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## P³ & beyond: Solving energies with higher order cliques (2007)

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Venue: | IN COMPUTER VISION AND PATTERN RECOGNITION |

Citations: | 102 - 17 self |

### Citations

3482 | Conditional random fields: Probabilistic models for segmenting and labeling sequence data
- Lafferty, McCallum, et al.
- 2001
(Show Context)
Citation Context ...ion (BP) [16, 25]. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical models such as Markov Random Fields (MRF) and Conditional Random Fields (CRF) =-=[13]-=-. α-expansion and αβ-swap are two popular move making algorithms for approximate energy minimization which were proposed in [5]. They are extremely efficient and have been shown to produce good result... |

2120 | R.: Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
(Show Context)
Citation Context ...screte optimization has emerged as an important tool in solving Computer Vision problems. This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts =-=[5, 11]-=-, treereweighted message passing [10, 24] and variants of belief propagation (BP) [16, 25]. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical model... |

1129 | Grabcut: Interactive foreground extraction using iterated graph cuts
- Rother, Kolmogorov, et al.
(Show Context)
Citation Context ... by (approximately) minimizing the corresponding Gibbs energy. Pairwise CRF : For the problem of segmentation, it is common practice to assume a pairwise CRF where the cliques are of size at most two =-=[1, 4, 20]-=-. In this case, the Gibbs energy of the CRF is of the form: E(x) = ∑ ψi(xi) + ∑ ∑ ψij(xi, xj), (39) i i j∈Ni where Ni is the neighbourhood of pixel Di (defined as the 8-neighbourhood). The unary poten... |

1009 | Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ... by (approximately) minimizing the corresponding Gibbs energy. Pairwise CRF : For the problem of segmentation, it is common practice to assume a pairwise CRF where the cliques are of size at most two =-=[1, 4, 20]-=-. In this case, the Gibbs energy of the CRF is of the form: E(x) = ∑ ψi(xi) + ∑ ∑ ψij(xi, xj), (39) i i j∈Ni where Ni is the neighbourhood of pixel Di (defined as the 8-neighbourhood). The unary poten... |

488 | Convergent tree-reweighted message passing for energy minimization
- Kolmogorov
(Show Context)
Citation Context ...portant tool in solving Computer Vision problems. This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts [5, 11], treereweighted message passing =-=[10, 24]-=- and variants of belief propagation (BP) [16, 25]. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical models such as Markov Random Fields (MRF) and ... |

474 | Generalized belief propagation
- Yedidia, Freeman, et al.
- 2000
(Show Context)
Citation Context ... This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts [5, 11], treereweighted message passing [10, 24] and variants of belief propagation (BP) =-=[16, 25]-=-. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical models such as Markov Random Fields (MRF) and Conditional Random Fields (CRF) [13]. α-expansion... |

415 | C.: A comparative study of energy minimization methods for Markov random fields with smoothness-based priors.
- Szeliski, Zabih, et al.
- 2008
(Show Context)
Citation Context ...re two popular move making algorithms for approximate energy minimization which were proposed in [5]. They are extremely efficient and have been shown to produce good results for a number of problems =-=[22]-=-. These algorithms minimize an energy function by starting from an initial labelling and making a series of changes (moves) which decrease the energy iteratively. Convergence is achieved when the ener... |

292 | Field of experts: A framework for learning image priors.
- Roth, Black
- 2005
(Show Context)
Citation Context ...ex interactions of random variables and thus could overcome this problem. Researchers have long recognized this fact and have used higher order models to improve the expressive power of MRFs and CRFs =-=[14, 18, 19]-=-. The initial work in this regard has been quite promising and higher order cliques have been shown to improve results. However their use has been quite limited due to the unavailability of efficient ... |

217 | Interactive image segmentation using an adaptive gmmrf model
- Blake, Rother, et al.
- 2004
(Show Context)
Citation Context ... by (approximately) minimizing the corresponding Gibbs energy. Pairwise CRF : For the problem of segmentation, it is common practice to assume a pairwise CRF where the cliques are of size at most two =-=[1, 4, 20]-=-. In this case, the Gibbs energy of the CRF is of the form: E(x) = ∑ ψi(xi) + ∑ ∑ ψij(xi, xj), (39) i i j∈Ni where Ni is the neighbourhood of pixel Di (defined as the 8-neighbourhood). The unary poten... |

