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## What energy functions can be minimized via graph cuts? (2004)

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Venue: | IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE |

Citations: | 1047 - 23 self |

### Citations

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- 1984
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Citation Context ...restricted form they can be solved efficiently using dynamic programming [2]. However, researchers typically have needed to rely on general purpose optimization techniques such as simulated annealing =-=[3, 10]-=-, which is extremely slow in practice. In the last few years, however, a new approach has been developed based on graph cuts. The basic technique is to construct a specialized graph for the energy fun... |

2120 | R.: Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
(Show Context)
Citation Context ... locally). The minimum cut in turn can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration =-=[7, 8, 12, 14]-=-, stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with s... |

2040 |
Network Flows: Theory, Algorithms, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ... theorem of Ford and Fulkerson [9] this is equivalent to computing the maximum flow from the source to sink. There are many algorithms which solve this problem in polynomial time with small constants =-=[1, 11]. It i-=-s convenient to denote a cut C = S, T by a labeling f mapping from the set of the nodes V−{s, t} to {0, 1} where f(v) =0meansthatv ∈ S, and f(v) =1meansthatv ∈ T . We will use this notation late... |

1546 | A taxonomy and evaluation of dense two-frame stereo correspondence algorithms
- Scharstein, Szeliski
(Show Context)
Citation Context ... the global minimum. The experimental results produced by these algorithms are also quite good, as documented in two recent evaluations of stereo algorithms using real imagery with dense ground truth =-=[22, 24]-=-. Minimizing an energy function via graph cuts, however, remains a technically difficult problem. Each paper constructs its own graph specifically for its individual energy function, and in some of th... |

1403 |
Geometric Algorithms and Combinatorial Optimization
- Grötschel, Lovász, et al.
- 1988
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Citation Context ...s. Submodular functions have received significant attention in the combinatorial optimization literature. A remarkable fact about submodular functions is that they can be minimized in polynomial time =-=[19]-=-, [23], [38]. A function g is assumed to be given as a value oracle, i.e., a “black box” which, for any input subset X returns the value gðXÞ. Unfortunately, algorithms for minimizing arbitrary submod... |

1313 | An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,”
- Boykov, Kolmogorov
- 2004
(Show Context)
Citation Context ...em of Ford and Fulkerson [14], this is equivalent to computing the maximum flow from the source to sink. There are many algorithms that solve this problem in polynomial time with small constants [1], =-=[7]-=-, [17]. It is convenient to note a cut C S;T by a labeling f mapping from the set of the vertices V fs; tg to f0; 1g, where fðvÞ 0 means that v 2 S and fðvÞ 1 means that v 2 T. We will use this no... |

1102 | Cognitive networks
- Thomas, DaSilva, et al.
- 2005
(Show Context)
Citation Context ... the sum of costs of all edges that go from S to T: cðS;TÞ X cðu; vÞ: u2S;v2T;ðu;vÞ2E The minimum s-t-cut problem is to find a cut C with the smallest cost. Due to the theorem of Ford and Fulkerson =-=[14]-=-, this is equivalent to computing the maximum flow from the source to sink. There are many algorithms that solve this problem in polynomial time with small constants [1], [7], [17]. It is convenient t... |

1009 | Interactive graph cuts for optimal boundary and region segmentation of objects in n-d images
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ...ide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging =-=[5, 6, 15]. -=-The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [7, 12–14, 20] it is the global minimum, in other cases a local minimum in a... |

672 | A new approach to the maximum-flow problem,”
- Goldberg, Tarjan
- 1988
(Show Context)
Citation Context ... theorem of Ford and Fulkerson [9] this is equivalent to computing the maximum flow from the source to sink. There are many algorithms which solve this problem in polynomial time with small constants =-=[1, 11]. It i-=-s convenient to denote a cut C = S, T by a labeling f mapping from the set of the nodes V−{s, t} to {0, 1} where f(v) =0meansthatv ∈ S, and f(v) =1meansthatv ∈ T . We will use this notation late... |

