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49
Beyond pairwise energies: Efficient optimization for higherorder MRFs
 in IEEE Conference on Computer Vision and Pattern Recognition : CVPR
, 2009
"... In this paper, we introduce a higherorder MRF optimization framework. On the one hand, it is very general; we thus use it to derive a generic optimizer that can be applied to almost any higherorder MRF and that provably optimizes a dual relaxation related to the input MRF problem. On the other han ..."
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Cited by 55 (8 self)
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In this paper, we introduce a higherorder MRF optimization framework. On the one hand, it is very general; we thus use it to derive a generic optimizer that can be applied to almost any higherorder MRF and that provably optimizes a dual relaxation related to the input MRF problem. On the other hand, it is also extremely flexible and thus can be easily adapted to yield far more powerful algorithms when dealing with subclasses of highorder MRFs. We thus introduce a new powerful class of highorder potentials, which are shown to offer enough expressive power and to be useful for many vision tasks. To address them, we derive, based on the same framework, a novel and extremely efficient messagepassing algorithm, which goes beyond the aforementioned generic optimizer and is able to deliver almost optimal solutions of very high quality. Experimental results on vision problems demonstrate the extreme effectiveness of our approach. For instance, we show that in some cases we are even able to compute the global optimum for NPhard higherorder MRFs in a very efficient manner. 1.
Fast approximate energy minimization with label costs
, 2010
"... The αexpansion algorithm [7] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higherorder terms. Our main contribution is to extend αexpansion so that it can simult ..."
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Cited by 49 (6 self)
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The αexpansion algorithm [7] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higherorder terms. Our main contribution is to extend αexpansion so that it can simultaneously optimize “label costs ” as well. An energy with label costs can penalize a solution based on the set of labels that appear in it. The simplest special case is to penalize the number of labels in the solution. Our energy is quite general, and we prove optimality bounds for our algorithm. A natural application of label costs is multimodel fitting, and we demonstrate several such applications in vision: homography detection, motion segmentation, and unsupervised image segmentation. Our C++/MATLAB implementation is publicly available.
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 49 (9 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.
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 39 (3 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.
A global perspective on map inference for lowlevel vision
 In Microsoft Research Technical Report
, 2009
"... In recent years the Markov Random Field (MRF) has become the de facto probabilistic model for lowlevel vision applications. However, in a maximum a posteriori (MAP) framework, MRFs inherently encourage delta function marginal statistics. By contrast, many lowlevel vision problems have heavy tailed ..."
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Cited by 24 (3 self)
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In recent years the Markov Random Field (MRF) has become the de facto probabilistic model for lowlevel vision applications. However, in a maximum a posteriori (MAP) framework, MRFs inherently encourage delta function marginal statistics. By contrast, many lowlevel vision problems have heavy tailed marginal statistics, making the MRF model unsuitable. In this paper we introduce a more general Marginal Probability Field (MPF), of which the MRF is a special, linear case, and show that convex energy MPFs can be used to encourage arbitrary marginal statistics. We introduce a flexible, extensible framework for effectively optimizing the resulting NPhard MAP problem, based around dualdecomposition and a modified mincost flow algorithm, and which achieves global optimality in some instances. We use a range of applications, including image denoising and texture synthesis, to demonstrate the benefits of this class of MPF over MRFs. 1.
Joint optimization of segmentation and appearance models
, 2009
"... Many interactive image segmentation approaches use an objective function which includes appearance models as an unknown variable. Since the resulting optimization problem is NPhard the segmentation and appearance are typically optimized separately, in an EMstyle fashion. One contribution of this p ..."
