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## A comparative study of energy minimization methods for Markov random fields (2006)

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### Other Repositories/Bibliography

Venue: | IN ECCV |

Citations: | 415 - 36 self |

### Citations

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Citation Context ...for a minimum energy solution, but instead computes the marginal probability distribution of each node in the graph. The belief propagation algorithm was originally designed for graphs without cycles =-=[25]-=-, in which case it produces the exact result for our energy. However, there is nothing in the formulation of BP that prevents it from being tried on graphs with loops. In general, LPB is not guarantee... |

5126 | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
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Citation Context ...data, while the other term enforces spatial coherence. One of the reasons this framework is so popular is that it can be justified in terms of maximum a posteriori estimation of a Markov Random Field =-=[1, 2]-=-. Despite the elegance and power of the energy minimization approach, its early adoption was slowed by computational considerations. The algorithms that were originally used, such as ICM [1] or simula... |

2120 | Fast approximate energy minimization via graph cuts
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Citation Context ...nealing [3], proved to be extremely inefficient. In the last few years, energy minimization approaches have had a renaissance, primarily due to powerful new optimization algorithms such as graph cuts =-=[4, 5]-=- and loopy belief propagation (LBP) [6]. The results, especially in stereo, have been dramatic; according to the widely-used Middlebury stereo benchmarks [7], almost all the top-performing stereo meth... |

1546 | A taxonomy and evaluation of dense two-frame stereo correspondence algorithms
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Citation Context ...imization algorithms such as graph cuts [4, 5] and loopy belief propagation (LBP) [6]. The results, especially in stereo, have been dramatic; according to the widely-used Middlebury stereo benchmarks =-=[7]-=-, almost all the top-performing stereo methods rely on graph cuts or LBP. Moreover, these methods give substantially more accurate results than were previously possible. Simultaneously, the range of a... |

1315 | An experimental comparison of min-cut/maxflow algorithms for energy minimization in vision.
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Citation Context ...atic placement of seams in digital photomontages [12]. Relatively little attention has been paid, however, to the relative performance of various optimization algorithms. Among the few exceptions are =-=[14]-=-, which compared the efficiency of several different max flow algorithms for graph cuts, and [13], which compared graph cuts with LBP. [13] also noted a particulary impressive demonstration of the eff... |

1130 | Grabcut: Interactive foreground extraction using iterated graph cuts
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Citation Context ...ramatically, moving from early applications such as image restoration [1], texture modeling [8], image labeling [9], and stereo matching [3, 4], to applications such as interactive photo segmentation =-=[10, 11]-=- and the automatic placement of seams in digital photomontages [12]. Relatively little attention has been paid, however, to the relative performance of various optimization algorithms. Among the few e... |

1047 | What energy functions can be minimized via graph cuts
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Citation Context ... original work of [4] the swap move algorithm was shown to be applicable to any energy where Vpq is a semi-metric, and the expansion move algorithm to any energy where Vpq is a metric. The results of =-=[5]-=- imply that the expansion move algorithm can be used if for all labels α,β,and γ, Vpq(α, α)+Vpq(β,γ) ≤ Vpq(α, γ)+Vpq(β,α). The swap move algorithm can be used if for all labels α,β Vpq(α, α)+Vpq(β,β) ... |

1010 | Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects
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Citation Context ...ramatically, moving from early applications such as image restoration [1], texture modeling [8], image labeling [9], and stereo matching [3, 4], to applications such as interactive photo segmentation =-=[10, 11]-=- and the automatic placement of seams in digital photomontages [12]. Relatively little attention has been paid, however, to the relative performance of various optimization algorithms. Among the few e... |

898 | Visual Reconstruction. - Blake, Zisserman - 1987 |

699 | Image quilting for texture synthesis and transfer,‖ in
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Citation Context ...re different pairings of adjacent labels can lead to different costs. This is important in a number of applications, ranging from stereo matching (§8.2 of [4]) to image stitching and texture quilting =-=[12,17,18]-=-. A more restricted form of the smoothness energy is Es = � {p,q}∈N wpq · V (|lp − lq|), where the smoothness terms are the product of spatially varying per-pairing weights wpq and a non-decreasing fu... |

579 | Learning low-level vision.
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Citation Context ...ation of BP that prevents it from being tried on graphs with loops. Indeed BP has been applied with great succes to loopy graphs first for error-correcting code problems [31] and then in early vision =-=[32]-=-. Detailed descriptions of the LBP algorithm can be found in [32] and [25]. In general, LPB is not guaranteed to converge, and may go into an infinite loop switching between two labelings. [25] presen... |

