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## Fast approximate energy minimization via graph cuts (2001)

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### Other Repositories/Bibliography

Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 2098 - 61 self |

### Citations

5036 |
Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...hat do not have this problem are called discontinuity-preserving. A large number of discontinuity-preserving energy functions have been proposed (see for example [7]). Geman and Geman's seminal paper =-=[3]-=- gave aBayesian interpretation of many energy functions, and proposed a discontinuitypreserving energy function based on Markov Random Fields (MRF's). The major di culty with energy minimization for e... |

3873 | Snakes: Active contour models
- Kass, Witkin, et al.
- 1987
(Show Context)
Citation Context ...apidly computed via dynamic programming [2]. However, dynamic programming is restricted essentially to energy functions in one-dimensional settings. This includes some important cases, such as snakes =-=[26]-=-. In general, the two-dimensional energy functions that arise in early vision cannot be solved eciently via dynamic programming. Graph cut techniques from combinatorial optimization 5 can be used tosn... |

3724 | Normalized cuts and image segmentation
- Shi, Malik
- 2000
(Show Context)
Citation Context ...ent graph cut terminology. For example, [47] computes a globally minimum cut. The minimum is computed among all cuts that sever the graph into two non-empty parts. The terminals need not be specied. [=-=38-=-] introduces normalized cuts by proposing a new denition of the cut cost. Although normalized cuts are formulated as a graph partitioning problem the actual approximate optimization is performed via n... |

2365 | Determining optical flow
- Horn, Schunck
- 1981
(Show Context)
Citation Context ...p) 2 ,whereipis the observed intensity of the pixel p. The choice of Esmooth is a critical issue, and many different functions have been proposed. For example, in some regularization-based approaches =-=[18, 26], -=-Esmooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do 1 In our approach we assume that L is finite. 2 p∈Psnot have this problem are called disc... |

1984 |
Network Flows: Theory, Algorithms, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...he cut C, denoted jCj, equals the sum of its edge weights. The minimum cut problem is to nd the cut with smallest cost. There are many algorithms for this problem with low-order polynomial complexity =-=[1]-=-; in practice they run in near-linear time for our graphs. Step 3.1 uses a single minimum cut on a graph whose size is O(jPj). The graph is dynamically updated after each iteration. The details of thi... |

1601 |
Spatial interaction and the statistical analysis of lattice LIKELIHOODS 33 systems
- Besag
- 1974
(Show Context)
Citation Context ...set difference, fNp denotes all labels of sites in Np, andFdenotes the set of all possible labelings. The easiest way to specify an MRFs is by the joint distribution using Hammersley-Clifford theorem =-=[3]-=-. The theorem proves the equivalence between MRFs and Gibbs random fields. Before defining Gibbs random fields we need to define a clique. A set of sites is called a clique if each member of the set i... |

1291 | An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization
- Boykov, Kolmogorov
- 2004
(Show Context)
Citation Context ...n the terminals, according to a theorem due to Ford and Fulkerson [15]. Our experimental results make use of a new max-flow algorithm that has the best speed on our graphs over many modern algorithms =-=[10]-=-. The running time is nearly linear in practice. 4 FINDING THE OPTIMAL SWAP MOVE Given an input labeling f (partition P) and a pair of labels ; , we wish to find a labeling ^ f that minimizes E over a... |

1078 | Cognitive Networks
- Thomas, DaSilva, et al.
- 2005
(Show Context)
Citation Context ...rithms with dierent low-order polynomial complexities [1]. For example, a minimum cut can be found by computing the maximumsow between the terminals, according to a theorem due to Ford and Fulkerson [=-=15]-=-. Our experimental results make use of a new max- ow algorithm that has the best speed on our graphs over many modern algorithms [10]. The running time is nearly linear in practice. 6 To avoid confusi... |

885 | Visual Reconstruction
- Blake, Zisserman
- 1987
(Show Context)
Citation Context ...tually more general than energy minimization methods, see [23] and [32]. There are also methods that have optimality guarantees in certain cases. Continuation methods, such as graduated non-convexity =-=[8]-=-, are an example. These methods involve approximating an intractable (non-convex) energy function by a sequence of energy functions, beginning with a tractable (convex) approximation. There are circum... |

505 | Markov Random Field Modeling in Computer Vision
- Li
- 1995
(Show Context)
Citation Context ...nsion jPj, which is many thousands. The energy functions that we consider in this paper arise in a variety of dierent contexts, including the Bayesian labeling ofsrst-order Markov Random Fields (see [=-=30]-=- for details). We consider energies of the form E(f) = X fp;qg2N V p;q (f p ; f q ) + X p2P D p (f p ); (1) where N is the set of interacting pairs of pixels. Typically N consists of adjacent pixels, ... |

