### Citations

2120 | R.: Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
(Show Context)
Citation Context ... the maximum a posteriori (MAP) solution of a Markov Random Field (MRF). Such optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts=-=[5, 10, 14]-=-, belief propagation[7, 21], and tree-reweighted message passing[13]. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in ... |

1047 | What energy functions can be minimized via graph cuts
- Kolmogorov, Zabih
- 2004
(Show Context)
Citation Context ... the maximum a posteriori (MAP) solution of a Markov Random Field (MRF). Such optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts=-=[5, 10, 14]-=-, belief propagation[7, 21], and tree-reweighted message passing[13]. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in ... |

954 | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics
- Martin, Fowlkes, et al.
- 2001
(Show Context)
Citation Context ...gy and the mean peak signal-to-noise ratio (PSNR) for denoising results over the same set of 10 images that were also used in both [17] and [23]. The images are from the Berkeley segmentation database=-=[20]-=-, grayscaled and reduced in size, as well as added the Gaussian noise with σ = 10, 20. The test images and the FoE model was kindly provided by Stefan Roth, one of the authors of [17, 25]. We minimize... |

515 | Efficient belief propagation for early vision.
- FELZENSZWALB, HUTTENLOCHER
- 2006
(Show Context)
Citation Context ... solution of a Markov Random Field (MRF). Such optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts[5, 10, 14], belief propagation=-=[7, 21]-=-, and tree-reweighted message passing[13]. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in terms of unary and pairwise... |

488 | Convergent tree-reweighted message passing for energy minimization
- Kolmogorov
(Show Context)
Citation Context ...ch optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts[5, 10, 14], belief propagation[7, 21], and tree-reweighted message passing=-=[13]-=-. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in terms of unary and pairwise clique potentials, with a few exceptions... |

292 | Field of experts: A framework for learning image priors.
- Roth, Black
- 2005
(Show Context)
Citation Context ...ics of natural scenes cannot be captured by such limited potentials[21]. Higher order cliques can model more complex interactions and reflect the natural statistics better. This has long been realized=-=[11, 22, 25]-=-, but with the recent success of the new energy optimization methods, there is a renewed emphasis on the effort to find an efficient way to optimize MRFs of higher order. For instance, belief propagat... |

216 | Exact optimization for Markov random fields with convex prio rs
- Ishikawa
(Show Context)
Citation Context ... the maximum a posteriori (MAP) solution of a Markov Random Field (MRF). Such optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts=-=[5, 10, 14]-=-, belief propagation[7, 21], and tree-reweighted message passing[13]. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in ... |

171 | M.: Optimizing binary MRFs via extended roof duality
- Rother, Kolmogorov, et al.
- 2007
(Show Context)
Citation Context ...abels. Second, a recent innovation allows optimization of firstorder non-submodular functions. This method by Boros, Hammer, and their co-workers [2, 4, 9] is variously called QPBO[15] or roof-duality=-=[26]-=-. If the function is submodular, QPBO is guaranteed to find the global minimum. Even if it is not submodular, QPBO returns a solution assigning either 0, 1, or −1 to each pixel, with the guarantee tha... |

145 | Minimizing nonsubmodular functions with graph cuts – A review. - Kolmogorov, Rother - 2007 |

80 | Beyond pairwise energies: Efficient optimization of higher-order MRFs
- Komodakis, Paragios
- 2009
(Show Context)
Citation Context ... in polynomial time. Most recently, there are at least three papers just in this CVPR, including this one, addressing the problem of minimizing higherorder Markov random fields: Komodakis and Paragios=-=[16]-=- employ a master-slave decomposition framework to solve a dual relaxation to the MRF problem; Rother et al.[27] use a soft-pattern-based representation of higher-order functions that may for some ener... |

74 | Minimizing sparse higher order energy functions of discrete variables
- Rother, Kohli, et al.
- 2009
(Show Context)
Citation Context ...essing the problem of minimizing higherorder Markov random fields: Komodakis and Paragios[16] employ a master-slave decomposition framework to solve a dual relaxation to the MRF problem; Rother et al.=-=[27]-=- use a soft-pattern-based representation of higher-order functions that may for some energies lead to very compact first-order functions with small number of non-submodular terms, as well as addressin... |

67 | Texture synthesis via a noncausal nonparametric multiscale Markov random field.
- Paget, Longstaff
- 1998
(Show Context)
Citation Context ...ics of natural scenes cannot be captured by such limited potentials[21]. Higher order cliques can model more complex interactions and reflect the natural statistics better. This has long been realized=-=[11, 22, 25]-=-, but with the recent success of the new energy optimization methods, there is a renewed emphasis on the effort to find an efficient way to optimize MRFs of higher order. For instance, belief propagat... |