516 | Markov Random Field Modeling in Computer Vision,
- Li
- 1995
(Show Context)
Citation Context ...is denoted by l, the number of pixels is n, and the number of labels is m. The energy function E, which can also be viewed as the log likelihood of the posterior distribution of a Markov Random Field =-=[2, 16]-=-, is composed of a data energy Ed and smoothness energy Es, E = Ed+λEs. The data energy is simply the sum of a set of per-pixel data costs dp(l), Ed = � p dp(lp). In the MRF framework, the data energy... |

515 | Efficient belief propagation for early vision.
- FELZENSZWALB, HUTTENLOCHER
- 2006
(Show Context)
Citation Context ...The class of energy functions we are considering is quite broad, and not all energy minimization methods can handle the entire class. For example, acceleration techniques based on distance transforms =-=[22]-=- can significantly speed up message-passing algorithms such as LBP or TRW, yet these methods are only applicable for certain smoothness costs V . Other algorithms, such as graph cuts, only have good t... |

490 | BOBICK A.: Graphcut textures: Image and video synthesis using graph cuts.
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Citation Context ...re different pairings of adjacent labels can lead to different costs. This is important in a number of applications, ranging from stereo matching (§8.2 of [4]) to image stitching and texture quilting =-=[12,17,18]-=-. A more restricted form of the smoothness energy is Es = � {p,q}∈N wpq · V (|lp − lq|), where the smoothness terms are the product of spatially varying per-pairing weights wpq and a non-decreasing fu... |

489 | Convergent Tree-reweighted Message Passing for Energy Minimization”,
- Kolmogorov
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Citation Context ...rgy. The original TRW algorithm does not necessarily converge, and does not, in fact, guarantee that the lower bound always increases with time. In this paper we use an improved version of TRW due to =-=[23]-=-, which is called sequential TRW, or TRW-S. In this version, the lower bound estimate is guaranteed not to decrease, which results in certain convergence properties. In TRW-S we first select an arbitr... |

474 | Generalized belief propagation. In
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Citation Context ...cient. In the last few years, energy minimization approaches have had a renaissance, primarily due to powerful new optimization algorithms such as graph cuts [4, 5] and loopy belief propagation (LBP) =-=[6]-=-. The results, especially in stereo, have been dramatic; according to the widely-used Middlebury stereo benchmarks [7], almost all the top-performing stereo methods rely on graph cuts or LBP. Moreover... |

429 |
Exact maximum a posteriori estimation for binary images.
- Greig, Porteous, et al.
- 1989
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Citation Context ...of important special cases have fast exact algorithms. If there are only two labels, the natural Potts model smoothness cost can be solved exactly with graph cuts (this was first applied to images by =-=[19]-=-). If the labels are the integers starting with 0 and the smoothness cost is an arbitrary convex function, [20] gives a graph cut construction. An algorithm due to [21] can be used with V (∆l)=∆l (L1 ... |

317 | Multi-Camera Scene Reconstruction via Graph-Cuts,”
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Citation Context ...nsider. We hope to investigate different grid topologies (such as the 8-connected topology for 2D, or 26-connected for 3D), as well as non-local topologies such as those used with multiple depth maps =-=[43, 44]-=-. Finally, we will expand our set of benchmarks to include both more images and more applications, and release our benchmarks and algorithms on the Web for the use of the research community. Reference... |

296 | Correctness of belief propagation in Gaussian graphical models of arbitrary topology,”
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Citation Context ...cient. In the last few years, energy minimization approaches have had a renaissance, primarily due to powerful new optimization algorithms such as graph cuts [5, 6] and Loopy Belief Propagation (LBP) =-=[7, 8]-=-. The results, especially in stereo, have been dramatic; according to the widely-used Middlebury stereo benchmarks [9], almost all the top-performing stereo methods rely on graph cuts or LBP. Moreover... |

274 |
Optimization by Simulated Annealing. Science 220
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- 1983
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Citation Context ...elegance and power of the energy minimization approach, its early adoption was slowed by computational considerations. The algorithms that were originally used, such as ICM [1] or simulated annealing =-=[3, 4]-=-, proved to be extremely inefficient. In the last few years, energy minimization approaches have had a renaissance, primarily due to powerful new optimization algorithms such as graph cuts [5, 6] and ... |

