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59
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 128 (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].
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 70 (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 67 (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).
Measuring uncertainty in graph cut solutions  efficiently computing minmarginal energies using dynamic graph cuts
 In ECCV
, 2006
"... Abstract. In recent years the use of graphcuts has become quite popular in computer vision. However, researchers have repeatedly asked the question whether it might be possible to compute a measure of uncertainty associated with the graphcut solutions. In this paper we answer this particular questi ..."
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Cited by 66 (10 self)
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Abstract. In recent years the use of graphcuts has become quite popular in computer vision. However, researchers have repeatedly asked the question whether it might be possible to compute a measure of uncertainty associated with the graphcut solutions. In this paper we answer this particular question by showing how the minmarginals associated with the label assignments in a MRF can be efficiently computed using a new algorithm based on dynamic graph cuts. We start by reporting the discovery of a novel relationship between the minmarginal energy corresponding to a latent variable label assignment, and the flow potentials of the node representing that variable in the graph used in the energy minimization procedure. We then proceed to show how the minmarginal energy can be computed by minimizing a projection of the energy function defined by the MRF. We propose a fast and novel algorithm based on dynamic graph cuts to efficiently minimize these energy projections. The minmarginal energies obtained by our proposed algorithm are exact, as opposed to the ones obtained from other inference algorithms like loopy belief propagation and generalized belief propagation. We conclude by showing how minmarginals can be used to compute a confidence measure for label assignments in labelling problems such as image segmentation. 1
A convex formulation of continuous multilabel problems
 In ECCV, pages III: 792–805
, 2008
"... Abstract. We propose a spatially continuous formulation of Ishikawa’s discrete multilabel problem. We show that the resulting nonconvex variational problem can be reformulated as a convex variational problem via embedding in a higher dimensional space. This variational problem can be interpreted a ..."
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Cited by 65 (13 self)
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Abstract. We propose a spatially continuous formulation of Ishikawa’s discrete multilabel problem. We show that the resulting nonconvex variational problem can be reformulated as a convex variational problem via embedding in a higher dimensional space. This variational problem can be interpreted as a minimal surface problem in an anisotropic Riemannian space. In several stereo experiments we show that the proposed continuous formulation is superior to its discrete counterpart in terms of computing time, memory efficiency and metrication errors. 1
Exact Inference in Multilabel CRFs with Higher Order Cliques
, 2008
"... This paper addresses the problem of exactly inferring the maximum a posteriori solutions of discrete multilabel MRFs or CRFs with higher order cliques. We present a framework to transform special classes of multilabel higher order functions to submodular second order boolean functions (referred to ..."
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Cited by 49 (11 self)
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This paper addresses the problem of exactly inferring the maximum a posteriori solutions of discrete multilabel MRFs or CRFs with higher order cliques. We present a framework to transform special classes of multilabel higher order functions to submodular second order boolean functions (referred to as F 2 s), which can be minimized exactly using graph cuts and we characterize those classes. The basic idea is to use two or more boolean variables to encode the states of a single multilabel variable. There are many ways in which this can be done and much interesting research lies in finding ways which are optimal or minimal in some sense. We study the space of possible encodings and find the ones that can transform the most general class of functions to F 2 s. Our main contributions are twofold. First, we extend the subclass of submodular energy functions that can be minimized exactly using graph cuts. Second, we show how higher order potentials can be used to improve single view 3D reconstruction results. We believe that our work on exact minimization of higher order energy functions will lead to similar improvements in solutions of other labelling problems. 1.
Graph Cut Based Optimization for MRFs with Truncated Convex Priors
 In CVPR
, 2007
"... Optimization with graph cuts became very popular in recent years. Progress in problems such as stereo correspondence, image segmentation, etc., can be attributed, in part, to the development of efficient graph cut based optimization. Recent evaluation of optimization techniques shows that the popula ..."
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Cited by 44 (6 self)
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Optimization with graph cuts became very popular in recent years. Progress in problems such as stereo correspondence, image segmentation, etc., can be attributed, in part, to the development of efficient graph cut based optimization. Recent evaluation of optimization techniques shows that the popular expansion and swap graph cut algorithms perform extremely well for energies where the underlying MRF has the Potts prior, which corresponds to the assumption that the true labeling is piecewise constant. For more general priors, however, such as corresponding to piecewise smoothness assumption, both swap and expansion algorithms do not perform as well. We develop several optimization algorithms for truncated convex priors, which imply piecewise smoothness assumption. Both expansion and swap algorithms are based on moves that give each pixel a choice of only two labels. Our insight is that to obtain a good approximation under piecewise smoothness assumption, a pixel should have a choice among more than two labels. We develop new “range ” moves which act on a larger set of labels than the expansion and swap algorithms. We evaluate our method on problems of image restoration, inpainting, and stereo correspondence. Our results show that we are able to get more accurate answers, both in terms of the energy, which is the direct goal, and in terms of accuracy, which is an indirect, but more important goal. 1.
Reduce, reuse & recycle: Efficiently solving multilabel MRFs
 In CVPR
, 2008
"... In this paper, we present novel techniques that improve the computational and memory efficiency of algorithms for solving multilabel energy functions arising from discrete MRFs orCRFs. These methods are motivated by the observations that the performance of minimization algorithms depends on: (a) th ..."
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Cited by 36 (2 self)
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In this paper, we present novel techniques that improve the computational and memory efficiency of algorithms for solving multilabel energy functions arising from discrete MRFs orCRFs. These methods are motivated by the observations that the performance of minimization algorithms depends on: (a) the initialization used for the primal and dual variables; and (b) the number of primal variables involved in the energy function. Our first method (dynamic αexpansion) works by ‘recycling ’ results from previous problem instances. The second method simplifies the energy function by ‘reducing ’ the number of unknown variables, and can also be used to generate a good initialization for the dynamic αexpansion algorithm by ‘reusing ’ dual variables. We test the performance of our methods on energy functions encountered in the problems of stereo matching, and colour and object based segmentation. Experimental results show that our methods achieve a substantial improvement in the performance of αexpansion, as well as other popular algorithms such as sequential treereweighted message passing, and maxproduct belief propagation. In most cases we achieve a 1015 times speedup in the computation time. Our modified αexpansion algorithm provides similar performance to FastPD [15]. However, it is much simpler and can be made orders of magnitude faster by using the initialization schemes proposed in the paper. † 1.