61 | Energy minimization via graph cuts: Settling what is possible
- Freedman, Drineas
- 2005
(Show Context)
Citation Context ...t consider triples[6, 14, 29]. This limitation severely reTable 1. Graph cut applicability Order Binary labels Multiple labels First Mincut[14], QPBO[9] → binary (α-exp. [5]), [10] Second → 1st order =-=[14, 8]-=- → binary (fusion [29]) Higher → 1st order (this paper) → binary (fusion, this paper) stricts the representational power of the models: the rich statistics of natural scenes cannot be captured by such... |

57 | Fusionflow: Discrete-continuous optimization for optical flow estimation,
- Lempitsky, Roth, et al.
- 2008
(Show Context)
Citation Context ...nary variables are reduced to first-order ones and minimized iteratively. For our approach, two recent advances in graph cut are essential. First, there is a recent generalization called “fusion move”=-=[18, 19]-=- of the α-expansion algorithm[5]. The energy is minimized in α-expansion by starting from an 1 Hiroshi Ishikawa Higher-Order Clique Reduction in Binary Graph Cut initial labeling and iteratively makin... |

52 | Logcut - efficient graph cut optimization for markov random fields - Lempitsky, Rother, et al. - 2007 |

50 | Exact inference in multi-label crfs with higher order cliques.
- Ramalingam, Kohli, et al.
- 2008
(Show Context)
Citation Context ...move-making algorithms, it can of course be used directly for binary-label problems. It can also be used to optimize the binary-label function converted from a multi-label one using the techniques in =-=[28]-=-. It can also be used with other algorithms that minimizes quadratic pseudo-Boolean optimization problems, such as BP and TRW. We plan to make our research code for the reduction publicly available at... |

39 | Efficient belief propagation for vision using linear constraint nodes
- Potetz
- 2007
(Show Context)
Citation Context ...he recent success of the new energy optimization methods, there is a renewed emphasis on the effort to find an efficient way to optimize MRFs of higher order. For instance, belief propagation variants=-=[17, 23]-=- have been introduced to do inference based on higher-order clique potentials. In graph cut, Kolmogorov and Zabih[14] found a reduction that can reduce second-order binary-label potentials into pairwi... |

30 | Preprocessing of unconstrained quadratic binary optimization”, RUTCOR Research Report
- Boros, Hammer, et al.
- 2006
(Show Context)
Citation Context ...ne of the two at each pixel to generate a new map of labels. Second, a recent innovation allows optimization of firstorder non-submodular functions. This method by Boros, Hammer, and their co-workers =-=[2, 4, 9]-=- is variously called QPBO[15] or roof-duality[26]. If the function is submodular, QPBO is guaranteed to find the global minimum. Even if it is not submodular, QPBO returns a solution assigning either ... |

27 |
p3 and beyond: Move making algorithms for solving higher order functions
- Kohli, Kumar, et al.
- 2009
(Show Context)
Citation Context ...orov and Zabih[14] found a reduction that can reduce second-order binary-label potentials into pairwise ones, followed by an algebraic simplification by Freedman and Drineas[8] (Table 1.) Kohli et al.=-=[12]-=- extend the class of energies for which the optimal α-expansion and αβ-swap moves can be computed in polynomial time. Most recently, there are at least three papers just in this CVPR, including this o... |

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 ...the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in terms of unary and pairwise clique potentials, with a few exceptions that consider triples=-=[6, 14, 29]-=-. This limitation severely reTable 1. Graph cut applicability Order Binary labels Multiple labels First Mincut[14], QPBO[9] → binary (α-exp. [5]), [10] Second → 1st order [14, 8] → binary (fusion [29]... |

19 | Reduction of bivalent maximization to the quadratic case. Cahiers du Centre d’Etudes de Recherche Operationnelle - Rosenberg - 1975 |

11 | Optimizing binary MRFs with higher order cliques
- Ali, Farag, et al.
- 2008
(Show Context)
Citation Context ...lent problem for quadratic pseudo-Boolean function. The method was proposed by Rosenberg[24] more than 30 years ago; it has since been recalled by Boros and Hammer[3] and, more recently, by Ali et al.=-=[1]-=-. In this reduction, the product xy of two variables x, y in the function is replaced by a new variable z, which is forced to have the same value as xy at any minimum of the function by adding penalty... |

7 |
Rethinking the prior model for stereo
- Ishikawa, Geiger
(Show Context)
Citation Context ...ics of natural scenes cannot be captured by such limited potentials[21]. Higher order cliques can model more complex interactions and reflect the natural statistics better. This has long been realized=-=[11, 22, 25]-=-, but with the recent success of the new energy optimization methods, there is a renewed emphasis on the effort to find an efficient way to optimize MRFs of higher order. For instance, belief propagat... |

6 |
Globally Optimal Solutions for Energy Minimization
- Meltzer, Yanover, et al.
- 2005
(Show Context)
Citation Context ... solution of a Markov Random Field (MRF). Such optimization schemes have become quite popular, largely owing to the success of optimization techniques such as graph cuts[5, 10, 14], belief propagation=-=[7, 21]-=-, and tree-reweighted message passing[13]. However, because of the lack of efficient algorithms to optimize energies with higher-order interactions, most are represented in terms of unary and pairwise... |