241 | On the optimality of solutions of the max-product belief propagation algorithm in arbitrary graphs. - Weiss, Freeman - 2006 |

217 |
On the statistical analysis of dirty pictures (with discussions):
- Besag
- 1986
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Citation Context ...data, while the other term enforces spatial coherence. One of the reasons this framework is so popular is that it can be justified in terms of maximum a posteriori estimation of a Markov Random Field =-=[1, 2]-=-. Despite the elegance and power of the energy minimization approach, its early adoption was slowed by computational considerations. The algorithms that were originally used, such as ICM [1] or simula... |

216 | Exact optimization for markov random fields with convex priors.
- Ishikawa
- 2003
(Show Context)
Citation Context ...oothness cost can be solved exactly with graph cuts (this was first applied to images by [19]). If the labels are the integers starting with 0 and the smoothness cost is an arbitrary convex function, =-=[20]-=- gives a graph cut construction. An algorithm due to [21] can be used with V (∆l)=∆l (L1 smoothness) and convexsA Comparative Study of Energy Minimization Methods for MRF’s 19 data costs. However, the... |

210 | Markov random fields with efficient approximations,”
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Citation Context ...cost. This resulted in significantly better performance. 3.2 Graph cuts The two most popular graph cuts algorithms, called the swap move algorithm and the expansion move algorithm, were introduced in =-=[27, 5]-=-. These algorithms 5s6 rapidly compute a local minimum, in the sense that no “permitted move” will produce a labeling with lower energy. For a pair of labels α, β, a swap move takes some subset of the... |

207 | A pixel dissimilarity measure that is insensitive to image sampling,” - Birchfield, Tomasi - 1998 |

191 | MAP estimation via agreement on trees : Message-passing and linear programming. - Wainwright, Jaakkola, et al. - 2005 |

189 | Multi-view stereo via volumetric graph-cuts. - VOGIATZIS, TORR, et al. - 2005 |

181 |
Multiprocessor Scheduling with the Aid of Network Flow Algorithms,”
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Citation Context ...a number of important special cases have fast exact algorithms. If there are only two labels, the natural Potts model smoothness cost can be solved exactly with graph cuts (this was first observed by =-=[21]-=- in the context of a scheduling problem, and first applied to images by [22]). If the labels are the integers starting with 0 and the smoothness cost is an arbitrary convex function, [23] gives a grap... |

172 | Comparison of graph cuts with belief propagation for stereo, using identical mrf parameters. ICCV, Vol II:900-906, - Tappen, Freeman - 2003 |

172 | Optimizing binary mrfs via extended roof duality. - Rother, Kolmogorov, et al. - 2007 |

150 | A survey of very large-scale neighborhood search techniques,
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Citation Context ...p moves) are so strong that there are much fewer minima in high dimensional spaces compared to standard moves. (These algorithms are “very large neighborhood search techniques”, in the terminology of =-=[28]-=-.) In the original work of [5] the swap move algorithm was shown to be applicable to any energy where Vpq is a semi-metric, and the expansion move algorithm to any energy where Vpq is a metric. The re... |

147 | A.S.: Map estimation via agreement on (hyper)trees: Message-passing and linear-programming approaches.
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Citation Context ...ementation uses the distance transform method described in [22], which significantly reduces the running time of the algorithm. Tree-reweighted message passing (TRW) — Tree-reweighted message passing =-=[30]-=- is a message-passing algorithm similar, on the surface, to LBP. Let Mt p→q be the message that pixel p sends to its neighbor q at iteration t;thisisavector of size m (the number of labels). The messa... |

145 | A revolution: Belief propagation in graphs with cycles’,
- Frey, MacKay
- 1997
(Show Context)
Citation Context ...there is nothing in the formulation of BP that prevents it from being tried on graphs with loops. Indeed BP has been applied with great succes to loopy graphs first for error-correcting code problems =-=[31]-=- and then in early vision [32]. Detailed descriptions of the LBP algorithm can be found in [32] and [25]. In general, LPB is not guaranteed to converge, and may go into an infinite loop switching betw... |

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

116 |
The theory and practice of Bayesian image labeling.
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Citation Context ...Simultaneously, the range of applications of pixel labeling problems has also expanded dramatically, moving from early applications such as image restoration [1], texture modeling [8], image labeling =-=[9]-=-, and stereo matching [3, 4], to applications such as interactive photo segmentation [10, 11] and the automatic placement of seams in digital photomontages [12]. Relatively little attention has been p... |