427 |
Exact maximum a posteriori estimation for binary images
- Greig, Porteous, et al.
- 1989
(Show Context)
Citation Context ...al cases such energies can be minimized exactly. If the number of possible labels is jLj = 2 then the exact solution can be found in polynomial time by computing a minimum cost cut on a certain graph =-=[4]-=-. If L is a nite 1D set and the interaction potential is V (fp;fq) =jfp ,fqj then the exact minimum can also be found e ciently via graph cuts [5, 2]. In general, however, the problem is NP-hard [8]. ... |

408 |
Scene labeling by relaxation operations
- Rosenfeld, Hummel, et al.
- 1976
(Show Context)
Citation Context ...m. 4 To apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including =-=[12, 36, 41]-=-. In relaxation labeling, combinatorial optimization is converted into continuous optimization with linear constraints. Then some form of gradient descent which gives the solution satisfying the const... |

367 |
Nonuniversal critical dynamics in Monte Carlo simulations
- Swendsen, Wang
- 1987
(Show Context)
Citation Context ... of simulated annealing give results that are very far from the global optimum even in the relatively simple case of binary labelings. Trying to improve the rate of convergence of simulated annealing =-=[3-=-9, 3] developed sampling algorithms for the Potts model that can make larger moves similar to our --swaps. The main dierence is that wesnd the best move among all possible --swaps, while [39, 3] rando... |

360 | Computing Visual Correspondence with Occlusions using Graph Cuts
- Kolmogorov, Zabih
- 2001
(Show Context)
Citation Context ...ch does not treat the images symmetrically and allows inconsistent disparities. For example, two pixels in thesrst image may be assigned to one pixel in the second image. Occlusions are also ignored. =-=[28-=-] presents a stereo algorithm based on expansion moves that addresses these problems. 25 data terms D p (f p ) in Section 8.1. We use dierent interactions V p;q (f p ; f q ) and we state them for each... |

353 |
Using dynamic programming for solving variational problems in vision
- Amini, Weymouth, et al.
- 1990
(Show Context)
Citation Context ...anseld approximation and other minimization methods like graduated non-convexity. There are a few interesting energy functions where the global minimum can be rapidly computed via dynamic programming =-=[2]-=-. However, dynamic programming is restricted essentially to energy functions in one-dimensional settings. This includes some important cases, such as snakes [26]. In general, the two-dimensional energ... |

349 | An optimal graph theoretic approach to data clustering: theory and its application to image segmentation
- Wu, Leahy
- 1993
(Show Context)
Citation Context ...importantly they require V to be convex [25], and hence their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering =-=[47, 16, 44-=-]. Unlike clustering, we assume that there is a natural set of labels (e.g. intensities or disparities), and a data penalty function D p () which makes some pixel-label assignments more likely than ot... |

297 |
On the foundations of relaxation labeling processes
- Hummel, Zucker
- 1983
(Show Context)
Citation Context ...traints. Then some form of gradient descent which gives the solution satisfying the constraints is used. Relaxation labeling techniques are actually more general than energy minimization methods, see =-=[23]-=- and [32]. There are also methods that have optimality guarantees in certain cases. Continuation methods, such as graduated non-convexity [8], are an example. These methods involve approximating an in... |

292 |
Statistical field theory
- Parisi
- 1988
(Show Context)
Citation Context ...estimating the partition function from which the minimum of the energy can be deduced. However computing the partition function is computationally intractable, and saddle point approximations (Parisi =-=[24]-=-) are used. Geiger and Yuille [14] provide an interesting connection between mean field approximation and other minimization methods like graduated non-convexity. There are a few interesting energy fu... |

279 |
Computational vision and regularization theory
- Poggio, Torre, et al.
- 1985
(Show Context)
Citation Context ... 2 , where ip is the observed intensity of the pixel p. The choice of Esmooth is a critical issue, and many di erent functions have been proposed. For example, in standard regularization-based vision =-=[6]-=-, Esmooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity-preserving. A large number of discontinuit... |

276 |
Regularization of inverse visual problems involving discontinuities
- Terzopoulos
- 1986
(Show Context)
Citation Context ... object boundaries. Energy functions that do not have this problem are called discontinuity-preserving. A large number of discontinuity-preserving energy functions have been proposed (see for example =-=[7]-=-). Geman and Geman's seminal paper [3] gave aBayesian interpretation of many energy functions, and proposed a discontinuitypreserving energy function based on Markov Random Fields (MRF's). The major d... |