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Cited by 20 (3 self)
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Many interactive image segmentation approaches use an objective function which includes appearance models as an unknown variable. Since the resulting optimization problem is NPhard the segmentation and appearance are typically optimized separately, in an EMstyle fashion. One contribution of this paper is to express the objective function purely in terms of the unknown segmentation, using higherorder cliques. This formulation reveals an interesting bias of the model towards balanced segmentations. Furthermore, it enables us to develop a new dual decomposition optimization procedure, which provides additionally a lower bound. Hence, we are able to improve on existing optimizers, and verify that for a considerable number of real world examples we even achieve global optimality. This is important since we are able, for the first time, to analyze the deficiencies of the model. Another contribution is to establish a property of a particular dual decomposition approach which involves convex functions depending on foreground area. As a consequence, we show that the optimal decomposition for our problem can be computed efficiently via a parametric maxflow algorithm. 1.
Energy Minimization for Linear Envelope MRFs
"... Markov random fields with higher order potentials have emerged as a powerful model for several problems in computer vision. In order to facilitate their use, we propose a new representation for higher order potentials as upper and lower envelopes of linear functions. Our representation concisely mod ..."
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Cited by 16 (5 self)
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Markov random fields with higher order potentials have emerged as a powerful model for several problems in computer vision. In order to facilitate their use, we propose a new representation for higher order potentials as upper and lower envelopes of linear functions. Our representation concisely models several commonly used higher order potentials, thereby providing a unified framework for minimizing the corresponding Gibbs energy functions. We exploit this framework by converting lower envelope potentials to standard pairwise functions with the addition of a small number of auxiliary variables. This allows us to minimize energy functions with lower envelope potentials using conventional algorithms such as BP, TRW and αexpansion. Furthermore, we show how the minimization of energy functions with upper envelope potentials leads to a difficult minmax problem. We address this difficulty by proposing a new message passing algorithm that solves a linear programming relaxation of the problem. Although this is primarily a theoretical paper, we demonstrate the efficacy of our approach on the binary (fg/bg) segmentation problem. 1.
Approximate Inference in Graphical Models using LP Relaxations
, 2010
"... Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the vari ..."
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Cited by 11 (1 self)
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Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the variables. We present new approaches to approximate inference based on linear programming (LP) relaxations. Our algorithms optimize over the cycle relaxation of the marginal polytope, which we show to be closely related to the first lifting of the SheraliAdams hierarchy, and is significantly tighter than the pairwise LP relaxation. We show how to efficiently optimize over the cycle relaxation using a cuttingplane algorithm that iteratively introduces constraints into the relaxation. We provide a criterion to determine which constraints would be most helpful in tightening the relaxation, and give efficient algorithms for solving the search problem of finding the best cycle constraint to add according to this criterion.
Beyond Trees: MRF Inference via OuterPlanar Decomposition
, 2010
"... Maximum a posteriori (MAP) inference in Markov Random Fields (MRFs) is an NPhard problem, and thus research has focussed on either finding efficiently solvable subclasses (e.g. trees), or approximate algorithms (e.g. Loopy Belief Propagation (BP) and Treereweighted (TRW) methods). This paper prese ..."
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Cited by 10 (1 self)
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Maximum a posteriori (MAP) inference in Markov Random Fields (MRFs) is an NPhard problem, and thus research has focussed on either finding efficiently solvable subclasses (e.g. trees), or approximate algorithms (e.g. Loopy Belief Propagation (BP) and Treereweighted (TRW) methods). This paper presents a unifying perspective of these approximate techniques called “Decomposition Methods”. These are methods that decompose the given problem over a graph into tractable subproblems over subgraphs and then employ message passing over these subgraphs to merge the solutions of the subproblems into a global solution. This provides a new way of thinking about BP and TRW as successive steps in a hierarchy of decomposition methods. Using this framework, we take a principled first step towards extending this hierarchy beyond trees. We leverage a new class of graphs amenable to exact inference, called outerplanar graphs, and propose an approximate inference algorithm called OuterPlanar Decomposition (OPD). OPD is a strict generalization of BP and TRW, and contains both of them as special cases. Our experiments show that this extension beyond trees is indeed very powerful – OPD outperforms current stateofart inference methods on hard nonsubmodular synthetic problems and is competitive on real computer vision applications.