216 | Exact optimization for Markov random fields with convex prio rs
- Ishikawa
(Show Context)
Citation Context ...ization can be translated to an st-mincut problem. The class of multi-label submodular functions which can be translated into an st-mincut problem has also been characterized independently by [2] and =-=[8]-=-. 2.2. Metric and Semi-metric Potential functions In this subsection we provide the constraints for pairwise potentials to define a metric or a semi-metric. Definition 3. A potential function ψij(a, b... |

179 | Pseudoboolean optimization.
- Boros, Hammer
- 2002
(Show Context)
Citation Context ... xmap = arg max x∈L 2.1. Submodular Energy Functions Pr(x|D) = arg min E(x). (3) x∈L Submodular set functions play an important role in energy minimization as they can be minimized in polynomial time =-=[3, 9]-=-. In this paper we will explain their properties in terms of functions of binary random variables which can be seen as set functions [11]. Definition 1. A projection of a function f : L n → R on s var... |

173 | Texture classification: Are filter banks necessary?
- Varma, Zisserman
- 2003
(Show Context)
Citation Context ...cifically, we use textons to represent the appearance of image patches. Unlike the distributions Ha which describe the potential for one variable Xi, textons capture rich statistics of natural images =-=[15, 23]-=-. In this work, we define textons to be n × n patches. Using the segmented keyframes, we obtain a dictionary of textons for each segment s = 1, . . . , k (denoted by Ps). Note, however, that our frame... |

130 | Recognizing Surfaces Using Three-Dimensional Textons
- Leung, Malik
- 1999
(Show Context)
Citation Context ...cifically, we use textons to represent the appearance of image patches. Unlike the distributions Ha which describe the potential for one variable Xi, textons capture rich statistics of natural images =-=[15, 23]-=-. In this work, we define textons to be n × n patches. Using the segmented keyframes, we obtain a dictionary of textons for each segment s = 1, . . . , k (denoted by Ps). Note, however, that our frame... |

94 |
Image-Based rendering using image-based priors,”
- Fitzgibbon, Wexler, et al.
- 2003
(Show Context)
Citation Context ...rated on the texture based video segmentation problem. The P n Potts model potentials can be used to solve many other Computer Vision problems such as object recognition [14] and novel view synthesis =-=[5]-=- as will be shown in forthcoming works. We conclude with the observation that the optimal moves for many interesting clique potentials such as those that preserve planarity are NP-hard to compute [10]... |

81 | Efficient Belief Propagation with learned higher-order Markov random fields
- Lan, Roth, et al.
- 2006
(Show Context)
Citation Context ...ted in terms of unary and pairwise clique potentials. This assumption severely restricts the representational power of these models making them unable to capture the rich statistics of natural scenes =-=[14]-=-. Higher order clique potentials have the capability to model complex interactions of random variables and thus could overcome this problem. Researchers have long recognized this fact and have used hi... |

67 | Texture synthesis via a noncausal nonparametric multiscale Markov random field.
- Paget, Longstaff
- 1998
(Show Context)
Citation Context ...ex interactions of random variables and thus could overcome this problem. Researchers have long recognized this fact and have used higher order models to improve the expressive power of MRFs and CRFs =-=[14, 18, 19]-=-. The initial work in this regard has been quite promising and higher order cliques have been shown to improve results. However their use has been quite limited due to the unavailability of efficient ... |

61 | Energy minimization via graph cuts: Settling what is possible
- Freedman, Drineas
- 2005
(Show Context)
Citation Context ... the class of functions defined by cliques of size at most n as P n . It should be noted that this class is different from the class F n of energy functions which involve only binary random variables =-=[7, 12]-=-. Higher order cliques Most energy minimization based methods for solving Computer Vision problems assume that the energy can be represented in terms of unary and pairwise clique potentials. This assu... |