265 |
Image Analysis, Random Fields and Dynamic Monte Carlo Methods, volume 27 of Appl. of Mathematics
- Winkler
- 1995
(Show Context)
Citation Context ...ties. Informally, a discontinuity preserving interaction term should have a bound on the largest possible penalty. This avoids overpenalizing sharp jumps between the labels of neighboring pixels; see =-=[46, 3-=-0] and our experimental results in Section 8.6. Examples of discontinuity preserving interaction penalties for a one-dimensional label set L include the truncated quadratic V (;s) = min(K; jsj 2 ) (a ... |

258 | A maximum-flow formulation of the n-camera stereo correspondence problem
- Roy, Cox
- 1998
(Show Context)
Citation Context ...ation of the Ising model to the case of more than 2 labels. [14] develop a method optimal to within a factor of two for the Potts model; however the data energy they use is very restrictive. Recently =-=[37, 24, 11]-=- used graph cuts tosnd the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly they require V to be convex... |

210 |
On the statistical analysis of dirty pictures (with discussion
- Besag
- 1986
(Show Context)
Citation Context ...is not necessarily the desired solution for image restoration. [9, 20] analyze the performance of simulated annealing in cases with a known global minimum. 4 which is a greedy technique introduced in =-=[4]-=-. For each pixel, the label which gives the largest decrease of the energy function is chosen, until convergence to a local minimum. Another example of an algorithm using standard moves is simulated a... |

209 | Markov random fields with efficient approximations
- Boykov, Zabih
- 1998
(Show Context)
Citation Context ...on of the Ising model. Ferrari et al. [12] develop a method optimal to within a factor of two for the Potts model energy function; however the data energy they use is very restrictive. Recently [19], =-=[9]-=-, and [29] used graph cuts to find the exact global minimum of a certain type of energy functions. However, these energy functions apply only if the labels are one-dimentional which rules out motion e... |

206 | A pixel dissimilarity measure that is insensitive to image sampling
- Birchfield, Tomasi
- 1998
(Show Context)
Citation Context ...l accuracy, as we do here. If a pixel overlaps a scene patch with high intensity gradient, then the corresponding pixels may have signicantly dierent intensities. For stereo we use the technique of [6] to develop a D p that is insensitive to image sampling. First we measure how well psts into the real valued range of disparities (d 1 2 ; d + 1 2 ) by C fwd (p; d) = min d 1 2 xd+ 1 2 jI p I 0 p+x ... |

202 |
Some generalized order-disorder transformations
- Potts
- 1952
(Show Context)
Citation Context ...te a local minimum whose energy is within a known factor from the global minimum. 7 The Potts model An interesting special case of the energy in equation (1) arises when V is given by the Potts model =-=[35-=-] E P (f) = X fp;qg2N u fp;qg T (f p 6= f q ) + X p2P D p (f p ): (18) Geman et al. [18] were thesrst to use this model in computer vision. In this case, discontinuities between any pair of labels ar... |

202 | Bayesian Modeling of Uncertainty in Low-Level Vision
- Szeliski
- 1989
(Show Context)
Citation Context ...m. 4 To apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including =-=[12, 36, 41]-=-. In relaxation labeling, combinatorial optimization is converted into continuous optimization with linear constraints. Then some form of gradient descent which gives the solution satisfying the const... |

194 | Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields
- Kleinberg, Tardos
- 1999
(Show Context)
Citation Context ...(f ∗ ) appears 2c times on the right side of (13). Therefore equation (14) can be rewritten to get the bound of 2c: E( ˆ f)+EB( ˆ f)≤E(f ∗ )+(2c−1)EB(f ∗ ) ≤ 2cE(f ∗ ). Note that Klein=-=berg and Tardos [22]-=- develop an algorithm for minimizing E which also has optimality properties. For the Potts model V discussed in the next section, their algorithm has a bound of 2, which is the same bound as we have. ... |

186 | Depth discontinuities by pixelto-pixel stereo
- Birchfield, Tomasi
- 1998
(Show Context)
Citation Context ... are much more likely to have the same disparity if we know that I(p) I(q). Most methods for computing correspondence do not make use of this kind of contextual information. Some exceptions include [=-=5, 33, 45]-=-. We can easily incorporate contextual information into our framework by allowing u fp;qg to vary depending on the intensities I p and I q . Let u fp;qg = U(jI p I q j): (19) Each u fp;qg represents a... |

176 | A unified mixture framework for motion segmentation: incorporating spatial coherence and estimating the number of models
- Weiss, Adelson
- 1996
(Show Context)
Citation Context ... are much more likely to have the same disparity if we know that I(p) I(q). Most methods for computing correspondence do not make use of this kind of contextual information. Some exceptions include [=-=5, 33, 45]-=-. We can easily incorporate contextual information into our framework by allowing u fp;qg to vary depending on the intensities I p and I q . Let u fp;qg = U(jI p I q j): (19) Each u fp;qg represents a... |

