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172
A comparative study of energy minimization methods for Markov random fields
 IN ECCV
, 2006
"... One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Ran ..."
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Cited by 415 (36 self)
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One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Random Fields (MRF’s), the resulting energy minimization problems were widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the topperforming stereo methods. Unfortunately, most papers define their own energy function, which is minimized with a specific algorithm of their choice. As a result, the tradeoffs among different energy minimization algorithms are not well understood. In this paper we describe a set of energy minimization benchmarks, which we use to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods—graph cuts, LBP, and treereweighted message passing—as well as the wellknown older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching and interactive segmentation. We also provide a generalpurpose software interface that allows vision researchers to easily switch between optimization methods with minimal overhead. We expect that the availability of our benchmarks and interface will make it significantly easier for vision researchers to adopt the best method for their specific problems. Benchmarks, code, results and images are available at
Global Stereo Reconstruction under Second Order Smoothness Priors
"... Secondorder priors on the smoothness of 3D surfaces are a better model of typical scenes than firstorder priors. However, stereo reconstruction using global inference algorithms, such as graphcuts, has not been able to incorporate secondorder priors because the triple cliques needed to express t ..."
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Cited by 127 (8 self)
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Secondorder priors on the smoothness of 3D surfaces are a better model of typical scenes than firstorder priors. However, stereo reconstruction using global inference algorithms, such as graphcuts, has not been able to incorporate secondorder 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 optimized. Our optimization strategy is a development of recent extensions to αexpansion, based on the “QPBO ” algorithm [5, 14, 26]. 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 αexpansion, or indeed any existing stereo algorithm, with arbitrary parameter settings. Experimental results demonstrate the usefulness of the secondorder prior and the efficacy of our optimization framework. An implementation of our stereo framework is available online [34].
Feature Correspondence via Graph Matching: Models and Global Optimization
"... Abstract. In this paper we present a new approach for establishing correspondences between sparse image features related by an unknown nonrigid mapping and corrupted by clutter and occlusion, such as points extracted from a pair of images containing a human figure in distinct poses. We formulate th ..."
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Cited by 120 (1 self)
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Abstract. In this paper we present a new approach for establishing correspondences between sparse image features related by an unknown nonrigid mapping and corrupted by clutter and occlusion, such as points extracted from a pair of images containing a human figure in distinct poses. We formulate this matching task as an energy minimization problem by defining a complex objective function of the appearance and the spatial arrangement of the features. Optimization of this energy is an instance of graph matching, which is in general a NPhard problem. We describe a novel graph matching optimization technique, which we refer to as dual decomposition (DD), and demonstrate on a variety of examples that this method outperforms existing graph matching algorithms. In the majority of our examples DD is able to find the global minimum within a minute. The ability to globally optimize the objective allows us to accurately learn the parameters of our matching model from training examples. We show on several matching tasks that our learned model yields results superior to those of stateoftheart methods. 1
MRF energy minimization and beyond via dual decomposition
 IN: IEEE PAMI. (2011
"... This paper introduces a new rigorous theoretical framework to address discrete MRFbased optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first ..."
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Cited by 105 (9 self)
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This paper introduces a new rigorous theoretical framework to address discrete MRFbased optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first decomposing it into a set of appropriately chosen subproblems and then combining their solutions in a principled way. In order to determine the limits of this method, we analyze the conditions that these subproblems have to satisfy and we demonstrate the extreme generality and flexibility of such an approach. We thus show that, by appropriately choosing what subproblems to use, one can design novel and very powerful MRF optimization algorithms. For instance, in this manner we are able to derive algorithms that: 1) generalize and extend stateoftheart messagepassing methods, 2) optimize very tight LPrelaxations to MRF optimization, 3) and take full advantage of the special structure that may exist in particular MRFs, allowing the use of efficient inference techniques such as, e.g, graphcut based methods. Theoretical analysis on the bounds related with the different algorithms derived from our framework and experimental results/comparisons using synthetic and real data for a variety of tasks in computer vision demonstrate the extreme potentials of our approach.
Learning to Discover Social Circles in Ego Networks
"... Our personal social networks are big and cluttered, and currently there is no good way to organize them. Social networking sites allow users to manually categorize their friends into social circles (e.g. ‘circles ’ on Google+, and ‘lists ’ on Facebook and Twitter), however they are laborious to cons ..."
