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## An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision (2001)

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

Citations: | 1291 - 53 self |

### Citations

2098 | Fast Approximate Energy Minimization via Graph Cuts
- Boykov, Veksler, et al.
(Show Context)
Citation Context ...s Corporate Research, Imaging & Visualization, Princeton NJ 08540, USA yuri@scr.siemens.com 2 Cornell University, Computer Science, Upson Hall, Ithaca NY 14853, USA vnk@cs.cornell.edu Abstract. After =-=[10, 15, 12, 2, 4]-=- minimum cut/maximumsow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides... |

1529 | A taxonomy and evaluation of dense two-frame stereo correspondence algorithms
- Scharstein, Szeliski
- 2002
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Citation Context ...thods provide arguably some of the most accurate solutions for the specified applications. For example, consider two recent evaluations of stereo algorithms using real imagery with dense ground truth =-=[34, 37]-=-. Greig et al. constructed a two terminal graph such that the minimum cost cut of the graph gives a globally optimal binary labeling L in case of the Potts model of interaction in (1). Previously, exa... |

1078 | Cognitive Networks
- Thomas, DaSilva, et al.
- 2005
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Citation Context ... is the maximum \amount of water" that can be sent from the source to the sink by interpreting graph edges as directed \pipes" with capacities equal to edge weights. The theorem of Ford and =-=Fulkerson [8]-=- states that a maximumsow from s to t saturates a set of edges in the graph dividing the nodes into two disjoint parts fS; T g corresponding to a minimum cut. Thus, min-cut and max- ow problems are eq... |

1065 |
Level Set Methods and Dynamic Implicit Surfaces
- Osher, Fedkiw
(Show Context)
Citation Context ...e. The results in [3] established a link between two standard energy minimization approaches frequently used in vision: combinatorial graph-cut methods and geometric methods based on level-sets (e.g. =-=[35, 29, 33, 28]-=-). 0 0 0 0 1 0 0 0 0 1 0 2 1 0 0 2 1 0 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1sIn IEEE Transactions on PAMI, Vol. 26, No. 9, pp. 1124-1137, Sept. 2004 p.4 A growing number of publications in vision use graph-... |

1035 | What Energy Functions can be Minimized via Graph Cuts
- Kolmogorov, Zabih
- 2004
(Show Context)
Citation Context ...on PAMI, Vol. 26, No. 9, pp. 1124-1137, Sept. 2004 p.1 An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision Yuri Boykov and Vladimir Kolmogorov ∗ Abstract After =-=[15, 31, 19, 8, 25, 5]-=- minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provid... |

999 | Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ...ect segmentation (Section 4.3). We chose formulations where certain appropriate versions of energy (1) can be minimized via graph cuts. The corresponding graph structures were previously described by =-=[10, 12, 2, 4, 14, 3]-=- in detail. These (or very similar) structures are used in all computer vision applications with graph cuts (that we are aware of) to date. Note that we could not test all known min-cut/max- ow algori... |

969 |
Level Set Methods and Fast Marching Methods
- Sethian
- 1999
(Show Context)
Citation Context ...e. The results in [3] established a link between two standard energy minimization approaches frequently used in vision: combinatorial graph-cut methods and geometric methods based on level-sets (e.g. =-=[35, 29, 33, 28]-=-). 0 0 0 0 1 0 0 0 0 1 0 2 1 0 0 2 1 0 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1sIn IEEE Transactions on PAMI, Vol. 26, No. 9, pp. 1124-1137, Sept. 2004 p.4 A growing number of publications in vision use graph-... |

664 | A new approach to the maximum flow problem
- Goldberg, Tarjan
- 1988
(Show Context)
Citation Context ... 2 we provide basic facts about graphs, min-cut and max-flow problems, and some standard combinatorial optimization algorithms for them. We consider both Goldberg-Tarjan style push-relabel algorithms =-=[14]-=- as well as methods based on augmenting paths a la Ford-Fulkerson [13]. Note that in the course of our experiments with standard augmenting path techniques we developed some new algorithmic ideas that... |