145 |
Boundary detection by constrained optimization
- Geman, Geman, et al.
- 1990
(Show Context)
Citation Context ...otts model An interesting special case of the energy in equation (1) arises when V is given by the Potts model [35] E P (f) = X fp;qg2N u fp;qg T (f p 6= f q ) + X p2P D p (f p ): (18) Geman et al. [=-=18]-=- were thesrst to use this model in computer vision. In this case, discontinuities between any pair of labels are penalized equally. This is in some sense the simplest discontinuity preserving model an... |

128 |
Computational vision and regularization theory. Nature 317
- Poggio, Torre, et al.
- 1985
(Show Context)
Citation Context ... p I p ) 2 , where I p is the observed intensity of p. The choice of E smooth is a critical issue, and many dierent functions have been proposed. For example, in some regularization-based approaches [=-=22, 34]-=-, E smooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinui... |

116 |
The theory and practice of Bayesian image labeling
- Chou, Brown
- 1990
(Show Context)
Citation Context ...m. 4 To apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including =-=[12, 36, 41]-=-. In relaxation labeling, combinatorial optimization is converted into continuous optimization with linear constraints. Then some form of gradient descent which gives the solution satisfying the const... |

111 | Efficient Graph-based Energy Minimization Methods in Computer Vision
- Veksler
- 1999
(Show Context)
Citation Context ...d to terminate in asnite number of cycles. In fact, under the assumptions that V and D p in equation (1) are constants independent of the image size P we can easily prove termination in O(jPj) cycles =-=[43]-=-. These assumptions are quite reasonable in practice. However, in the experiments we report in Section 8, the algorithm stops after a few cycles, and most of the improvements occur during thesrst cycl... |

105 | Segmentation by grouping junctions
- Ishikawa, Geiger
- 1998
(Show Context)
Citation Context ...me by computing a minimum cost cut on a certain graph [4]. If L is a nite 1D set and the interaction potential is V (fp;fq) =jfp ,fqj then the exact minimum can also be found e ciently via graph cuts =-=[5, 2]-=-. In general, however, the problem is NP-hard [8]. In this paper we develop algorithms that approximately minimize energy E(f) for an arbitrary nite set of labels L under two fairly general classes of... |

102 | Occlusions, discontinuities, and epipolar lines in stereo
- Ishikawa, Geiger
- 1998
(Show Context)
Citation Context ...ation of the Ising model to the case of more than 2 labels. [14] develop a method optimal to within a factor of two for the Potts model; however the data energy they use is very restrictive. Recently =-=[37, 24, 11]-=- used graph cuts tosnd the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly they require V to be convex... |

97 |
A common framework for image segmentation
- Geiger, Yuille
- 1991
(Show Context)
Citation Context ...ng the partition function, from which the minimum of the energy can be deduced. However computing the partition function is computationally intractable, and saddle point approximations [31] are used. =-=[17]-=- provides an interesting connection between meanseld approximation and other minimization methods like graduated non-convexity. There are a few interesting energy functions where the global minimum ca... |

87 | Selforganization in vision: Stochastic clustering for image segmentation, perceptual grouping, and image database organization
- Gdalyahu, Weinshall, et al.
(Show Context)
Citation Context ...importantly they require V to be convex [25], and hence their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering =-=[47, 16, 44-=-]. Unlike clustering, we assume that there is a natural set of labels (e.g. intensities or disparities), and a data penalty function D p () which makes some pixel-label assignments more likely than ot... |

68 |
The complexity of multiway cuts
- Dahlhaus, Johnson, et al.
- 1992
(Show Context)
Citation Context ...r subset of C separates the terminals in G(C). The cost of the multiway cut C is denoted by jCj and equals the sum of its edge weights. The multiway cut problem is tosnd the minimum cost multiway cut =-=[13]-=-. In [13] they also show 22 that the multiway cut problem is NP-complete. Note that the multiway cut problem is a generalization of the standard two-terminal graph cut problem described in Section 3.3... |

62 |
Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems, Theor
- Kant, He
- 1997
(Show Context)
Citation Context ...bed G in a grid of pixels, which happens in two stages. In the first stage we convert G into a planar graph of degree 4. In the second stage we embed this graph in the grid by using a method given in =-=[20]. -=-This embedding can be done in polynomial time; after it is done, each vertex v ∈Gcorresponds to a connected set of pixels S(v) in the grid, and the adjacency relationships among vertices in G has be... |

61 |
Determining optical
- Horn, Schunck
- 1981
(Show Context)
Citation Context ... p I p ) 2 , where I p is the observed intensity of p. The choice of E smooth is a critical issue, and many dierent functions have been proposed. For example, in some regularization-based approaches [=-=22, 34]-=-, E smooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinui... |