515 | Markov random field modeling in computer vision
- Li
- 1995
(Show Context)
Citation Context ... that have not yet been minimized using graph cuts. All of the previous work with graph cuts involves a neighborhood system that is defined on pairs of pixels. In the language of Markov Random Fields =-=[10, 19], -=-these methods consider firstorder MRF’s. The associated energy functions lie in F 2 . Our results allow for the minimization of energy functions in the larger class F 3 , and thus for neighborhood s... |

429 |
Exact maximum a posteriori estimation for binary images.
- Greig, Porteous, et al.
- 1989
(Show Context)
Citation Context ... locally). The minimum cut in turn can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration =-=[7, 8, 12, 14]-=-, stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with s... |

365 | Computing visual correspondence with occlusions using graph cuts.
- Kolmogorov, Zabih
- 2001
(Show Context)
Citation Context ... can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion =-=[4, 7, 8, 13, 16, 20, 21]-=-, voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee... |

357 |
Using dynamic programming for solving variational problems in Vision,”
- Amini, Weymouth, et al.
- 1990
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Citation Context ...lt, as it generally requires minimizing a non-convex function in a space with thousands of dimensions. If the functions have a restricted form they can be solved efficiently using dynamic programming =-=[2]-=-. However, researchers typically have needed to rely on general purpose optimization techniques such as simulated annealing [3, 10], which is extremely slow in practice. In the last few years, however... |

317 | Multi-Camera Scene Reconstruction via Graph-Cuts,”
- Kolmogorov, Zabih
- 2002
(Show Context)
Citation Context ...successfully used for a wide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction =-=[18] a-=-nd medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [7, 12–14, 20] it is the global minimum, in o... |

291 | Computational vision and regularization theory - Poggio, Torre, et al. - 1985 |

258 | A Maximum-Flow Formulation of the N-Camera Stereo Correspondence Problem,”
- Roy, Cox
- 1998
(Show Context)
Citation Context ... can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion =-=[4, 7, 8, 13, 16, 20, 21]-=-, voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee... |

250 | Computing geodesics and minimal surfaces via graph cuts
- Boykov, Kolmogorov
(Show Context)
Citation Context ...or a wide variety of vision problems, including image restoration [9], [10], [18], [21], stereo and motion [4], [9], [10], [20], [24], [27], [32], [35], [36], image synthesis [29], image segmentation =-=[8]-=-, voxel occupancy [39], multicamera scene reconstruction [28], and medical imaging [5], [6], [25], [26]. The output of these algorithms is generally a solution with some interesting theoretical qualit... |

231 |
Submodular functions and optimization,
- Fujishige
- 2005
(Show Context)
Citation Context ...bmodular functions. 7 Let S be a finite set and g :2 S !Rbe a real-valued function defined on the set of all subsets of S. g is called submodular if for any X; Y S gðXÞþgðY Þ gðX [ Y ÞþgðX \ Y Þ: See =-=[15]-=-, for example, for a discussion of submodular functions. An equivalent definition of submodular functions is that g is called submodular if, for any X S and i; j 2S X, gðX [fjgÞ gðXÞ gðX [fi; jgÞ gðX ... |

216 | Exact optimization for Markov random fields with convex prio rs
- Ishikawa
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Citation Context ...ls are consecutive integers and V is the L1 distance. The construction is due to [20] and is a modified version of [36]. This construction has been further generalized to handle an arbitrary convex V =-=[22]-=-. However, a convex V is not discontinuity preserving and optimizing an energy function with such a V leads to oversmoothing at the borders of objects. The ability to find the global minimum efficient... |

210 | Markov random fields with efficient approximations,”
- Boykov, Veksler, et al.
- 1998
(Show Context)
Citation Context ... locally). The minimum cut in turn can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration =-=[7, 8, 12, 14]-=-, stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with s... |