488 | Graphcut Textures: Image and Video Synthesis Using Graph Cuts
- Kwatra, Schödl, et al.
- 2003
(Show Context)
Citation Context ...low algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et al. and by most later graph-based methods (e.g. =-=[32, 18, 4, 17, 8, 2, 30, 39, 21, 36, 38, 6, 23, 24, 9, 26]-=-) can be represented as 1 E(L) = � Dp(Lp) + � p∈P (p,q)∈N Vp,q(Lp, Lq), (1) where L = {Lp |p ∈ P} is a labeling of image P, Dp(·) is a data penalty function, Vp,q is an interaction potential, and N is... |

427 |
Exact maximum a posteriori estimation for binary images
- Greig, Porteous, et al.
- 1989
(Show Context)
Citation Context ...s Corporate Research, Imaging & Visualization, Princeton NJ 08540, USA yuri@scr.siemens.com 2 Cornell University, Computer Science, Upson Hall, Ithaca NY 14853, USA vnk@cs.cornell.edu Abstract. After =-=[10, 15, 12, 2, 4]-=- minimum cut/maximumsow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides... |

360 | Computing Visual Correspondence with Occlusions using Graph Cuts
- Kolmogorov, Zabih
- 2001
(Show Context)
Citation Context ...ow algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et. al. and by most later graph based methods (e.g. =-=[15, 12, 2, 11, 4, 1, 18, 13, 16, 17, 3, 14-=-]) can be represented as a posterior energy in MAP-MRF 1 framework: E(L) = X p2P D p (L p ) + X (p;q)2N V p;q (L p ; L q ); (1) where L = fL p jp 2 Pg is a labeling of image P , D p () is a data penal... |

349 | An optimal graph theoretic approach to data clustering: theory and its application to image segmentation
- Wu, Leahy
- 1993
(Show Context)
Citation Context ...r almost 10 years mainly because binary image restoration looked very limited as an application. Early attempts to use combinatorial graph cut algorithms in vision were restricted to image clustering =-=[40]-=-. In the late 90’s a large number of new computer vision techniques appeared that figured how to use min-cut/max-flow algorithms on graphs for solving more interesting non-binary problems. [32] was th... |

322 |
Geometric Partial Differential Equations and Image Analysis
- Sapiro
- 2001
(Show Context)
Citation Context ...on in [5] can be generalized to find geodesics and minimum surfaces in Riemannian metric spaces. This result links graph-cut segmentation methods with popular geometric techniques based on level-sets =-=[35, 29, 33, 28]-=-. The technique in [5] finds a globally optimal binary segmentation of N-dimensional image under appropriate constraints. The computation is done in one pass of a max-flow/min-cut algorithm on a certa... |

314 | Multi-camera scene reconstruction via graph cuts
- Kolmogorov, Zabih
- 2002
(Show Context)
Citation Context ...No. 9, pp. 1124-1137, Sept. 2004 p.25 4.3.3 Multi-camera scene reconstruction In this section we consider a graph cuts based algorithm for reconstructing a shape of an object taken by several cameras =-=[24]-=-. Suppose we are given n calibrated images of the same scene taken from different viewpoints (or at different moments of time). Let Pi be the set of pixels in the camera i, and let P = P1∪. . .∪Pn be ... |

269 | Combinatorial Optimization
- Cook, Cunningham, et al.
- 1998
(Show Context)
Citation Context ...ush it to neighboring nodes. Push-relabel techniques are harder to describe in just a few sentences and we would rather refer the reader to our favorite text-book on basic graph theory and algorithms =-=[6]-=-. For our experimental tests on graph-based energy minimization methods in vision we selected the following standard algorithms. DINIC: Algorithm of Dinic [7]. H PRF: Push-Relabel algorithm [9] with t... |