61 |
Comparison of the efficiency of deterministic and stochastic algorithms for visual reconstruction
- Blake
- 1989
(Show Context)
Citation Context ...idly computed, it is possible to separate these issues. For example, [16] points out that the global minimum of an Ising energy function is not necessarily the desired solution for image restoration. =-=[8, 16]-=- analyze the performance of simulated annealing in cases with a known global minimum. 5sdescent which gives the solution satisfying the constraints is used. 3 Overview of our algorithms The most impor... |

52 |
Organizational complexity
- Dooley
- 2002
(Show Context)
Citation Context ...subset of C separates the terminals in G…C†. The cost of the multiway cut C is denoted by jCj and equals the sum of its edge weights. The multiway cut problem is to find the minimum cost multiway cut =-=[13]-=-. In [13], they also show that the multiway cut problem is NP-complete. Note that the multiway cut problem is a generalization of the standard two-terminal graph cut problem described in Section 3.3. ... |

51 | An experimental comparison of stereo algorithms
- Szeliski, Zabih
- 1999
(Show Context)
Citation Context ...rithms for K = 20, where K is the parameter in equation (20). Figs. 10(e) and (f) show the results of normalized correlation and simulated annealing. Comparisons with other algorithms can be found in =-=[40-=-]. Note, however, that [40] conrms that for this imagery the best previous algorithm is simulated annealing, which outperforms (among others) correlation, robust estimation, scanline-based dynamic pro... |

47 |
Parallel integration of vision modules
- Poggio, Gamble, et al.
- 1988
(Show Context)
Citation Context ... are much more likely to have the same disparity if we know that I(p) I(q). Most methods for computing correspondence do not make use of this kind of contextual information. Some exceptions include [=-=5, 33, 45]-=-. We can easily incorporate contextual information into our framework by allowing u fp;qg to vary depending on the intensities I p and I q . Let u fp;qg = U(jI p I q j): (19) Each u fp;qg represents a... |

39 | The dynamics of nonlinear relaxation labeling processes
- Pelillo
- 1997
(Show Context)
Citation Context ...Then some form of gradient descent which gives the solution satisfying the constraints is used. Relaxation labeling techniques are actually more general than energy minimization methods, see [23] and =-=[32]-=-. There are also methods that have optimality guarantees in certain cases. Continuation methods, such as graduated non-convexity [8], are an example. These methods involve approximating an intractable... |

35 | Image segmentation by nested cuts
- Veksler
- 2000
(Show Context)
Citation Context ...importantly they require V to be convex [25], and hence their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering =-=[47, 16, 44-=-]. Unlike clustering, we assume that there is a natural set of labels (e.g. intensities or disparities), and a data penalty function D p () which makes some pixel-label assignments more likely than ot... |

25 | Depth and Motion Discontinuities
- Birchfield
- 1999
(Show Context)
Citation Context ...ively. The results in Fig. 19(b,c) were histogram equalized to reveal oversmoothing in (c), which does not happen in (b). Similar oversmoothing for the absolute difference model occurs in stereo, see =-=[43, 7]-=-. 9 Conclusions We consider a wide class of energy functions with various discontinuity preserving smoothness constraints. While it is NP-hard to compute the exact minimum, we developed two algorithms... |

21 | Global Optimization Using Embedded Graphs
- ISHIKAWA
(Show Context)
Citation Context ...used graph cuts tosnd the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly they require V to be convex =-=[25]-=-, and hence their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering [47, 16, 44]. Unlike clustering, we assume t... |

15 |
Markov random with ecient approximations
- Boykov, Veksler, et al.
(Show Context)
Citation Context ...ation of the Ising model to the case of more than 2 labels. [14] develop a method optimal to within a factor of two for the Potts model; however the data energy they use is very restrictive. Recently =-=[37, 24, 11]-=- used graph cuts tosnd the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly they require V to be convex... |

15 | Fast approximate maximum a posteriori restoration of multicolour images
- Ferrari, Frigessi, et al.
- 1995
(Show Context)
Citation Context ... the global minimum in this case by a single graph cut computation. Note that the Potts model we discuss in Section 7 is a natural generalization of the Ising model to the case of more than 2 labels. =-=[14]-=- develop a method optimal to within a factor of two for the Potts model; however the data energy they use is very restrictive. Recently [37, 24, 11] used graph cuts tosnd the exact global minimum of a... |

11 |
Spin-Glass Theory and
- Mézard, Parisi, et al.
- 1987
(Show Context)
Citation Context ...sed on estimating the partition function, from which the minimum of the energy can be deduced. However computing the partition function is computationally intractable, and saddle point approximations =-=[31]-=- are used. [17] provides an interesting connection between meanseld approximation and other minimization methods like graduated non-convexity. There are a few interesting energy functions where the gl... |