48 |
Extending pictorial structures for object recognition
- Kumar, Torr, et al.
- 2004
(Show Context)
Citation Context ... order cliques provide a probabilistic formulation for exemplar based techniques. Exemplars are used in a wide range of computer vision applications, from 3D reconstruction [17] to object recognition =-=[12, 21]-=-. We consider one such problem, i.e. video segmentation, which can be stated as follows. Given a video and a small number of keyframes which have been segmented into k segments, the task is to segment... |

46 | Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation,” in
- Meltzer, Yanover, et al.
- 2005
(Show Context)
Citation Context ... This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts [5, 11], treereweighted message passing [10, 24] and variants of belief propagation (BP) =-=[16, 25]-=-. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical models such as Markov Random Fields (MRF) and Conditional Random Fields (CRF) [13]. α-expansion... |

35 | A new framework for approximate labeling via graph cuts. ICCV - Komodakis, Tziritas - 2005 |

33 |
What energy functions can be minimizedvia graph cuts
- Kolmogorov, Zabih
(Show Context)
Citation Context ...screte optimization has emerged as an important tool in solving Computer Vision problems. This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts =-=[5, 11]-=-, treereweighted message passing [10, 24] and variants of belief propagation (BP) [16, 25]. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical model... |

31 | Tree-based reparameterization for approximate inference on loopy graphs.
- Wainwright, Jaakola, et al.
- 2001
(Show Context)
Citation Context ...portant tool in solving Computer Vision problems. This has primarily been the result of the increasing use of energy minimization algorithms such as graph cuts [5, 11], treereweighted message passing =-=[10, 24]-=- and variants of belief propagation (BP) [16, 25]. These algorithms allow us to perform approximate inference (i.e. obtain the MAP estimate) on graphical models such as Markov Random Fields (MRF) and ... |

25 | Two strongly polynomial cut canceling algorithms for minimum cost network flow
- Ervolina, McCormick
- 1993
(Show Context)
Citation Context ... xmap = arg max x∈L 2.1. Submodular Energy Functions Pr(x|D) = arg min E(x). (3) x∈L Submodular set functions play an important role in energy minimization as they can be minimized in polynomial time =-=[3, 9]-=-. In this paper we will explain their properties in terms of functions of binary random variables which can be seen as set functions [11]. Definition 1. A projection of a function f : L n → R on s var... |

25 | Single-histogram class models for image segmentation.
- Schroff, Criminisi, et al.
- 2006
(Show Context)
Citation Context ... order cliques provide a probabilistic formulation for exemplar based techniques. Exemplars are used in a wide range of computer vision applications, from 3D reconstruction [17] to object recognition =-=[12, 21]-=-. We consider one such problem, i.e. video segmentation, which can be stated as follows. Given a video and a small number of keyframes which have been segmented into k segments, the task is to segment... |

8 |
3D texture reconstruction from extensive BTF data
- Neubeck, Zalesny, et al.
- 2005
(Show Context)
Citation Context ...ues. We observe that higher order cliques provide a probabilistic formulation for exemplar based techniques. Exemplars are used in a wide range of computer vision applications, from 3D reconstruction =-=[17]-=- to object recognition [12, 21]. We consider one such problem, i.e. video segmentation, which can be stated as follows. Given a video and a small number of keyframes which have been segmented into k s... |

3 |
Strukturelle bilderkennung
- Flach
- 2002
(Show Context)
Citation Context ...se minimization can be translated to an st-mincut problem. The class of multi-label submodular functions which can be translated into an st-mincut problem has also been characterized independently by =-=[6]-=- and [8]. 2.2. Metric and Semi-metric Potential functions In this subsection we provide the constraints for pairwise potentials to define a metric or a semi-metric. Definition 3. A potential function ... |

1 |
Strukturelle bilderkennung
- Boris
- 2002
(Show Context)
Citation Context ...se minimization can be translated to an st-mincut problem. The class of multi-label submodular functions which can be translated into an st-mincut problem has also been characterized independently by =-=[2]-=- and [8]. 2.2. Metric and Semi-metric Potential functions In this subsection we provide the constraints for pairwise potentials to define a metric or a semi-metric. Definition 3. A potential function ... |