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## Higher-Order Clique Reduction Without Auxiliary Variables

Citations: | 1 - 0 self |

### Citations

2120 | R.: Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
(Show Context)
Citation Context ...proximately minimized by “move-making” algorithms which iteratively update the solution by devising a labeling problem with binary labels and solving it with graph cuts. Popular -expansion algorithm =-=[4]-=- and more general fusion-move algorithms [19] fall into this category. To extend these methods to the higher-order case, reduction of higher-order potentials with binary labels into firstorder cases h... |

259 | Robust higher order potentials for enforcing label consistency. In: CVPR
- Kohli, Ladicky, et al.
- 2008
(Show Context)
Citation Context ...gies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized =-=[15, 16]-=-, minimization of general higher-order energy is needed to utilize sophisticated priors, especially if it is learned from data as in the case of Fields of Experts (FoEs) [23]. One approach that can mi... |

145 | Minimizing nonsubmodular functions with graph cuts – A review.
- Kolmogorov, Rother
- 2007
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Citation Context ...e average over about 180 fusionmove iterations while denoising 10 test images. cuts can globally minimize submodular energy with binary labels [13]. Even non-submodular energy can be partially solved =-=[3, 2, 14]-=-. Energies with more labels can be approximately minimized by “move-making” algorithms which iteratively update the solution by devising a labeling problem with binary labels and solving it with graph... |

126 | Global stereo reconstruction under second order smoothness priors
- Woodford, Torr, et al.
- 2009
(Show Context)
Citation Context ...have seen an increasing emphasis on higher-order Markov Random Fields in vision. Although their importance was long understood and there were earlier attempts on utilizing them [5, 18, 21], their use =-=[9, 27]-=- and research into minimization of various classes of higherorder MRF energies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While there are useful higher... |

81 | Efficient Belief Propagation with learned higher-order Markov random fields
- Lan, Roth, et al.
- 2006
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Citation Context ...roduction Recent years have seen an increasing emphasis on higher-order Markov Random Fields in vision. Although their importance was long understood and there were earlier attempts on utilizing them =-=[5, 18, 21]-=-, their use [9, 27] and research into minimization of various classes of higherorder MRF energies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While ther... |

80 | Beyond pairwise energies: Efficient optimization of higher-order MRFs
- Komodakis, Paragios
- 2009
(Show Context)
Citation Context ...ugh their importance was long understood and there were earlier attempts on utilizing them [5, 18, 21], their use [9, 27] and research into minimization of various classes of higherorder MRF energies =-=[1, 6, 8, 11, 12, 17, 20, 24, 25, 26]-=- have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized [15, 16], minimization of general higher-o... |

74 | Minimizing sparse higher order energy functions of discrete variables
- Rother, Kohli, et al.
- 2009
(Show Context)
Citation Context ...ugh their importance was long understood and there were earlier attempts on utilizing them [5, 18, 21], their use [9, 27] and research into minimization of various classes of higherorder MRF energies =-=[1, 6, 8, 11, 12, 17, 20, 24, 25, 26]-=- have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized [15, 16], minimization of general higher-o... |

61 | Energy minimization via graph cuts: Settling what is possible
- Freedman, Drineas
- 2005
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Citation Context ... [6] proved a novel identity that turns the coefficient of the highest-order term to negative, allowing the use of the preferable reduction above that only applies to terms with negative coefficients =-=[7, 13]-=-. They also proved a more general formula for reducing multiple higher-order terms at once, with which the coefficients of any number of higher-order terms can be turned negative with only one new var... |

53 |
Higher-order clique reduction in binary graph cut.
- Ishikawa
- 2009
(Show Context)
Citation Context ... the first-order energy after the reduction. It can be seen that the proposed ELC reduction produces an energy with much less variables. From left to right: ELC reduction followed by the reduction in =-=[10, 11]-=-, ELC reduction followed by the reduction in Fix et al.[6], approximate ELC reduction, the reduction in [10, 11], and the reduction in [6]. The numbers are the average over about 180 fusionmove iterat... |

52 | Logcut - efficient graph cut optimization for markov random fields
- Lempitsky, Rother, et al.
- 2007
(Show Context)
Citation Context ...ithms which iteratively update the solution by devising a labeling problem with binary labels and solving it with graph cuts. Popular -expansion algorithm [4] and more general fusion-move algorithms =-=[19]-=- fall into this category. To extend these methods to the higher-order case, reduction of higher-order potentials with binary labels into firstorder cases have been proposed [6, 7, 8, 11, 13]. The idea... |

50 | Exact inference in multi-label crfs with higher order cliques.
- Ramalingam, Kohli, et al.
- 2008
(Show Context)
Citation Context ...ugh their importance was long understood and there were earlier attempts on utilizing them [5, 18, 21], their use [9, 27] and research into minimization of various classes of higherorder MRF energies =-=[1, 6, 8, 11, 12, 17, 20, 24, 25, 26]-=- have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized [15, 16], minimization of general higher-o... |