102 | Exact Voxel Occupancy with Graph Cuts
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Citation Context ...ting certain terms, but it continues to work well in practice. Binary image segmentation — Binary MRF’s are also widely used in medical image segmentation [10], stereo matching using minimal surfaces =-=[27,28]-=-, and video segmentation using stereo disparity cues [29] As previously mentioned, for the natural Ising model smoothness costs, the global minimum can be computed rapidly via a graph cuts [19]; this ... |

84 |
Stochastic Stereo Matching over Scale”,
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Citation Context ...elegance and power of the energy minimization approach, its early adoption was slowed by computational considerations. The algorithms that were originally used, such as ICM [1] or simulated annealing =-=[3]-=-, proved to be extremely inefficient. In the last few years, energy minimization approaches have had a renaissance, primarily due to powerful new optimization algorithms such as graph cuts [4, 5] and ... |

83 |
Markov random field image models and their applications to computer vision. In:
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Citation Context ...reviously possible. Simultaneously, the range of applications of pixel labeling problems has also expanded dramatically, moving from early applications such as image restoration [1], texture modeling =-=[8]-=-, image labeling [9], and stereo matching [3, 4], to applications such as interactive photo segmentation [10, 11] and the automatic placement of seams in digital photomontages [12]. Relatively little ... |

78 | Bi-layer segmentation of binocular stereo video, in
- Kolmogorov, Criminisi, et al.
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Citation Context ...e. Binary image segmentation — Binary MRF’s are also widely used in medical image segmentation [10], stereo matching using minimal surfaces [27,28], and video segmentation using stereo disparity cues =-=[29]-=- As previously mentioned, for the natural Ising model smoothness costs, the global minimum can be computed rapidly via a graph cuts [19]; this result has been generalized to other smoothness costs by ... |

77 | Generalizing swendsen-wang to sampling arbitrary posterior probabilities
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Citation Context ...is probably room for improvement in our implementation of LBP, especially in terms of the schedule of message updates. We plan to implement several other modern algorithms, particularly Swendsen-Wang =-=[41]-=-, tree-based belief propagation [42], and the method of [40], along with simulated annealing. In addition, we plan to investigate the use of hierarchical methods, which have produced some quite promis... |

77 | P.H.: Dynamic graph cuts for efficient inference in Markov random fields. - Kohli, Torr - 2007 |

76 | Surfaces with occlusions from layered stereo. - Lin, Tomasi - 2002 |

74 | Panoramic video textures. - Agarwala, Zheng, et al. - 2005 |

72 | Approximate Labeling via Graph-Cuts Based on Linear Programming - Komodakis, Tziritas - 2007 |

67 | The direct cost of virtual functions calls in C - Driesen, Hölzle |

50 | An efficient algorithm for image segmentation, Markov random fields and related problems
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Citation Context ... was first applied to images by [19]). If the labels are the integers starting with 0 and the smoothness cost is an arbitrary convex function, [20] gives a graph cut construction. An algorithm due to =-=[21]-=- can be used with V (∆l)=∆l (L1 smoothness) and convexsA Comparative Study of Energy Minimization Methods for MRF’s 19 data costs. However, the NP-hardness result proved in [4] applies if there are mo... |

48 | Some network flow problems solved with pseudo-Boolean programming. - Hammer - 1965 |

47 | Bayesian 3D modeling from images using multiple depth maps.
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- 2005
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Citation Context ...nsider. We hope to investigate different grid topologies (such as the 8-connected topology for 2D, or 26-connected for 3D), as well as non-local topologies such as those used with multiple depth maps =-=[43, 44]-=-. Finally, we will expand our set of benchmarks to include both more images and more applications, and release our benchmarks and algorithms on the Web for the use of the research community. Reference... |

46 | Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation,” in
- Meltzer, Yanover, et al.
- 2005
(Show Context)
Citation Context ...t as an integer program, which can be solved as a linear program; if the linear program’s solutions happen to be integers, they are the global minimum. This is the basis for the approach was taken by =-=[31]-=-, who demonstrated that they could compute the global minimum for several common energy functions on the Middlebury images. The global minimum has only slightly lower energy than that produced by grap... |

45 | Probabilistic fusion of stereo with color and contrast for bilayer segmentation - Kolmogorov, Criminisi, et al. |

44 | Locally adapted hierarchical basis preconditioning. - Szeliski - 2006 |

44 | Graph cut based optimization for MRFs with truncated convex priors - Veksler - 2007 |