10 |
Approximation algorithms for classi problems with pairwise relationships: Metric labeling and markov random
- Kleinberg, Tardos
- 2000
(Show Context)
Citation Context ...pears 2c times on the right side of (17). Therefore equation (17) can be rewritten to get the bound of 2c: E( ^ f) + E( ^ f jB) E(f ) + (2c 1)EB (f ) 2cE(f ): Note that Kleinberg and Tardos [27] develop an algorithm for minimizing E which also has optimality properties. For the Potts model V discussed in the next section, their algorithm has a bound of 2. This is the same bound as we obtain ... |

10 |
ªSnakes: Active Contour Models,º Int'l
- Kass, Witkin, et al.
- 1988
(Show Context)
Citation Context ...apidly computed via dynamic programming [2]. However, dynamic programming is restricted essentially to energy functions in one-dimensional settings. This includes some important cases, such as snakes =-=[26]-=-. In general, the two-dimensional energy functions that arise in early vision cannot be solved efficiently via dynamic programming. 4. Note that in continuous cases, the labels near to f in (5) are no... |

9 |
Comparison of the eciency of deterministic and stochastic algorithms for visual reconstruction
- Blake
- 1989
(Show Context)
Citation Context ...it is possible to separate these issues. For example, [20] points out that the global minimum of a special case of Ising energy function is not necessarily the desired solution for image restoration. =-=[9, 20]-=- analyze the performance of simulated annealing in cases with a known global minimum. 4 which is a greedy technique introduced in [4]. For each pixel, the label which gives the largest decrease of the... |

9 |
One dimensional regularization with discontinuities
- Lee, Pavlidis
- 1988
(Show Context)
Citation Context ... object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinuity preserving energy functions have been proposed (see for example =-=[21, 29, 42]-=-). The major diculty with energy minimization lies in the enormous computational costs. Typically these energy functions have many local minima (i.e., they are non-convex). Worse still, the space of p... |

8 |
ªDetermining Optical Flow,º
- Horn, Schunck
- 1981
(Show Context)
Citation Context ...rmally fp Ip , where Ip is the observed intensity of p. The choice of Esmooth is a critical issue and many different functions have been proposed. For example, in some regularization-based approaches =-=[22]-=-, [34], Esmooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discon... |

6 |
An experimental comparison of min-cut/max- algorithms for energy minimization in vision
- Boykov, Kolmogorov
(Show Context)
Citation Context ...en the terminals, according to a theorem due to Ford and Fulkerson [15]. Our experimental results make use of a new max- ow algorithm that has the best speed on our graphs over many modern algorithms =-=[10-=-]. The running time is nearly linear in practice. 6 To avoid confusion, we would like to mention that some clustering based segmentation techniques in vision use dierent graph cut terminology. For exa... |

5 |
ªOn the Statistical Analysis of Dirty Pictures,º
- Besag
- 1986
(Show Context)
Citation Context ...der the local minimum with respect to standard moves shown in Fig. 1c. An example of a local method using standard moves is Iterated Conditional Modes (ICM), which is a greedy technique introduced in =-=[4]-=-. For each pixel, the label which gives the largest decrease of the energy function is chosen, until convergence to a local minimum. Another example of an algorithm using standard moves is simulated a... |

5 |
ªNormalized Cuts and Image Segmentation,º
- Shi, Malik
- 2000
(Show Context)
Citation Context ...cut terminology. For example, [47] computes a globally minimum cut. The minimum is computed among all cuts that sever the graph into two nonempty parts. The terminals need not be specified. Recently, =-=[38]-=- introduced normalized cuts by proposing a new definition of the cut cost. Although normalized cuts are formulated as a graph partitioning problem, the actual approximate optimization is performed via... |

4 |
Unsupervised Image Segmentation
- Barker, Rayner
- 1998
(Show Context)
Citation Context ... of simulated annealing give results that are very far from the global optimum even in the relatively simple case of binary labelings. Trying to improve the rate of convergence of simulated annealing =-=[3-=-9, 3] developed sampling algorithms for the Potts model that can make larger moves similar to our --swaps. The main dierence is that wesnd the best move among all possible --swaps, while [39, 3] rando... |

4 |
Computng visual correspondence with occlusions via graph cuts
- Kolmogorov, Zabih
- 2001
(Show Context)
Citation Context ... does not treat the images symmetrically and allows inconsistent disparities. For example, two pixels in the first image may be assigned to one pixel in the second image. Occlusions are also ignored. =-=[28]-=- presents a stereo algorithm based on expansion moves that addresses these problems. our data terms Dp…fp† in Section 8.1. We use different interactions Vp;q…fp;fq† and we state them for each example.... |

