## Optimizing binary MRFs via extended roof duality (2007)

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Venue: | In Proc. CVPR |

Citations: | 100 - 10 self |

### BibTeX

@TECHREPORT{Rother07optimizingbinary,

author = {Carsten Rother and Vladimir Kolmogorov and Victor Lempitsky and Martin Szummer},

title = {Optimizing binary MRFs via extended roof duality},

institution = {In Proc. CVPR},

year = {2007}

}

### Years of Citing Articles

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### Abstract

Many computer vision applications rely on the efficient optimization of challenging, so-called non-submodular, binary pairwise MRFs. A promising graph cut based approach for optimizing such MRFs known as “roof duality” was recently introduced into computer vision. We study two methods which extend this approach. First, we discuss an efficient implementation of the “probing ” technique introduced recently by Boros et al. [5]. It simplifies the MRF while preserving the global optimum. Our code is 400-700 faster on some graphs than the implementation of [5]. Second, we present a new technique which takes an arbitrary input labeling and tries to improve its energy. We give theoretical characterizations of local minima of this procedure. We applied both techniques to many applications, including image segmentation, new view synthesis, superresolution, diagram recognition, parameter learning, texture restoration, and image deconvolution. For several applications we see that we are able to find the global minimum very efficiently, and considerably outperform the original roof duality approach. In comparison to existing techniques, such as graph cut, TRW, BP, ICM, and simulated annealing, we nearly always find a lower energy. 1.

### Citations

1569 |
Network Flows, Theory Algorithms and Applications
- Ahuja, Magnanti, et al.
(Show Context)
Citation Context ...zing binary MRFs introduced in [15]. The idea is to solve a particular linear programming (LP) relaxation of the energy where integer constraints xp ∈ {0, 1} are replaced with linear constraints xp ∈ =-=[0, 1]-=-. It can be shown that this LP has a half-integer optimal solution ¯x, i.e. ¯xp ∈ {0, 1, 1 2 } for every node p. It is convenient to define the corresponding partial labeling x of the integer problem ... |

1509 | Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
- 2001
(Show Context)
Citation Context ...+I is the labeling x = f(y). P+I In some scenarios an input labeling x for energy E is available, and it is desirable that the method does not increase it. (An example is the expansion move algorithm =-=[8]-=-; the input labeling (0, . . .,0) corresponds to the current configuration.) This can be achieved by “tracking” the input solution during QPBOP; details are given in [20]. 4. Experiments In this secti... |

880 | An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision
- Boykov, Kolmogorov
(Show Context)
Citation Context ... the algorithm can work with integer capacities, whereas maintaining the symmetry condition requires floating point numbers (even if the original costs are integers). We used the maxflow algorithm in =-=[7]-=-, and reused flow and search trees as described in [13]. We modified the code so that it maintains a list of visited nodes; thus, set U can be traversed without going through the entire graph. Using t... |

716 | Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ...amples in table 1). Image segmentation An important issue for interactive image segmentation is the combination of boundary constraints, as in intelligent scissors [17], and region constraints, as in =-=[6]-=-. Here we show that this is possible by including a few non-submodular terms, see fig. 6. We have tested our system for many images, where two examples are listed in table 1. The conclusion is that QP... |

481 | Learning low-level vision
- Freeman, Pasztor, et al.
(Show Context)
Citation Context ... new methods (P+BP+I and BP+I) attain the lowest energy, and QPBOP confirms that this is indeed the global minimum. Super-resolution and new view synthesis For superresolution we used the approach of =-=[11]-=- where a node label corresponds to a patch from a reference patch dictionary. The MRF pairwise terms encode the compatibility of overlapping patches of neighboring nodes. The amountsApplications Sim. ... |

326 | Convergent tree-reweighted message passing for energy minimization
- Kolmogorov
(Show Context)
Citation Context ...des. BP+I First we run QPBO, then max-product BP (only for unlabeled nodes) and finally improve the solution using QPBOI with random permutations of nodes. We used a “sequential” schedule of BP as in =-=[14]-=-. Before starting BP, we reparameterized the energy so that θpq;00 = θpq;11, and θpq;01 = θpq;10 for each edge (p, q). (Note that it does not make sense to start with the reparameterization obtained a... |

272 | A comprehensive study of energy minimization methods for markov random fields with smoothness-based priors
- Szeliski, Zabih, et al.
- 2008
(Show Context)
Citation Context ...urations in an MRF) has proven to be extremely successful for many vision applications such as stereo, image segmentation, image denoising, superresolution, new view synthesis and others. We refer to =-=[24]-=- for an overview of MRF optimization techniques in vision. Binary MRFs In this paper we focus on a special class of MRFs. Namely, we consider the problem of minimizing an 2 University College London v... |