200 | A combinatorial algorithm minimizing submodular functions in strongly polynomial time - Schrijver |

181 | Multiprocessor Scheduling with the Aid of Network Flow Algorithms,” - Stone - 1977 |

138 | Multiway cut for stereo and motion with slanted surfaces.
- Birchfield, Tomasi
- 1999
(Show Context)
Citation Context ... can be computed very efficiently by max flow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion =-=[4, 7, 8, 13, 16, 20, 21]-=-, voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee... |

107 | Segmentation by grouping junctions. In:
- Ishikawa, Geiger
- 1998
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Citation Context |

103 | Occlusions, discontinuities, and epipolar lines in stereo,”
- Ishikawa, Geiger
- 1998
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Citation Context |

102 | Exact Voxel Occupancy with Graph Cuts
- Snow, Viola, et al.
- 2000
(Show Context)
Citation Context ...ow algorithms. These methods have been successfully used for a wide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy =-=[23]-=-, multicamera scene reconstruction [18] and medical imaging [5, 6, 15]. The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [7, 12... |

98 | A combinatorial strongly polynomial algorithm for minimizing submodular functions,
- Iwata, Fleischer, et al.
- 2001
(Show Context)
Citation Context ...modular functions have received significant attention in the combinatorial optimization literature. A remarkable fact about submodular functions is that they can be minimized in polynomial time [19], =-=[23]-=-, [38]. A function g is assumed to be given as a value oracle, i.e., a “black box” which, for any input subset X returns the value gðXÞ. Unfortunately, algorithms for minimizing arbitrary submodular f... |

84 |
Stochastic Stereo Matching over Scale”,
- Barnard
- 1985
(Show Context)
Citation Context ...restricted form they can be solved efficiently using dynamic programming [2]. However, researchers typically have needed to rely on general purpose optimization techniques such as simulated annealing =-=[3, 10]-=-, which is extremely slow in practice. In the last few years, however, a new approach has been developed based on graph cuts. The basic technique is to construct a specialized graph for the energy fun... |

80 | Interactive organ segmentation using graph cuts
- Boykov, Jolly
- 2000
(Show Context)
Citation Context ...ide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging =-=[5, 6, 15]. -=-The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [7, 12–14, 20] it is the global minimum, in other cases a local minimum in a... |

76 | Surfaces with occlusions from layered stereo.
- Lin, Tomasi
- 2002
(Show Context)
Citation Context ...low algorithms. These methods have been successfully used for a wide variety of vision problems, including image restoration [9], [10], [18], [21], stereo and motion [4], [9], [10], [20], [24], [27], =-=[32]-=-, [35], [36], image synthesis [29], image segmentation [8], voxel occupancy [39], multicamera scene reconstruction [28], and medical imaging [5], [6], [25], [26]. The output of these algorithms is gen... |

70 |
Stereo without epipolar lines: A maximumflow formulation,”
- Roy
- 1999
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Citation Context |

67 | The complexity of multiway cuts - Dahlhaus, Johnson, et al. - 1992 |

51 | An experimental comparison of stereo algorithms.
- Szeliski, Zabih
- 1999
(Show Context)
Citation Context ... the global minimum. The experimental results produced by these algorithms are also quite good, as documented in two recent evaluations of stereo algorithms using real imagery with dense ground truth =-=[22, 24]-=-. Minimizing an energy function via graph cuts, however, remains a technically difficult problem. Each paper constructs its own graph specifically for its individual energy function, and in some of th... |

45 | A MRF-based Approach for Real-Time Subway Monitoring”, - Paragios, Ramesh - 2001 |

36 | Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions - Iwata, Fleischer - 2001 |