41 | Fast approximately optimal solutions for single and dynamic MRFs - Komodakis, Tziritas |

35 | A new framework for approximate labeling via graph cuts. ICCV
- Komodakis, Tziritas
- 2005
(Show Context)
Citation Context ...ent several other modern algorithms, as well as additional benchmarks. We are particularly interested in [31], whose method could potentially achieve the global minimum on some of our benchmarks, and =-=[32]-=-, who generalize the expansion move algorithm so that (like TRW) it also computes a lower bound on the energy. We also plan to increase the class of energy functions we consider. We hope to investigat... |

33 |
Tree-reweighted belief propagation and approximate ML estimation by pseudo-moment matching,”
- Wainwright, Jaakkola, et al.
- 2003
(Show Context)
Citation Context ...implementation uses the distance transform method described in [25], which significantly reduces the running time of the algorithm.s3.4 Tree-reweighted message passing Tree-reweighted message passing =-=[33]-=- is a message-passing algorithm similar, on the surface, to the loopy BP. Let M t p→q be the message that pixel p sends to its neighbor q at iteration t; this is a vector of size m (the number of labe... |

28 | C.: Comparison of energy minimization algorithms for highly connected graphs
- Kolmogorov, Rother
(Show Context)
Citation Context ...re a huge improvement over older techniques such as simulated annealing, less is known about the efficiency vs. accuracy tradeoff among more recently developed algorithms. Concurrently with our work, =-=[15]-=- compared tree-reweighted message passing, LBP and graph cuts for highly connected graphs. In this paper, we evaluate a number of different energy minimization algorithms for pixel labeling problems. ... |

8 |
et al. Interactive digital photomontage
- Agarwala
- 2004
(Show Context)
Citation Context ...1], texture modeling [8], image labeling [9], and stereo matching [3, 4], to applications such as interactive photo segmentation [10, 11] and the automatic placement of seams in digital photomontages =-=[12]-=-. Relatively little attention has been paid, however, to the relative performance of various optimization algorithms. Among the few exceptions are [14], which compared the efficiency of several differ... |

6 | Globally Optimal Solutions for Energy Minimization - Meltzer, Yanover, et al. - 2005 |

5 |
et al. Lazy snapping
- Li, Sun, et al.
- 2004
(Show Context)
Citation Context ...faster while achieving nearly the same level of performance. Our benchmark consists of a segmentation problem, inspired by the interactive segmentation algorithm of [12] or its more recent extensions =-=[13, 14]-=-. As with our Photomontage stitching example, this application requires user interaction; we handle this issue as above, by saving the user interactions to a file and using it to derive the data costs... |

4 | Minimal Surfaces for Stereo Vision - Buehler, Gortler, et al. - 2002 |

4 | A Revolution: Belief Propagation - Frey, MacKay - 1997 |

3 |
et al., “Panoramic video textures
- Agarwala, Zheng, et al.
- 2005
(Show Context)
Citation Context ...here are clear winners. On the Photomontage benchmark, expansion moves perform best, which provides some justification for the fact that this algorithm is used by various image stitching applications =-=[12, 26]-=-. On the stereo benchmark, the two best methods seem to be TRW-S and expansion moves. There are also some obvious paired comparisons; for instance, there never seems to be any reason to use swap moves... |

2 | Tree Consistency and Bounds on the - Wainwright, Jaakkola, et al. - 2004 |

1 |
et al.: Minimal surfaces for stereo vision
- Buehler
(Show Context)
Citation Context ...ting certain terms, but it continues to work well in practice. Binary image segmentation — Binary MRF’s are also widely used in medical image segmentation [10], stereo matching using minimal surfaces =-=[27,28]-=-, and video segmentation using stereo disparity cues [29] As previously mentioned, for the natural Ising model smoothness costs, the global minimum can be computed rapidly via a graph cuts [19]; this ... |

1 |
de Freitas, N.: From fields to trees
- Hamze
- 2004
(Show Context)
Citation Context ...our implementation of LBP, especially in terms of the schedule of message updates. We plan to implement several other modern algorithms, particularly Swendsen-Wang [41], tree-based belief propagation =-=[42]-=-, and the method of [40], along with simulated annealing. In addition, we plan to investigate the use of hierarchical methods, which have produced some quite promising recent results for LBP [25]. We ... |

1 | Richard Szeliski received the PhD degree in computer science from - Yedidia, Freeman, et al. - 2000 |