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Cited by 95 (5 self)
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Our personal social networks are big and cluttered, and currently there is no good way to organize them. Social networking sites allow users to manually categorize their friends into social circles (e.g. ‘circles ’ on Google+, and ‘lists ’ on Facebook and Twitter), however they are laborious to construct and must be updated whenever a user’s network grows. We define a novel machine learning task of identifying users ’ social circles. We pose the problem as a node clustering problem on a user’s egonetwork, a network of connections between her friends. We develop a model for detecting circles that combines network structure as well as user profile information. For each circle we learn its members and the circlespecific user profile similarity metric. Modeling node membership to multiple circles allows us to detect overlapping as well as hierarchically nested circles. Experiments show that our model accurately identifies circles on a diverse set of data from Facebook, Google+, and Twitter for all of which we obtain handlabeled groundtruth. 1
Minimizing Sparse Higher Order Energy Functions of Discrete Variables
"... Higher order energy functions have the ability to encode high level structural dependencies between pixels, which have been shown to be extremely powerful for image labeling problems. Their use, however, is severely hampered in practice by the intractable complexity of representing and minimizing su ..."
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Cited by 74 (13 self)
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Higher order energy functions have the ability to encode high level structural dependencies between pixels, which have been shown to be extremely powerful for image labeling problems. Their use, however, is severely hampered in practice by the intractable complexity of representing and minimizing such functions. We observed that higher order functions encountered in computer vision are very often “sparse”, i.e. many labelings of a higher order clique are equally unlikely and hence have the same high cost. In this paper, we address the problem of minimizing such sparse higher order energy functions. Our method works by transforming the problem into an equivalent quadratic function minimization problem. The resulting quadratic function can be minimized using popular message passing or graph cut based algorithms for MAP inference. Although this is primarily a theoretical paper, it also shows how higher order functions can be used to obtain impressive results for the binary texture restoration problem.
Motion Detail Preserving Optical Flow Estimation
"... We discuss the cause of a severe optical flow estimation problem that fine motion structures cannot always be correctly reconstructed in the commonly employed multiscale variational framework. Our major finding is that significant and abrupt displacement transition wrecks smallscale motion structur ..."
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Cited by 72 (1 self)
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We discuss the cause of a severe optical flow estimation problem that fine motion structures cannot always be correctly reconstructed in the commonly employed multiscale variational framework. Our major finding is that significant and abrupt displacement transition wrecks smallscale motion structures in the coarsetofine refinement. A novel optical flow estimation method is proposed in this paper to address this issue, which reduces the reliance of the flow estimates on their initial values propagated from the coarser level and enables recovering many motion details in each scale. The contribution of this paper also includes adaption of the objective function and development of a new optimization procedure. The effectiveness of our method is borne out by experiments for both large and smalldisplacement optical flow estimation.
A convex relaxation approach for computing minimal partitions
 In Proc. of CVPR
, 2009
"... In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of conve ..."
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Cited by 69 (16 self)
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In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of convex constraints on the dual variables. For minimization we propose an efficient primal dual projected gradient algorithm which also allows a fast implementation on parallel hardware. Although our approach does not guarantee to find global minimizers of the Potts model we can give a tight bound on the energy between the computed solution and the true minimizer. Furthermore we show that our relaxation approach dominates recently proposed relaxations. As a consequence, our approach allows to compute solutions closer to the true minimizer. For many practical problems we even find the global minimizer. We demonstrate the excellent performance of our approach on several multilabel image segmentation and stereo problems. 1.
Fusion Moves for Markov Random Field Optimization
"... The efficient application of graph cuts to Markov Random Fields (MRFs) with multiple discrete or continuous labels remains an open question. In this paper, we demonstrate one possible way of achieving this by using graph cuts to combine pairs of suboptimal labelings or solutions. We call this combi ..."
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Cited by 68 (5 self)
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The efficient application of graph cuts to Markov Random Fields (MRFs) with multiple discrete or continuous labels remains an open question. In this paper, we demonstrate one possible way of achieving this by using graph cuts to combine pairs of suboptimal labelings or solutions. We call this combination process the fusion move. By employing recently developed graph cut based algorithms (socalled QPBOgraph cut), the fusion move can efficiently combine two proposal labelings in a theoretically sound way, which is in practice often globally optimal. We demonstrate that fusion moves generalize many previous graph cut approaches, which allows them to be used as building block within a broader variety of optimization schemes than were considered before. In particular, we propose new optimization schemes for computer vision MRFs with applications to image restoration, stereo, and optical flow, among others. Within these schemes the fusion moves are used 1) for the parallelization of MRF optimization into several threads; 2) for fast MRF optimization by combining cheaptocompute solutions; and 3) for the optimization of highly nonconvex continuouslabeled MRFs with 2D labels. Our final example is a nonvision MRF concerned with cartographic label placement, where fusion moves can be used to improve the performance of a standard inference method (loopy belief propagation).