258 | A maximum-flow formulation of the n-camera stereo correspondence problem
- Roy, Cox
- 1998
(Show Context)
Citation Context ...ow algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et al. and by most later graph-based methods (e.g., =-=[32]-=-, [18], [4], [17], [8], [2], [30], [39], [21], [36], [38], [6], [23], [24], [9], [26]) can be represented as 1 EðLÞ X DpðLpÞþ X Vp;qðLp;LqÞ; ð1Þ p2P ðp;qÞ2N where L fLp jp 2Pgis a labeling of image... |

248 | Computing geodesics and minimal surfaces via graph cuts
- Boykov, Kolmogorov
- 2003
(Show Context)
Citation Context ...ple cliques. In fact, full potential of graph-cut techniques in multi-label cases is still not entirely understood. Geometric properties of segments produced by graph-cut methods were investigated in =-=[3]-=-. This work studied cut metric on regular grid-graphs and showed that discrete topology of graphcuts can approximate any continuous Riemannian metric space. The results in [3] established a link betwe... |

213 | Exact optimization for markov random fields with convex priors
- Ishikawa
- 2003
(Show Context)
Citation Context ... 26, No. 9, pp. 1124-1137, Sept. 2004 p.20 (a) Diamond, 54 labels (b) Bell Quad, 32 labels (c) Diamond, 100x100 pix (d) Bell Quad, 125x125 pix Figure 7: Running times for “multi-layered” graphs (e.g. =-=[31, 19]-=-). The results are obtained in the context of image restoration with linear interaction potentials (see Section 4.2.2). In (a) and (b) we fixed the number of allowed labels (graph layers) and tested e... |

209 | Markov random fields with efficient approximations
- Boykov, Zabih
- 1998
(Show Context)
Citation Context ...low algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et al. and by most later graph-based methods (e.g. =-=[32, 18, 4, 17, 8, 2, 30, 39, 21, 36, 38, 6, 23, 24, 9, 26]-=-) can be represented as 1 E(L) = � Dp(Lp) + � p∈P (p,q)∈N Vp,q(Lp, Lq), (1) where L = {Lp |p ∈ P} is a labeling of image P, Dp(·) is a data penalty function, Vp,q is an interaction potential, and N is... |

208 | On implementing the push-relabel method for the maximum flow problem
- Cherkassky, Goldberg
- 1997
(Show Context)
Citation Context ...re our new algorithm presented in Section 3 and standard algorithms of combinatorial optimization introduced in Section 2.2: DINIC, H PRF, and Q PRF. Many experimental tests, including the results in =-=[5]-=-, show that the last two algorithms work consistently better than a large number of other mincut /max- ow algorithms of combinatorial optimization. For DINIC, H PRF, and Q PRF we took the implementati... |

199 | S.: Faster shortest-path algorithms for planar graphs
- Henzinger, Klein, et al.
- 1997
(Show Context)
Citation Context ...ses of graphs. Examples of interesting but inapplicable methods include randomized techniques for dense undirected graphs [20], methods for planar graphs assuming small number of terminal connections =-=[27, 16]-=-, and others. 3 New Min-Cut/Max-Flow Algorithm In this section we present a new algorithm developed during our attempts to improve empirical performance of standard augmenting path techniques on graph... |

167 |
Algorithm for Solution of a Problem of Maximum Flows in Networks with Power Estimation
- Dinic
- 1970
(Show Context)
Citation Context ...ithm. In Section 4 we tested this new augmenting-path style algorithm as well as three standard algorithms: the H PRF and Q PRF versions of the “push-relabel” method [14, 10], and the Dinic algorithm =-=[12]-=- that also uses augmenting paths. We selected several examples in image restoration, stereo, and segmentation where different forms of energy (1) are minimized via graph structures originally describe... |

139 | Large occlusion stereo
- Bobick, Intille
- 1999
(Show Context)
Citation Context ...pp. 1124-1137, Sept. 2004 p.24 problem tractable. Inevitably, such simplification can generate errors that range from minor inconsistencies to major misinterpretation of the scene geometry. Recently, =-=[1]-=- reported some progress in solving stereo with occlusions. [17] were first to suggest a graph-cut based solution for stereo that elegantly handles occlusions assuming monotonicity constraint. Here we ... |

