakness that decisions once made, are fixed and may later be shown to be suboptimal. GRASP improves over greedy approaches by progressively constructing a list of best candidate matches and randomly selecting one from them. The process is repeated until all matches are exhausted. A final local search over the neighborhood of obtained matches may be used to further optimize the solution. In @cite_23 , Poore and Robertson presented a Lagrangian relaxation method that progressively relaxes the original and intermediate recovery MiAPs to linear assignment problems, by incorporating constraints of each MiAP into its objective function via the Lagrangian. However, this method involves implicitly enumerative procedure, and is difficult to implement and analyze.
@cite_23
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We propose a method for global multi-target tracking that can incorporate higher-order track smoothness constraints such as constant velocity. Our problem formulation readily lends itself to path estimation in a trellis graph, but unlike previous methods, each node in our network represents a candidate pair of matching observations between consecutive frames. Extra constraints on binary flow variables in the graph result in a problem that can no longer be solved by min-cost network flow. We therefore propose an iterative solution method that relaxes these extra constraints using Lagrangian relaxation, resulting in a series of problems that ARE solvable by min-cost flow, and that progressively improve towards a high-quality solution to our original optimization problem. We present experimental results showing that our method outperforms the standard network-flow formulation as well as other recent algorithms that attempt to incorporate higher-order smoothness constraints. When the Lagrangian relaxation based methods are applied to solve power system unit commitment, the identical solutions to the subproblems associated with identical units may cause the dual solution to be far away from the optimal solution and serious solution oscillations. As a result, the quality of the feasible solution obtained may be very unsatisfactory. This issue has been long recognized as an inherent disadvantage of Lagrangian relaxation based methods. In this paper, the homogeneous solution issue is identified and analyzed through a simple example. Based on this analysis, a successive subproblem solving method is developed. The new method combines the concepts of augmented Lagrangian relaxation and surrogate subgradient to produce a good search direction at the high level. The low level subproblems including those corresponding to the identical units are solved successively so that the commitments of the identical units may not be homogeneous in the dual solution. Compared with the standard Lagrangian relaxation method, the new method can obtain better dual solutions and avoid the solution oscillations. Numerical testing shows the new method is efficient and the quality of the feasible solution is greatly improved. Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications. A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions". Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetry-breaking self-amplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and self-amplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variables. Simulation results are shown on 100 node random graphs and for a wide range of connectivities. This paper addresses the problem of finding the minimum number of vehicles required to visit a set of nodes subject to time window and capacity constraints. The fleet is homogeneous and is located at a common depot. Each node requires the same type of service. An exact method is introduced based on branch and cut. In the computations, ever increasing lower bounds on the optimal solution are obtained by solving a series of relaxed problems that incorporate newly found valid inequalities. Feasible solutions or upper bounds are obtained with the help of greedy randomized adaptive search procedure (GRASP). A wide variety of cuts is introduced to tighten the linear programming (LP) relaxation of the original mixed-integer program. To find violated cuts, it is necessary to solve a separation problem. A substantial portion of the paper is aimed at describing the heuristics developed for this purpose. A new approach for obtaining feasible solutions from the LP relaxation is also discussed. Numerical results for standard 50- and 100-node benchmark problems are reported. We propose a robust approach for performing automatic species-level recognition of fossil pollen grains in microscopy images that exploits both global shape and local texture characteristics in a patch-based matching methodology. We introduce a novel criteria for selecting meaningful and discriminative exemplar patches. We optimize this function during training using a greedy submodular function optimization framework that gives a near-optimal solution with bounded approximation error. We use these selected exemplars as a dictionary basis and propose a spatially-aware sparse coding method to match testing images for identification while maintaining global shape correspondence. To accelerate the coding process for fast matching, we introduce a relaxed form that uses spatially-aware soft-thresholding during coding. Finally, we carry out an experimental study that demonstrates the effectiveness and efficiency of our exemplar selection and classification mechanisms, achieving @math accuracy on a difficult fine-grained species classification task distinguishing three types of fossil spruce pollen. The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. In this paper, we propose a new approach, parallel iterative improvement (PII), to solving the stable matching problem. This approach treats