39 | Efficient belief propagation for vision using linear constraint nodes
- Potetz
- 2007
(Show Context)
Citation Context ...roduction Recent years have seen an increasing emphasis on higher-order Markov Random Fields in vision. Although their importance was long understood and there were earlier attempts on utilizing them =-=[5, 18, 21]-=-, their use [9, 27] and research into minimization of various classes of higherorder MRF energies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While ther... |

30 | Preprocessing of unconstrained quadratic binary optimization”, RUTCOR Research Report
- Boros, Hammer, et al.
- 2006
(Show Context)
Citation Context ...e average over about 180 fusionmove iterations while denoising 10 test images. cuts can globally minimize submodular energy with binary labels [13]. Even non-submodular energy can be partially solved =-=[3, 2, 14]-=-. Energies with more labels can be approximately minimized by “move-making” algorithms which iteratively update the solution by devising a labeling problem with binary labels and solving it with graph... |

29 | Transformation of general binary MRF minimization to the first order case
- Ishikawa
(Show Context)
Citation Context ...erence between the proposed approximation algorithm and Fix et al. [6]. Figure 3 shows the denoising examples. Each row shows, from left to right, the noise-added image, the denoised image using HOCR =-=[10, 11]-=-, the denoised image using Fix et al. [6], and the denoised image using the approximation algorithm proposed in 3.5. Each denoising was done only for 10 seconds on a machine with Intel Core i7-4770K C... |

27 |
p3 and beyond: Move making algorithms for solving higher order functions
- Kohli, Kumar, et al.
- 2009
(Show Context)
Citation Context ...gies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized =-=[15, 16]-=-, minimization of general higher-order energy is needed to utilize sophisticated priors, especially if it is learned from data as in the case of Fields of Experts (FoEs) [23]. One approach that can mi... |

25 | L (2006) Statistical priors for efficient combinatorial optimization via graph cuts. Computer Vision–ECCV 2006 pp 263–274
- Cremers, Grady
(Show Context)
Citation Context ...roduction Recent years have seen an increasing emphasis on higher-order Markov Random Fields in vision. Although their importance was long understood and there were earlier attempts on utilizing them =-=[5, 18, 21]-=-, their use [9, 27] and research into minimization of various classes of higherorder MRF energies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While ther... |

20 | R.: A graph cut algorithm for higher-order Markov random fields
- Fix, Grubner, et al.
- 2011
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19 |
Reduction of bivalent maximization to the quadratic case. Cahiers du Centre d’Etudes de Recherche Operationnelle
- Rosenberg
- 1975
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Citation Context ...hese algorithms in the case of higher-order energies, a number of techniques for reducing higher-order terms to first order have been developed. Until a few years ago, there was only one known method =-=[22]-=- that can reduce general higher-order terms into first order. However, this method is considered impractical, as it uses very large coefficients that makes the resulting energy very hard to minimize. ... |

18 |
G.: A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO). Discrete Optimization 5
- Boros, Hammer, et al.
- 2008
(Show Context)
Citation Context ...e average over about 180 fusionmove iterations while denoising 10 test images. cuts can globally minimize submodular energy with binary labels [13]. Even non-submodular energy can be partially solved =-=[3, 2, 14]-=-. Energies with more labels can be approximately minimized by “move-making” algorithms which iteratively update the solution by devising a labeling problem with binary labels and solving it with graph... |

16 | Triangleflow: Optical flow with triangulation-based higher-order likelihoods. ECCV
- Glocker, Heibel, et al.
- 2010
(Show Context)
Citation Context ...have seen an increasing emphasis on higher-order Markov Random Fields in vision. Although their importance was long understood and there were earlier attempts on utilizing them [5, 18, 21], their use =-=[9, 27]-=- and research into minimization of various classes of higherorder MRF energies [1, 6, 8, 11, 12, 17, 20, 24, 25, 26] have intensified significantly in the past few years. While there are useful higher... |

7 |
Generic cuts: An efficient algorithm for optimal inference in higher order MRF-MAP
- Arora, Banerjee, et al.
- 2012
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5 | Generalized roof duality for multi-label optimization: Optimal lower bounds and persistency
- Windheuser, Ishikawa, et al.
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4 |
Inference for order reduction
- Gallagher, Batra, et al.
- 2012
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3 |
Zabih,What energy functions can beminimized via graph cuts
- Kolmogorov, R
- 2004
(Show Context)
Citation Context ...in [10, 11], and the reduction in [6]. The numbers are the average over about 180 fusionmove iterations while denoising 10 test images. cuts can globally minimize submodular energy with binary labels =-=[13]-=-. Even non-submodular energy can be partially solved [3, 2, 14]. Energies with more labels can be approximately minimized by “move-making” algorithms which iteratively update the solution by devising ... |

2 |
Submodular relaxations for MRFs with high-order potentials
- Osokin, Vetrov
- 2012
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