105 | Segmentation by grouping junctions
- Ishikawa, Geiger
- 1998
(Show Context)
Citation Context ...s Corporate Research, Imaging & Visualization, Princeton NJ 08540, USA yuri@scr.siemens.com 2 Cornell University, Computer Science, Upson Hall, Ithaca NY 14853, USA vnk@cs.cornell.edu Abstract. After =-=[10, 15, 12, 2, 4]-=- minimum cut/maximumsow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides... |

102 | Occlusions, discontinuities, and epipolar lines in stereo - Ishikawa, Geiger - 1998 |

102 | Exact Voxel Occupancy with Graph Cuts
- Snow, Viola, et al.
- 2000
(Show Context)
Citation Context ...ow algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et. al. and by most later graph based methods (e.g. =-=[15, 12, 2, 11, 4, 1, 18, 13, 16, 17, 3, 14-=-]) can be represented as a posterior energy in MAP-MRF 1 framework: E(L) = X p2P D p (L p ) + X (p;q)2N V p;q (L p ; L q ); (1) where L = fL p jp 2 Pg is a labeling of image P , D p () is a data penal... |

100 | Random sampling in cut, flow, and network design problems
- Karger
- 1994
(Show Context)
Citation Context ...ialized min-cut/max-flow algorithms that are designed for some restricted classes of graphs. Examples of interesting but inapplicable methods include randomized techniques for dense undirected graphs =-=[20]-=-, methods for planar graphs assuming small number of terminal connections [27, 16], and others. 3 New Min-Cut/Max-Flow Algorithm In this section we present a new algorithm developed during our attempt... |

68 |
Stereo without epipolar lines: A maximum flow formulation
- Roy
- 1999
(Show Context)
Citation Context ...on PAMI, Vol. 26, No. 9, pp. 1124-1137, Sept. 2004 p.1 An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision Yuri Boykov and Vladimir Kolmogorov ∗ Abstract After =-=[15, 31, 19, 8, 25, 5]-=- minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provid... |

51 | An experimental comparison of stereo algorithms
- Szeliski, Zabih
- 1999
(Show Context)
Citation Context ...thods provide arguably some of the most accurate solutions for the specified applications. For example, consider two recent evaluations of stereo algorithms using real imagery with dense ground truth =-=[34, 37]-=-. Greig et al. constructed a two terminal graph such that the minimum cost cut of the graph gives a globally optimal binary labeling L in case of the Potts model of interaction in (1). Previously, exa... |

48 |
Flow in planar graphs with multiple sources and sinks
- Miller, Naor
- 1995
(Show Context)
Citation Context ...ses of graphs. Examples of interesting but inapplicable methods include randomized techniques for dense undirected graphs [20], methods for planar graphs assuming small number of terminal connections =-=[27, 16]-=-, and others. 3 New Min-Cut/Max-Flow Algorithm In this section we present a new algorithm developed during our attempts to improve empirical performance of standard augmenting path techniques on graph... |

35 | Image segmentation by nested cuts
- Veksler
- 2000
(Show Context)
Citation Context ...ow algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et. al. and by most later graph based methods (e.g. =-=[15, 12, 2, 11, 4, 1, 18, 13, 16, 17, 3, 14-=-]) can be represented as a posterior energy in MAP-MRF 1 framework: E(L) = X p2P D p (L p ) + X (p;q)2N V p;q (L p ; L q ); (1) where L = fL p jp 2 Pg is a labeling of image P , D p () is a data penal... |

29 | Minimal surfaces for stereo - Buehler, Gortler, et al. - 2002 |

27 |
A new approach to the maximum problem
- Goldberg, Tarjan
- 1988
(Show Context)
Citation Context ...developed while working with graphs in vision. In Section 4 we tested our new algorithm and three standard mincut /max- ow algorithms: H PRF and Q PRF versions of Goldberg-style \pushrelabel " me=-=thod [9, 5-=-], and the Dinic algorithm [7]. We selected several examples in image restoration, stereo, and segmentation where dierent forms of energy (1) are minimized via graph structures originally described in... |