243 | Intelligent scissors for image composition
- Mortensen, Barrett
- 1995
(Show Context)
Citation Context ...th very little extra runtime (see examples in table 1). Image segmentation An important issue for interactive image segmentation is the combination of boundary constraints, as in intelligent scissors =-=[17]-=-, and region constraints, as in [6]. Here we show that this is possible by including a few non-submodular terms, see fig. 6. We have tested our system for many images, where two examples are listed in... |

94 | Minimizing Nonsubmodular Functions with Graph Cuts—A Review
- Kolmogorov, Rother
- 2007
(Show Context)
Citation Context ...problem of minimizing non-submodular functions, which is a very challenging task (in general, NP-hard). A promising approach for this problem called roof duality was proposed in [12] (see a review in =-=[15]-=-). It produces part of an optimal solution. Boros et al. [3] give an efficient algorithm for computing a roof dual. It can be viewed as a generalization of the standard graph cut algorithm used in vis... |

79 |
Image-based rendering using image-based priors
- Fitzgibbon, Wexler, et al.
- 2005
(Show Context)
Citation Context ...o make it suitable for our purpose we use two labels and a 5 × 5 patch size which correspond to an 8-connected MRF (no overlap in the 3 × 3 center as in [11]). For New View Synthesis as introduced in =-=[10]-=- we may use the same MRF structure, where labels are now color modes derived from depth images (details omitted). We have tested several examples and parameter settings for both applications and may c... |

78 | Domino treewidth
- Bodlaender, Engelfriet
- 1997
(Show Context)
Citation Context ...d graph of size 100 × 100 for which QPBO failed to label 73% of nodes (runtime was 0.08 sec). We found that the unlabeled nodes induce a clique of size 45 (treewidth 44), based on the min-fill method =-=[4]-=-, with other triangulations being even larger. Thus the exact junction tree algorithm is completely infeasible here, requiring Terabytes of RAM just to store the assuming that the output of BP y satis... |

72 |
Roof duality, complementation and persistency in quadratic 0-1 optimization
- Hammer, Hansen, et al.
- 1984
(Show Context)
Citation Context ...fied. We focus on the problem of minimizing non-submodular functions, which is a very challenging task (in general, NP-hard). A promising approach for this problem called roof duality was proposed in =-=[12]-=- (see a review in [15]). It produces part of an optimal solution. Boros et al. [3] give an efficient algorithm for computing a roof dual. It can be viewed as a generalization of the standard graph cut... |

57 | Digital tapestry
- Rother, Kumar, et al.
- 2005
(Show Context)
Citation Context ...) is run with a random traversal order until convergence. Details of Belief Propagation (BP) are described in Sec. 3.3. Since graph cut (GC) cannot handle non-submodular terms we truncated them as in =-=[21]-=-. Simulated annealing (SA) is capable of producing high quality results with potentially long runtimes. We tweaked the parameters of SA for each individual problem, to achieve best results. Finally, T... |

54 | Efficiently solving dynamic markov random fields using graph cuts
- Kohli, Torr
- 2005
(Show Context)
Citation Context ...as maintaining the symmetry condition requires floating point numbers (even if the original costs are integers). We used the maxflow algorithm in [7], and reused flow and search trees as described in =-=[13]-=-. We modified the code so that it maintains a list of visited nodes; thus, set U can be traversed without going through the entire graph. Using the relaxed symmetry condition makes the update operatio... |

53 | On the optimality of tree-reweighted max-product message passing
- Kolmogorov, Wainwright
- 2005
(Show Context)
Citation Context ...high quality results with potentially long runtimes. We tweaked the parameters of SA for each individual problem, to achieve best results. Finally, TRW-S is guaranteed to give the same answer as QPBO =-=[16]-=-, therefore we omit it. (We verified experimentally that the result for labeled nodes is identical. Furthermore, running TRW-S until convergence of the lower bound is much slower than QPBO in practice... |

41 |
A linear time algorithm for testing the truth of certain quantified Boolean formulas
- Aspvall, Plass, et al.
- 1979
(Show Context)
Citation Context ...ollows: (i) If π(p) > π(¯p) then xp = 0. (ii) If π(p) < π(¯p) then xp = 1. (iii) If π(p) = π(¯p) then xp = ∅. If the symmetry property is satisfied, then this procedure is equivalent to the method in =-=[2]-=-. Let us show that the algorithm works correctly if the relaxed symmetry condition holds. Lemma 8. Define c ′ and f as in proposition 6. (a) If ca>0 for arc a=(u→v)∈A st then π(u) ≤ π(v). (b) If fa �=... |