3 |
Grimson and Theo Pavlidis. Discontinuity detection for visual surface reconstruction
- L
- 1985
(Show Context)
Citation Context ... object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinuity preserving energy functions have been proposed (see for example =-=[21, 29, 42]-=-). The major diculty with energy minimization lies in the enormous computational costs. Typically these energy functions have many local minima (i.e., they are non-convex). Worse still, the space of p... |

3 |
ªMarkov Random Fields with Efficient Approximations,º
- Boykov, Veksler, et al.
- 1998
(Show Context)
Citation Context ...l to the case of more than two labels. A method optimal to within a factor of two for the Potts model was developed in [14]; however, their energy data term is very restrictive. Recently, [37], [24], =-=[11]-=- used graph cuts to find the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly, they require V to be con... |

3 |
ªBoundary Detection by Constrained Optimization,º
- Geman, Geman, et al.
- 1990
(Show Context)
Citation Context ...he global minimum. 7 THE POTTS MODEL An interesting special case of the energy in (1) arises when V is given by the Potts model [35] EP …f† ˆ X ufp;qg T…fp 6ˆ fq†‡ X Dp…fp†: …18† fp;qg2N Geman et al. =-=[18]-=- were the first to use this model in computer vision. In this case, discontinuities between any pair of labels are penalized equally. This is, in some sense, the simplest discontinuity preserving mode... |

3 |
ªComputational Vision and Regularization Theory,º Nature
- Poggio, Torre, et al.
- 1985
(Show Context)
Citation Context ... fp Ip , where Ip is the observed intensity of p. The choice of Esmooth is a critical issue and many different functions have been proposed. For example, in some regularization-based approaches [22], =-=[34]-=-, Esmooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinuit... |

2 |
Depth and Motion Discontinuities
- Birch
- 1999
(Show Context)
Citation Context ...tively. The results in Fig. 19(b,c) were histogram equalized to reveal oversmoothing in (c), which does not happen in (b). Similar oversmoothing for the absolute dierence model occurs in stereo, see [=-=43, 7]-=-. 9 Conclusions We consider a wide class of energy functions with various discontinuity preserving smoothness constraints. While it is NP-hard to compute the exact minimum, we developed two algorithms... |

2 |
ªA Common Framework for Image Segmentation,º Int'l
- Geiger, Yuille
- 1991
(Show Context)
Citation Context ...onally intractable, and saddle point approximations [31] are used. There is also and interesting connection between mean field approximation and other minimization methods like graduated nonconvexity =-=[17]-=-. There are a few interesting energy functions where the global minimum can be rapidly computed via dynamic programming [2]. However, dynamic programming is restricted essentially to energy functions ... |

2 |
ªOn the Foundations of Relaxation Labeling Processes,º
- Hummel, Zucker
- 1983
(Show Context)
Citation Context ...raints. Then, some form of gradient descent which gives the solution satisfying the constraints is used. Relaxation labeling techniques are actually more general than energy minimization methods, see =-=[23]-=- and [32]. There are also methods that have optimality guarantees in certain cases. Continuation methods, such as graduated nonconvexity [8], are an example. These methods involve approximating an int... |

2 |
ªScene Labeling by Relaxation Operations,º
- Rosenfeld, Hummel, et al.
- 1976
(Show Context)
Citation Context ...o apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including [12], =-=[36]-=-, [41]. In relaxation labeling, combinatorial optimization is converted into continuous optimization with linear constraints. Then, some form of gradient descent which gives the solution satisfying th... |

2 |
ªNonuniversal Critical Dynamics in Monte Carlo Simulation,º
- Swendsen, Wang
- 1987
(Show Context)
Citation Context ... of simulated annealing give results that are very far from the global optimum even in the relatively simple case of binary labelings. Trying to improve the rate of convergence of simulated annealing =-=[39]-=-, [3] developed sampling algorithms for the Potts model that can make larger moves similar to our - -swaps. The main difference is that we find the best move among all possible - -swaps, while [39], [... |

1 |
Markov random elds with e cient approximations
- Boykov, Veksler, et al.
- 1998
(Show Context)
Citation Context ...me by computing a minimum cost cut on a certain graph [4]. If L is a nite 1D set and the interaction potential is V (fp;fq) =jfp ,fqj then the exact minimum can also be found e ciently via graph cuts =-=[5, 2]-=-. In general, however, the problem is NP-hard [8]. In this paper we develop algorithms that approximately minimize energy E(f) for an arbitrary nite set of labels L under two fairly general classes of... |

1 |
cient Graph-based Energy Minimization Methods in Computer Vision
- unknown authors
- 1999
(Show Context)
Citation Context ...h [4]. If L is a nite 1D set and the interaction potential is V (fp;fq) =jfp ,fqj then the exact minimum can also be found e ciently via graph cuts [5, 2]. In general, however, the problem is NP-hard =-=[8]-=-. In this paper we develop algorithms that approximately minimize energy E(f) for an arbitrary nite set of labels L under two fairly general classes of interaction potentials V : semi-metric and metri... |