the stable matching problem as an optimization problem with all possible matchings forming its solution space. Since a stable matching always exists for any stable matching problem instance, finding a stable matching is equivalent to finding a matching with the minimum number (which is always zero) of unstable pairs. A particular PII algorithm is presented to show the effectiveness of this approach by constructing a new matching from an existing matching and using techniques such as randomization and greedy selection to speedup the convergence process. Simulation results show that the PII algorithm has better average performance compared with the classical stable matching algorithms and converges in n iterations with high probability. The proposed algorithm is also useful for some real-time applications with stringent time constraint. We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art. Recent advances in scene understanding and related tasks have highlighted the importance of using regions to reason about high-level scene structure. Typically, the regions are selected beforehand and then an energy function is defined over them. This two step process suffers from the following deficiencies: (i) the regions may not match the boundaries of the scene entities, thereby introducing errors; and (ii) as the regions are obtained without any knowledge of the energy function, they may not be suitable for the task at hand. We address these problems by designing an efficient approach for obtaining the best set of regions in terms of the energy function itself. Each iteration of our algorithm selects regions from a large dictionary by solving an accurate linear programming relaxation via dual decomposition. The dictionary of regions is constructed by merging and intersecting segments obtained from multiple bottom-up over-segmentations. To demonstrate the usefulness of our algorithm, we consider the task of scene segmentation and show significant improvements over state of the art methods. The classical Hospitals Residents problem (HR) models the assignment of junior doctors to hospitals based on their preferences over one another. In an instance of this problem, a stable matching M is sought which ensures that no blocking pair can exist in which a resident r and hospital h can improve relative to M by becoming assigned to each other. Such a situation is undesirable as it could naturally lead to r and h forming a private arrangement outside of the matching. The original HR model assumes that preference lists are strictly ordered. However in practice, this may be an unreasonable assumption: an agent may find two or more agents equally acceptable, giving rise to ties in its preference list. We thus obtain the Hospitals Residents problem with Ties (HRT). In such an instance, stable matchings may have different sizes and MAX HRT, the problem of finding a maximum cardinality stable matching, is NP-hard. In this paper we describe an Integer Programming (IP) model for MAX HRT. We also provide some details on the implementation of the model. Finally we present results obtained from an empirical evaluation of the IP model based on real-world and randomly generated problem instances. Graph matching has a wide spectrum of computer vision applications such as finding feature point correspondences across images. The problem of graph matching is generally NP-hard, so most existing work pursues suboptimal solutions between two graphs. This paper investigates a more general problem of matching N attributed graphs to each other, i.e. labeling their common node correspondences such that a certain compatibility affinity objective is optimized. This multi-graph matching problem involves two key ingredients affecting the overall accuracy: a) the pairwise affinity matching score between two local graphs, and b) global matching consistency that measures the uniqueness and consistency of the pairwise matching results by different sequential matching orders. Previous work typically either enforces the matching consistency constraints in the beginning of iterative optimization, which may propagate matching error both over iterations and across different graph pairs; or separates score optimizing and consistency synchronization in two steps. This paper is motivated by the observation that affinity score and consistency are mutually affected and shall be tackled jointly to capture their correlation behavior. As such, we propose a novel multi-graph matching algorithm to incorporate the two aspects by iteratively approximating the global-optimal affinity score, meanwhile gradually infusing the consistency as a regularizer, which improves the performance of the initial solutions obtained by existing pairwise graph matching solvers. The proposed algorithm with a theoretically proven convergence shows notable efficacy on both synthetic and public image datasets. We study the object localization problem in images given a single hand-drawn example or a gallery of shapes as the object model. Although many shape matching algorithms have been proposed for the problem over the decades, chamfer matching remains to be the preferred method when speed and robustness are considered. In this paper, we significantly improve the accuracy of chamfer matching while reducing the computational time from linear to sublinear (shown empirically). Specifically, we incorporate edge orientation information in the matching algorithm such that the resulting cost function is piecewise smooth and the cost variation is tightly bounded. Moreover, we present a sublinear time algorithm for exact computation of the directional chamfer matching score using techniques from 3D distance transforms and directional integral images. In addition, the smooth cost function allows to bound the cost distribution of large neighborhoods and skip the bad hypotheses within. Experiments show that the proposed approach improves the speed of the original chamfer matching upto an order of 45x, and it is much faster than many state of art techniques while the accuracy is comparable. Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the "high noise" regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees. In this paper, we address the approximate nearest neighbor (ANN) search problem over large scale visual descriptors. We investigate a simple but very effective approach, neighborhood graph search, which constructs a neighborhood graph to index the data points and conducts a local search, expanding neighborhoods with a best-first manner, for ANN search. Our empirical analysis shows that neighborhood expansion is very efficient, with O(1) cost, for a new NN candidate location, and has high chances to locate true NNs and hence it usually performs well. However, it often gets sub-optimal solutions since local search only checks the neighborhood of the current solution, or conducts exhaustive and continuous neighborhood expansions to find better solutions, which deteriorates the query efficiency. In this paper, we propose a query-driven iterated neighborhood graph search approach to improve the performance. We follow the iterated local search (ILS) strategy, widely-used in combinatorial optimization, to find a solution beyond a local optimum. We handle the key challenge in making neighborhood graph search adapt to ILS, Perturbation, which generates a new pivot to restart a local search. To this end, we present a criterion to check if the local search over a neighborhood graph arrives at the local solution. Moreover, we exploit the query and search history to design the perturbation scheme, resulting in a more effective search. The major benefit is avoiding unnecessary neighborhood expansions and hence more efficiently finding true NNs. Experimental results on large scale SIFT matching, similar image search, and shape retrieval with non-metric distance measures, show that our approach performs much better than previous state-of-the-art ANN search approaches. Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve performance ratio 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze and Suen proved that the Modified Randomized Greedy algorithm achieves performance ratio 0.5 + e (where e = 1 400000) on arbitrary graphs in the mid-nineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012. In this paper, we revisit the Ranking algorithm using the LP framework. Special care is given to analyze the structural properties of the Ranking algorithm in order to derive the LP constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our LP. We use continuous LP relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous LP. Our work achieves the currently best theoretical performance ratio of [EQUATION] on arbitrary graphs. Moreover, experiments suggest that Ranking cannot perform better than 0.724 in general. We consider several vehicle routing problems (VRP) with profits, which seek to select a subset of customers, each one being associated with a profit, and to design service itineraries. When the sum of profits is maximized under distance constraints, the problem is usually called the team orienteering problem. The capacitated profitable tour problem seeks to maximize profits minus travel costs under capacity constraints. Finally, in the VRP with a private fleet and common carrier, some customers can be delegated to an external carrier subject to a cost. Three families of combined decisions must be taken: customer’s selection, assignment to vehicles, and sequencing of deliveries for each route.We propose a new neighborhood search for these problems, which explores an exponential number of solutions in pseudo-polynomial time. The search is conducted with standard VRP neighborhoods on an exhaustive solution representation, visiting all customers. Since visiting all customers is usually infeasible or suboptimal, an efficient select algorithm, based on resource constrained shortest paths, is repeatedly used on any new route to find the optimal subsequence of visits to customers. The good performance of these neighborhood structures is demonstrated by extensive computational experiments with a local search, an iterated local search, and a hybrid genetic algorithm. Intriguingly, even a local-improvement method to the first local optimum of this neighborhood achieves an average gap of 0.09 on classic team orienteering benchmark instances, rivaling with the current state-of-the-art metaheuristics. Promising research avenues on hybridizations with more standard routing neighborhoods are also open. As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm. As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm. This paper considers a family of metrics to compare images based on their local descriptors. It encompasses the vector or locally aggregated descriptors descriptor and matching techniques such as hamming embedding. Making the bridge between these approaches leads us to propose a match kernel that takes the best of existing techniques by combining an aggregation procedure with a selective match kernel. The representation underpinning this kernel is approximated, providing a large scale image search both precise and scalable, as shown by our experiments on several benchmarks. We show that the same aggregation procedure, originally applied per image, can effectively operate on groups of similar features found across multiple images. This method implicitly performs feature set augmentation, while enjoying savings in memory requirements at the same time. Finally, the proposed method is shown effective for place recognition, outperforming state of the art methods on a large scale landmark recognition benchmark.