17 | A new bayesian framework for object recognition
- Boykov, Huttenlocher
- 1999
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Citation Context |

15 |
Markov random with ecient approximations
- Boykov, Veksler, et al.
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Citation Context |

13 | MRF Solutions for Probabilistic Optical flow Formulations
- Roy, Govindu
- 2000
(Show Context)
Citation Context ...optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et al. and by most later graph-based methods (e.g., [32], [18], [4], [17], [8], [2], =-=[30]-=-, [39], [21], [36], [38], [6], [23], [24], [9], [26]) can be represented as 1 EðLÞ X DpðLpÞþ X Vp;qðLp;LqÞ; ð1Þ p2P ðp;qÞ2N where L fLp jp 2Pgis a labeling of image P, Dpð Þ is a data penalty funct... |

12 |
Incorporating spatial priors into an information theoretic approach for fMRI data analysis
- Kim, Fish, et al.
- 2000
(Show Context)
Citation Context ... we consider two examples of energy (1) with the Potts and linear models of interaction. Graph based methods for minimizing Potts energy were used in many dierent applications including segmentation [=-=13]-=-, stereo [2, 4], object recognition [1], shape reconstruction [16], and augmented reality [17]. Linear interaction energy was used for stereo [15] and segmentation [12]. The structures of the correspo... |

7 |
Fusion of color, shading and boundary information for factory pipe segmentation
- Thirion, Bascle, et al.
- 2000
(Show Context)
Citation Context |

3 |
Algorithm for solution of a problem of maximum in networks with power estimation
- Dinic
- 1970
(Show Context)
Citation Context ...phs in vision. In Section 4 we tested our new algorithm and three standard mincut /max- ow algorithms: H PRF and Q PRF versions of Goldberg-style \pushrelabel " method [9, 5], and the Dinic algor=-=ithm [7-=-]. We selected several examples in image restoration, stereo, and segmentation where dierent forms of energy (1) are minimized via graph structures originally described in [10, 12, 2, 4, 14, 3]. Such ... |

3 |
and Venu Govindu. MRF solutions for probabilistic optical flow formulations
- Roy
- 2000
(Show Context)
Citation Context ...low algorithms from combinatorial optimization can be used to minimize certain important energy functions in vision. The energies addressed by Greig et al. and by most later graph-based methods (e.g. =-=[32, 18, 4, 17, 8, 2, 30, 39, 21, 36, 38, 6, 23, 24, 9, 26]-=-) can be represented as 1 E(L) = � Dp(Lp) + � p∈P (p,q)∈N Vp,q(Lp, Lq), (1) where L = {Lp |p ∈ P} is a labeling of image P, Dp(·) is a data penalty function, Vp,q is an interaction potential, and N is... |

2 |
Optimal object extraction via constrained graph-cuts
- Boykov, Funka-Lea
- 2001
(Show Context)
Citation Context ...ect segmentation (Section 4.4). We chose formulations where certain appropriate versions of energy (1) can be minimized via graph cuts. The corresponding graph structures were previously described by =-=[15, 18, 4, 8, 23, 24, 5]-=- in detail. These (or very similar) structures are used in all computer vision applications with graph cuts (that we are aware of) to date.sIn IEEE Transactions on PAMI, Vol. 26, No. 9, pp. 1124-1137,... |

1 |
Graph-based Algorithms for Multi-camera Reconstruction Problem
- Kolmogorov
- 2003
(Show Context)
Citation Context ...age is finished. At this point q must have active status as it is located next to a free node p. 3.3 Algorithm tuning The proof of correctness of the algorithm presented above is straightforward (see =-=[22]-=-). At the same time, our description leaves many free choices in implementing certain details. For example, we found that the order of processing active nodes and orphans may have a significant effect... |