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20
A Comparative Study of Modern Inference Techniques for Discrete Energy Minimization Problem
"... Seven years ago, Szeliski et al. published an influential study on energy minimization methods for Markov random fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the ph ..."
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Seven years ago, Szeliski et al. published an influential study on energy minimization methods for Markov random fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenominal success of random field models means that the kinds of inference problems we solve have changed significantly. Specifically, the models today often include higher order interactions, flexible connectivity structures, large labelspaces of different cardinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of 24 stateofart techniques on a corpus of 2,300 energy minimization instances from 20 diverse computer vision applications. To ensure reproducibility, we evaluate all methods in the OpenGM2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types.
Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey
, 2013
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Higherorder segmentation via multicuts
 CORR ABS/1305.6387
"... Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, ..."
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Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, that cover a broad range of segmentation problems relevant to image analysis and computer vision. We show how to take into account such higherorder terms systematically in view of computational inference, and present results of a comprehensive and competitive numerical evaluation of a variety of dedicated cuttingplane algorithms. Our results reveal ways to evaluate a significant subset of models globally optimal, without compromising runtime. Polynomially solvable relaxations are studied as well, along with advanced rounding schemes for postprocessing.
Generalized Roof Duality for MultiLabel Optimization: Optimal
"... Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we ..."
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Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the firstorder case.
Maximum Persistency in Energy Minimization
"... We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variable ..."
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We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to
Partial optimality by pruning for MAPinference with general graphical models
 In CVPR
, 2014
"... We consider the energy minimization problem for undirected graphical models, also known as MAPinference problem for Markov random fields which is NPhard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal nonrelaxed integral solution. Our algorithm is initiali ..."
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We consider the energy minimization problem for undirected graphical models, also known as MAPinference problem for Markov random fields which is NPhard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal nonrelaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAPinference 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 MAPinference problem. 1.
A PrimalDual Algorithm for HigherOrder Multilabel Markov Random Fields
"... Graph cuts method such as αexpansion [4] and fusion moves [22] have been successful at solving many optimization problems in computer vision. Higherorder Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, especially for multilabel MRF’s ..."
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Graph cuts method such as αexpansion [4] and fusion moves [22] have been successful at solving many optimization problems in computer vision. Higherorder Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, especially for multilabel MRF’s (i.e. more than 2 labels). In this paper we propose a new primaldual energy minimization method for arbitrary higherorder multilabel MRF’s. Primaldual methods provide guaranteed approximation bounds, and can exploit information in the dual variables to improve their efficiency. Our algorithm generalizes the PD3 [19] technique for firstorder MRFs, and relies on a variant of maxflow that can exactly optimize certain higherorder binary MRF’s [14]. We provide approximation bounds similar to PD3 [19], and the method is fast in practice. It can optimize nonsubmodular MRF’s, and additionally can incorporate problemspecific knowledge in the form of fusion proposals. We compare experimentally against the existing approaches that can efficiently handle these difficult energy functions [6, 10, 11]. For higherorder denoising and stereo MRF’s, we produce lower energy while running significantly faster. 1. Higherorder MRFs There is widespread interest in higherorder MRF’s for problems like denoising [23]and stereo [30], yet the resulting energy functions have proven to be very difficult to minimize. The optimization problem for a higherorder MRF is defined over a hypergraph with vertices V and cliques C plus a label set L. We minimize the cost of the labeling f: LV  → < defined by f(x) =
Structured learning of sumofsubmodular higher order energy functions
, 1309
"... Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sumofsubmodular (SoS) functions [2, 17], which can be efficient ..."
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Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sumofsubmodular (SoS) functions [2, 17], which can be efficiently minimized via a variant of max flow called submodular flow [6]. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach [14, 33] and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cuttingplane algorithm. We also show how the stateoftheart max flow method for vision problems [10] can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique [27], which uses handtuned parameters instead of machine learning. On a standard dataset [11] our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels. 1.
MAPInference on Large Scale HigherOrder Discrete Graphical Models by Fusion Moves
"... Many computer vision problems can be cast into optimization problems over discrete graphical models also known as Markov or conditional random fields. Standard methods are able to solve those problems quite efficiently. However, problems with huge label spaces and or higherorder structure remain c ..."
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Many computer vision problems can be cast into optimization problems over discrete graphical models also known as Markov or conditional random fields. Standard methods are able to solve those problems quite efficiently. However, problems with huge label spaces and or higherorder structure remain challenging or intractable even for approximate methods. We reconsider the work of Lempitsky et al. 2010 on fusion moves and apply it to general discrete graphical models. We propose two alternatives for calculating fusion moves that outperform the standard in several applications. Our generic software framework allows us to easily use different proposal generators which spans a large class of inference algorithms and thus makes exhaustive evaluation feasible. Because these fusion algorithms can be applied to models with huge label spaces and higherorder terms, they might stimulate and support research of such models which may have not been possible so far due to the lack of adequate inference methods. 1
MBestDiverse Labelings for Submodular Energies and Beyond
"... Abstract We consider the problem of finding M best diverse solutions of energy minimization problems for graphical models. Contrary to the sequential method of Batra et al., which greedily finds one solution after another, we infer all M solutions jointly. It was shown recently that such jointly in ..."
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Abstract We consider the problem of finding M best diverse solutions of energy minimization problems for graphical models. Contrary to the sequential method of Batra et al., which greedily finds one solution after another, we infer all M solutions jointly. It was shown recently that such jointly inferred labelings not only have smaller total energy but also qualitatively outperform the sequentially obtained ones. The only obstacle for using this new technique is the complexity of the corresponding inference problem, since it is considerably slower algorithm than the method of Batra et al. In this work we show that the joint inference of M best diverse solutions can be formulated as a submodular energy minimization if the original MAPinference problem is submodular, hence fast inference techniques can be used. In addition to the theoretical results we provide practical algorithms that outperform the current stateoftheart and can be used in both submodular and nonsubmodular case.