37 | Heuristic algorithms for the unconstrained binary quadratic programming problem
- Beasley
- 1998
(Show Context)
Citation Context ...ques for obtaining a lower bound. A large number of heuristic ideas have also been applied to this problem, e.g. tabu search, scatter search, simulated annealing, evolutionary algorithms. We refer to =-=[1, 4]-=- and references therein for an overview of different exact and approximate methods. 2. Optimizing Binary MRFs: Roof duality In this section we give an overview of the roof duality approach for optimiz... |

19 | Preprocessing of unconstrained quadratic binary optimization
- Boros, Hammer, et al.
- 2006
(Show Context)
Citation Context ...recently introduced into computer vision. We study two methods which extend this approach. First, we discuss an efficient implementation of the “probing” technique introduced recently by Boros et al. =-=[5]-=-. It simplifies the MRF while preserving the global optimum. Our code is 400-700 faster on some graphs than the implementation of [5]. Second, we present a new technique which takes an arbitrary input... |

18 |
Network flows and minimization of quadratic pseudo-Boolean functions
- Boros, Hammer, et al.
- 1991
(Show Context)
Citation Context ...ry challenging task (in general, NP-hard). A promising approach for this problem called roof duality was proposed in [12] (see a review in [15]). It produces part of an optimal solution. Boros et al. =-=[3]-=- give an efficient algorithm for computing a roof dual. It can be viewed as a generalization of the standard graph cut algorithm used in vision: for submodular functions the two methods give the same ... |

17 | Learning statistical priors for efficient combinatorial optimization via graph cuts
- Cremers, Grady
- 2006
(Show Context)
Citation Context ...application we restore a noisy test image of a texture, based on an MRF model learned from a training image of the same texture type. We used the same learning procedure as described in [15] based on =-=[9]-=- with the only difference that QPBO is replaced by P+BP+I. We have done this forsone Brodatz texture D103 (see [15]) where the test error (averaged over 20 examples) reduces from 25.4 to 25.1 when usi... |

13 | A graph cut algorithm for generalized image deconvolution
- Raj, Zabih
- 2005
(Show Context)
Citation Context ...stead of QPBO. One example is listed in table 1 where P+BP+I achieved the global minimum. For this application BP+I achieved nearly the same result with a speed-up factor of 6. Image deconvolution In =-=[19]-=- image deconvolution was formulated as a labeling problem with a pairwise MRF and solved using graph cut based alpha-expansion. Given an n×n convolution kernel the MRF connectivity is (2n−1)× (2n −1) ... |

12 |
et al, “A comparative study of energy minimization methods for Markov random fields with smoothnessbased priors
- Szeliski
- 2008
(Show Context)
Citation Context ...urations in an MRF) has proven to be extremely successful for many vision applications such as stereo, image segmentation, image denoising, superresolution, new view synthesis and others. We refer to =-=[22]-=- for an overview of MRF optimization techniques in vision. Binary MRFs In this paper we focus on a special class of MRFs. Namely, we consider the problem of minimizing an energy function of the form E... |

12 | Contextual recognition of hand-drawn diagrams with conditional random fields
- Szummer, Yuan
(Show Context)
Citation Context ...ms is an application where the QPBOP method considerably outperforms standard QPBO. We tested 2700 diagram problems, with an average of 64 nodes and connectivity of 4.1. The MRF model is described in =-=[23]-=-. QPBOP returned the global minimum for all problems, whereas QPBOP (Global Min.) QPBO (37.1% unlabeled) Figure 4. Diagram recognition. Given a raw unlabeled handdrawing, the task is to classify wheth... |

9 | MRF’s for MRI’s: Bayesian reconstruction of MR images via graph cuts
- Raj, Singh, et al.
(Show Context)
Citation Context ...stands for quadratic pseudoboolean optimization - this is what the minimization problem (1) is called in [12, 3]. Recently it was successfully applied to vision applications such as MR reconstruction =-=[18]-=- and texture restoration [15]. Our contributions In some cases the roof duality approach does not work very well, i.e. it leaves many nodes unlabeled. We investigate two extensions of the roof duality... |

2 |
Local search heuristics for unconstrained quadratic binary optimization
- Boros, Hammer, et al.
- 2005
(Show Context)
Citation Context ...ques for obtaining a lower bound. A large number of heuristic ideas have also been applied to this problem, e.g. tabu search, scatter search, simulated annealing, evolutionary algorithms. We refer to =-=[1, 4]-=- and references therein for an overview of different exact and approximate methods. 2. Optimizing Binary MRFs: Roof duality In this section we give an overview of the roof duality approach for optimiz... |