Convergent Tree-Reweighted Message Passing for Energy Minimization

Algorithms for discrete energy minimization are of fundamental importance in computer vision. In this paper, we focus on the recent technique proposed by Wainwright et al. (Nov. 2005)- tree-reweighted max-product message passing (TRW). It was inspired by the problem of maximizing a lower bound on the energy. However, the algorithm is not guaranteed to increase this bound - it may actually go down. In addition, TRW does not always converge. We develop a modification of this algorithm which we call sequential tree-reweighted message passing. Its main property is that the bound is guaranteed not to decrease. We also give a weak tree agreement condition which characterizes local maxima of the bound with respect to TRW algorithms. We prove that our algorithm has a limit point that achieves weak tree agreement. Finally, we show that, our algorithm requires half as much memory as traditional message passing approaches. Experimental results demonstrate that on certain synthetic and real problems, our algorithm outperforms both the ordinary belief propagation and tree-reweighted algorithm in (M. J. Wainwright, et al., Nov. 2005). In addition, on stereo problems with Potts interactions, we obtain a lower energy than graph cuts

[1]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[3]  D. Greig,et al.  Exact Maximum A Posteriori Estimation for Binary Images , 1989 .

[4]  Arie M. C. A. Koster,et al.  The partial constraint satisfaction problem: Facets and lifting theorems , 1998, Oper. Res. Lett..

[5]  Olga Veksler,et al.  Fast approximate energy minimization via graph cuts , 2001, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[6]  Éva Tardos,et al.  Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[7]  Michael I. Jordan,et al.  Probabilistic Networks and Expert Systems , 1999 .

[8]  M. I. Schlesinger,et al.  Some solvable subclasses of structural recognition problems , 2000 .

[9]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[10]  W. Freeman,et al.  Generalized Belief Propagation , 2000, NIPS.

[11]  Marie-Pierre Jolly,et al.  Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images , 2001, ICCV.

[12]  Joseph Naor,et al.  Approximation algorithms for the metric labeling problem via a new linear programming formulation , 2001, SODA '01.

[13]  Y.Y. Boykov,et al.  Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[14]  Olga Veksler,et al.  Fast Approximate Energy Minimization via Graph Cuts , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Vladimir Kolmogorov,et al.  Multi-camera Scene Reconstruction via Graph Cuts , 2002, ECCV.

[16]  Nanning Zheng,et al.  Stereo Matching Using Belief Propagation , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Yee Whye Teh,et al.  On Improving the Efficiency of the Iterative Proportional Fitting Procedure , 2003, AISTATS.

[18]  Martin J. Wainwright,et al.  Tree-reweighted belief propagation algorithms and approximate ML estimation by pseudo-moment matching , 2003, AISTATS.

[19]  Irfan A. Essa,et al.  Graphcut textures: image and video synthesis using graph cuts , 2003, ACM Trans. Graph..

[20]  William T. Freeman,et al.  Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[21]  Mark A. Paskin.  Sample Propagation , 2003, NIPS.

[22]  Hiroshi Ishikawa,et al.  Exact Optimization for Markov Random Fields with Convex Priors , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  Martin J. Wainwright,et al.  Tree-based reparameterization framework for analysis of sum-product and related algorithms , 2003, IEEE Trans. Inf. Theory.

[24]  Martin J. Wainwright,et al.  Tree consistency and bounds on the performance of the max-product algorithm and its generalizations , 2004, Stat. Comput..

[25]  Vladimir Kolmogorov,et al.  What energy functions can be minimized via graph cuts? , 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  R. Zabih,et al.  What energy functions can be minimized via graph cuts , 2004 .

[27]  Vladimir Kolmogorov,et al.  An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision , 2001, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  William T. Freeman,et al.  Learning Low-Level Vision , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[29]  Daniel P. Huttenlocher,et al.  Efficient Belief Propagation for Early Vision , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[30]  Richard Szeliski,et al.  A Taxonomy and Evaluation of Dense Two-Frame Stereo Correspondence Algorithms , 2001, International Journal of Computer Vision.

[31]  Martin J. Wainwright,et al.  On the Optimality of Tree-reweighted Max-product Message-passing , 2005, UAI.

[32]  Nikos Komodakis,et al.  A new framework for approximate labeling via graph cuts , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[33]  William T. Freeman,et al.  Constructing free-energy approximations and generalized belief propagation algorithms , 2005, IEEE Transactions on Information Theory.

[34]  Martin J. Wainwright,et al.  MAP estimation via agreement on trees: message-passing and linear programming , 2005, IEEE Transactions on Information Theory.

[35]  Yair Weiss,et al.  Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[36]  Martin J. Wainwright,et al.  MAP estimation via agreement on (hyper)trees: Message-passing and linear programming , 2005, ArXiv.

[37]  Tomás Werner,et al.  A Linear Programming Approach to Max-Sum Problem: A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.