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..., please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-0097-0205. outside this class, then one has to use other techniques such as max-product belief propagation (BP) [6], =-=[22]-=-, [27], [35]. BP can be applied to any function of the form as in (1), but it has some drawbacks. First, it usually finds a solution with higher energy than graph cuts (in cases when graph cuts can be...

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...tensities. defines parameters of the energy: sð Þ is a unary data penalty function and stð ; Þ is a pairwise interaction potential. This energy is often derived in the context of Markov Random Fields =-=[8]-=-: A minimum of E corresponds to a maximum a posteriori (MAP) labeling x. In general, minimizing E is an NP-hard problem, so researchers have focused on approximate minimization algorithms. The two mos...

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...-hard problem, so researchers have focused on approximate minimization algorithms. The two most well-known techniques are graph cuts and belief propagation. The former one was introduced in the 1990s =-=[3]-=-, [9], [10], [15] and showed a major improvement over previously used simulated annealing [8]. To our knowledge, graph cuts are currently considered to be the most accurate minimization algorithm for ...

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...ing problem with Potts interactions [3]. The input is two images taken from different viewpoints and the goal is to find a disparity for every pixel in the left image. We used the four data sets from =-=[23]-=-—Tsukuba, Map, Sawtooth, and Venus. Fig. 5 shows the energy plots and Fig. 6 shows disparity maps for one of the data sets (Tsukuba). In addition to message passing techniques, we included the results...

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...1:07 105 nodes. Energy plots are shown in Fig. 5. The average running time for the TRW-S algorithm was 0.067 secs per iteration on a Pentium IV 2.8 GHz processor. For comparison, maxflow algorithm in =-=[1]-=- took 0.114 secs (on average). This shows that, for this problem, maxflow significantly outperforms message passing algorithms. 5.2.2 Stereo Matching We have tested the algorithms on the energy functi...

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...o researchers have focused on approximate minimization algorithms. The two most well-known techniques are graph cuts and belief propagation. The former one was introduced in the 1990s [3], [9], [10], =-=[15]-=- and showed a major improvement over previously used simulated annealing [8]. To our knowledge, graph cuts are currently considered to be the most accurate minimization algorithm for energy functions ...

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...uts are currently considered to be the most accurate minimization algorithm for energy functions arising in many vision applications, e.g., stereo [3], [14], image restoration [3], image segmentation =-=[2]-=-, texture synthesis [19]. In fact, for some functions, it finds a global minimum. However, graph cuts can be applied only to a limited class of energy functions [3], [10], [15]. If a function falls . ...

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...Note that any node can serve as a root. 2.2 Reparameterization If two parameter vectors and define the same energy function (i.e., Eðx j ÞEðx j Þ for all x 2X) then is called a reparameterization of =-=[5]-=-, [25], [26], [30], [32] (in [25] this notion was called equivalent transformations). We will write this as . Note that this condition does not necessarily imply that since there are various linear ...

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...n (4) is tight, but, on the other, do not have a “structured” set of labels so that maxflow-based techniques cannot be applied. It is interesting, for example, to test the problem of super resolution =-=[7]-=-. In our experiments, we noticed that TRW-S algorithm would always converge to a fixed point of TRW, although such convergence would usually take much longer than achieving a weak tree agreement. Howe...

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...ticle, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-0097-0205. outside this class, then one has to use other techniques such as max-product belief propagation (BP) =-=[6]-=-, [22], [27], [35]. BP can be applied to any function of the form as in (1), but it has some drawbacks. First, it usually finds a solution with higher energy than graph cuts (in cases when graph cuts ...

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... problem, so researchers have focused on approximate minimization algorithms. The two most well-known techniques are graph cuts and belief propagation. The former one was introduced in the 1990s [3], =-=[9]-=-, [10], [15] and showed a major improvement over previously used simulated annealing [8]. To our knowledge, graph cuts are currently considered to be the most accurate minimization algorithm for energ...

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...se send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-0097-0205. outside this class, then one has to use other techniques such as max-product belief propagation (BP) [6], [22], =-=[27]-=-, [35]. BP can be applied to any function of the form as in (1), but it has some drawbacks. First, it usually finds a solution with higher energy than graph cuts (in cases when graph cuts can be appli...

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...mulated annealing [8]. To our knowledge, graph cuts are currently considered to be the most accurate minimization algorithm for energy functions arising in many vision applications, e.g., stereo [3], =-=[14]-=-, image restoration [3], image segmentation [2], texture synthesis [19]. In fact, for some functions, it finds a global minimum. However, graph cuts can be applied only to a limited class of energy fu...

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...lem, so researchers have focused on approximate minimization algorithms. The two most well-known techniques are graph cuts and belief propagation. The former one was introduced in the 1990s [3], [9], =-=[10]-=-, [15] and showed a major improvement over previously used simulated annealing [8]. To our knowledge, graph cuts are currently considered to be the most accurate minimization algorithm for energy func...

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...ion for functions with metric pairwise terms; they proved certain performance guarantees, e.g., two-approximation bound for the Potts energy. (Their work extended the approach of Kleinberg and Tardos =-=[11]-=-). Wainwright et al. [33] studied this LP formulation in the context of the TRW algorithm for general energy functions of the form 1. Recently, Komodakis and Tziritas [16] showed that graph cuts (or, ...