1 |
ªUnsupervised Image Segmentation,º
- Barker, Rayner
- 1998
(Show Context)
Citation Context ...mulated annealing give results that are very far from the global optimum even in the relatively simple case of binary labelings. Trying to improve the rate of convergence of simulated annealing [39], =-=[3]-=- developed sampling algorithms for the Potts model that can make larger moves similar to our - -swaps. The main difference is that we find the best move among all possible - -swaps, while [39], [3] ra... |

1 |
ªDepth Discontinuities by Pixel-toPixel Stereo,º Int'l
- Birchfield, Tomasi
- 1999
(Show Context)
Citation Context ... q are much more likely to have the same disparity if we know that I…p† I…q†. Most methods for computing correspondence do not make use of this kind of contextual information. Some exceptions include =-=[5]-=-, [33], [45]. We can easily incorporate contextual information into our framework by allowing u fp;qg to vary depending on the intensities Ip and Iq. Let u fp;qg ˆ U…jIp Iqj†: …19† Each u fp;qg repres... |

1 |
ªDepth and Motion Discontinuities,º
- Birchfield
- 1999
(Show Context)
Citation Context ... Figs. 19b and 19c were histogram equalized to reveal oversmoothing in Fig. 19c, which does not happen in Fig. 19b. Similar oversmoothing for the absolute difference model occurs in stereo, see [43], =-=[7]-=-.s16 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 11, NOVEMBER 2001 Fig. 17. Flower garden sequence. (a) First image, 352x240, 8 labels. (b) Horizontal movement. Fig. 1... |

1 |
SaÂ, ªFast Approximate Maximum A Posteriori Restoration of Multicolour Images,º
- Ferrari, Frigessi, et al.
- 1995
(Show Context)
Citation Context ...Potts model we discuss in Section 7 is a natural generalization of the Ising model to the case of more than two labels. A method optimal to within a factor of two for the Potts model was developed in =-=[14]-=-; however, their energy data term is very restrictive. Recently, [37], [24], [11] used graph cuts to find the exact global minimum of a certain type of energy functions. However, these methods apply o... |

1 |
ªDiscontinuity Detection for Visual Surface Reconstruction,º Computer Vision
- Grimson, Pavlidis
- 1985
(Show Context)
Citation Context ...bject boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinuity preserving energy functions have been proposed (see, for example, =-=[21]-=-, [29], [42]). The major difficulty with energy minimization lies in the enormous computational costs. Typically, these energy functions have many local minima (i.e., they are nonconvex). Worse still,... |

1 |
ªGlobal Optimization Using Embedded Graphs,º
- Ishikawa
- 2000
(Show Context)
Citation Context ...d graph cuts to find the exact global minimum of a certain type of energy functions. However, these methods apply only if the labels are one-dimensional. Most importantly, they require V to be convex =-=[25]-=- and, hence, their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering [47], [16], [44]. Unlike clustering, we ass... |

1 |
ªParallel Integration of Vision Modules,º
- Poggio, Gamble, et al.
- 1988
(Show Context)
Citation Context ...e much more likely to have the same disparity if we know that I…p† I…q†. Most methods for computing correspondence do not make use of this kind of contextual information. Some exceptions include [5], =-=[33]-=-, [45]. We can easily incorporate contextual information into our framework by allowing u fp;qg to vary depending on the intensities Ip and Iq. Let u fp;qg ˆ U…jIp Iqj†: …19† Each u fp;qg represents a... |

1 |
ªSome Generalized Order-Disorder Transformation,º Proc
- Potts
- 1952
(Show Context)
Citation Context ...so generate a local minimum whose energy is within a known factor from the global minimum. 7 THE POTTS MODEL An interesting special case of the energy in (1) arises when V is given by the Potts model =-=[35]-=- EP …f† ˆ X ufp;qg T…fp 6ˆ fq†‡ X Dp…fp†: …18† fp;qg2N Geman et al. [18] were the first to use this model in computer vision. In this case, discontinuities between any pair of labels are penalized equ... |

1 |
ªImage Segmentation by Nested Cuts,º
- Veksler
- 2000
(Show Context)
Citation Context ...hey require V to be convex [25] and, hence, their energies are not discontinuity preserving, see Section 8.6. Note that graph cuts have also been used for segmentation based on clustering [47], [16], =-=[44]-=-. Unlike clustering, we assume that there is a natural set of labels (e.g., intensities or disparities), and a data penalty function Dp… † which makes some pixel-label assignments more likely than oth... |