17 | The ZpiM algorithm: a method for interferometric image reconstruction
- Dias, Leitao
- 2002
(Show Context)
Citation Context ...xpansion move is shown in Fig. 1. The expansion move algorithm cycles through the labels in some order (fixed or random) and finds the lowest energy 3. This is somewhat analogous to the Z M algorithm =-=[13]-=-, although that method produces a global minimum of a function with nonbinary variables by repeatedly minimizing a function with binary variables. -expansion move from the current labeling. If this ex... |

14 | Minimum cuts, modular functions, and matroid polyhedra - Cunningham - 1985 |

13 |
Optimal Task Assignment in Linear Array Networks
- Lee, Lee, et al.
- 1992
(Show Context)
Citation Context ...ce we can combine two terms Vp;q and Vq;p into one term. 2. It is interesting to note that some similar techniques have been developed by algorithms researchers working on the task assignment problem =-=[30]-=-, [33], [40].sKOLMOGOROV AND ZABIH: WHAT ENERGY FUNCTIONS CAN BE MINIMIZED VIA GRAPH CUTS? 149 Fig. 1. An example of an expansion move. The labeling on the right is a white-expansion move from the lab... |

12 |
Incorporating spatial priors into an information theoretic approach for fMRI data analysis
- Kim, Fish, et al.
- 2000
(Show Context)
Citation Context ...ide variety of vision problems including image restoration [7, 8, 12, 14], stereo and motion [4, 7, 8, 13, 16, 20, 21], voxel occupancy [23], multicamera scene reconstruction [18] and medical imaging =-=[5, 6, 15]. -=-The output ofsthese algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [7, 12–14, 20] it is the global minimum, in other cases a local minimum in a... |

10 | Submodular Functions and Optimization, volume 47 of Annals of Discrete Mathematics - Fujishige - 1991 |

7 |
Visual Correspondence Using Energy Minimization and
- Kim, Kolmogorov, et al.
- 2003
(Show Context)
Citation Context ...tly by max flow algorithms. These methods have been successfully used for a wide variety of vision problems, including image restoration [9], [10], [18], [21], stereo and motion [4], [9], [10], [20], =-=[24]-=-, [27], [32], [35], [36], image synthesis [29], image segmentation [8], voxel occupancy [39], multicamera scene reconstruction [28], and medical imaging [5], [6], [25], [26]. The output of these algor... |

5 |
Automatic segmentation of contrast-enhanced image sequences
- Kim, Zabih
- 2003
(Show Context)
Citation Context ...and motion [4], [9], [10], [20], [24], [27], [32], [35], [36], image synthesis [29], image segmentation [8], voxel occupancy [39], multicamera scene reconstruction [28], and medical imaging [5], [6], =-=[25]-=-, [26]. The output of these algorithms is generally a solution with some interesting theoretical quality guarantee. In some cases [9], [18], [20], [21], [35], it is the global minimum, in other cases,... |

4 |
Task assignment in distributed systems using network flow methods
- Milis
(Show Context)
Citation Context ...can combine two terms Vp;q and Vq;p into one term. 2. It is interesting to note that some similar techniques have been developed by algorithms researchers working on the task assignment problem [30], =-=[33]-=-, [40].sKOLMOGOROV AND ZABIH: WHAT ENERGY FUNCTIONS CAN BE MINIMIZED VIA GRAPH CUTS? 149 Fig. 1. An example of an expansion move. The labeling on the right is a white-expansion move from the labeling ... |

3 |
Minimum Cuts, Modular Functions
- Cunningham
- 1985
(Show Context)
Citation Context ...functions can be defined as functions that can be represented by a graph without the source and the sink and without auxiliary vertices. It is well-known that cut functions are submodular. Cunningham =-=[11]-=- characterizes cut functions and gives a general-purpose graph construction for them. It can be shown that the set of cut functions is a strict subset of F 2 . We allow a more general graph constructi... |

3 | Image labeling and grouping by minimizing linear functionals over cones - Schellewald, Schnorr - 2001 |

2 | Use of temporal information in a segmentation algorithm of ultrasound images - Herlin, Giraudon - 1993 |