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...5]. BP can be applied to any function of the form as in (1), but it has some drawbacks. First, it usually finds a solution with higher energy than graph cuts (in cases when graph cuts can be applied) =-=[28]-=-. Second, BP does not always converge—it often goes into a loop. 1.1 Tree-Reweighted Message Passing Recently, Wainwright et al. [33] introduced a new algorithm for energy minimization called max-prod...

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...l and Schlesinger [18], [25] showed how to find a subgradient direction assuming that a certain arc consistency condition is violated. (A description of their augmenting DAG algorithm can be found in =-=[34]-=-). Another algorithm with the same stopping criterion (namely, arc consistency) is max-sum diffusion. 1 Note that, after some transformations, arc consistency can be shown to be equivalent to the WTA ...

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...nnot write it as h ; xi since and x belong to different spaces. However, we can introduce mapping : X!IR d so that Eðx j Þh ; ðxÞi X ðxÞ: Mapping , called the canonical overcomplete representation =-=[30]-=-, [32], consists of the following functions : X!IR: 1. The max-sum diffusion algorithm was developed independently by Kovalevsky and Koval in 1975 and by B. Flach in 1998. Neither of the works was pub...

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...en widely studied in the literature in different contexts. Schlesinger [24] applied it to energy functions of the form 1 whose pairwise terms encode hard constraints: stð ; Þ 2 f0; þ1g. Koster et al. =-=[17]-=- formulated this LP relaxation for arbitrary functions Eðx j Þ (in their terminology, the problem is called partial constraint satisfaction). Chekuri et al. [4] used the formulation for functions with...

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...d by the number of directed edges in the graph). We report only the average values over 100 samples. Note that, for functions of binary variables, TRW computes that same solution as maxflow algorithm =-=[12]-=-. In particular, for attractive potentials, TRW is guaranteed to find a global minimum, while, for mixed potentials, it finds part of an optimal solution. It should also be noted that, in both cases, ...

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...rite it as h ; xi since and x belong to different spaces. However, we can introduce mapping : X!IR d so that Eðx j Þh ; ðxÞi X ðxÞ: Mapping , called the canonical overcomplete representation [30], =-=[32]-=-, consists of the following functions : X!IR: 1. The max-sum diffusion algorithm was developed independently by Kovalevsky and Koval in 1975 and by B. Flach in 1998. Neither of the works was published...

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...rtain linear programming (LP) relaxation of the energy function (described in more detail in Section 2.3). This relaxation has been widely studied in the literature in different contexts. Schlesinger =-=[24]-=- applied it to energy functions of the form 1 whose pairwise terms encode hard constraints: stð ; Þ 2 f0; þ1g. Koster et al. [17] formulated this LP relaxation for arbitrary functions Eðx j Þ (in thei...

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...oach of Kleinberg and Tardos [11]). Wainwright et al. [33] studied this LP formulation in the context of the TRW algorithm for general energy functions of the form 1. Recently, Komodakis and Tziritas =-=[16]-=- showed that graph cuts (or, more precisely, the expansion move method in [3]) have close links with this LP. Thus, the LP relaxation in [4], [17], [24], [33] plays a very important role in the theory...

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...s whether WTA always gives a global maximum of the bound. We show that this is not the case by providing a counterexample. Note that this is different from sum-product tree-reweighted message passing =-=[31]-=-: In the latter case, a fixed point of TRW is guaranteed to be the global maximum of the lower bound on the negative log partition function. TRW algorithms require some choice of trees covering the gr...

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...ny node can serve as a root. 2.2 Reparameterization If two parameter vectors and define the same energy function (i.e., Eðx j ÞEðx j Þ for all x 2X) then is called a reparameterization of [5], [25], =-=[26]-=-, [30], [32] (in [25] this notion was called equivalent transformations). We will write this as . Note that this condition does not necessarily imply that since there are various linear relations am...

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...17], [24], [33] plays a very important role in the theory of MRF optimization algorithms. Several authors developed specialized techniques that try to solve this linear program. Koval and Schlesinger =-=[18]-=-, [25] showed how to find a subgradient direction assuming that a certain arc consistency condition is violated. (A description of their augmenting DAG algorithm can be found in [34]). Another algorit...

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...satisfies WTA condition and the set of nodes with multiple minima consists of disjoint chains which are monotonic with respect to the ordering used, then the procedure will find a global minimum (see =-=[20]-=-). 5.1.1 Attractive Potentials The results for attractive potentials are shown in Figs. 4a and 4b. Note that, in this case, the global minimum can be found in polynomial time using the maxflow algorit...

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...24], [33] plays a very important role in the theory of MRF optimization algorithms. Several authors developed specialized techniques that try to solve this linear program. Koval and Schlesinger [18], =-=[25]-=- showed how to find a subgradient direction assuming that a certain arc consistency condition is violated. (A description of their augmenting DAG algorithm can be found in [34]). Another algorithm wit...

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...to node t would not modify T st;jk and T t;k except for a constant independent of j or k). 10. The idea of reusing messages in junction trees was used in the context of iterative proportional fitting =-=[29]-=- and Rao-Blackwellized sampling [21].KOLMOGOROV: CONVERGENT TREE-REWEIGHTED MESSAGE PASSING FOR ENERGY MINIMIZATION 1575 Fig. 2. The TRW-S algorithm for a graph with monotonic chains. . Assuming that...

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...d e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-0097-0205. outside this class, then one has to use other techniques such as max-product belief propagation (BP) [6], [22], [27], =-=[35]-=-. BP can be applied to any function of the form as in (1), but it has some drawbacks. First, it usually finds a solution with higher energy than graph cuts (in cases when graph cuts can be applied) [2...