MRF Energy Minimization and Beyond via Dual Decomposition

This paper introduces a new rigorous theoretical framework to address discrete MRF-based optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first decomposing it into a set of appropriately chosen subproblems, and then combining their solutions in a principled way. In order to determine the limits of this method, we analyze the conditions that these subproblems have to satisfy and demonstrate the extreme generality and flexibility of such an approach. We thus show that by appropriately choosing what subproblems to use, one can design novel and very powerful MRF optimization algorithms. For instance, in this manner we are able to derive algorithms that: 1) generalize and extend state-of-the-art message-passing methods, 2) optimize very tight LP-relaxations to MRF optimization, and 3) take full advantage of the special structure that may exist in particular MRFs, allowing the use of efficient inference techniques such as, e.g., graph-cut-based methods. Theoretical analysis on the bounds related with the different algorithms derived from our framework and experimental results/comparisons using synthetic and real data for a variety of tasks in computer vision demonstrate the extreme potentials of our approach.

[1]  Vladimir Kolmogorov,et al.  Convergent Tree-Reweighted Message Passing for Energy Minimization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

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

[4]  Francisco Barahona,et al.  The volume algorithm: producing primal solutions with a subgradient method , 2000, Math. Program..

[5]  Pushmeet Kohli,et al.  Dynamic Graph Cuts for Efficient Inference in Markov Random Fields , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  Nikos Komodakis,et al.  Fast, Approximately Optimal Solutions for Single and Dynamic MRFs , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[7]  William T. Freeman,et al.  Nonparametric belief propagation , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

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

[9]  William T. Freeman,et al.  On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs , 2001, IEEE Trans. Inf. Theory.

[10]  Riccardo Zecchina,et al.  Survey propagation: An algorithm for satisfiability , 2002, Random Struct. Algorithms.

[11]  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.

[12]  Dimitri P. Bertsekas,et al.  Incremental Subgradient Methods for Nondifferentiable Optimization , 2001, SIAM J. Optim..

[13]  Nikos Komodakis,et al.  MRF Optimization via Dual Decomposition: Message-Passing Revisited , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[14]  D. Bertsekas,et al.  A DESCENT NUMERICAL METHOD FOR OPTIMIZATION PROBLEMS WITH NONDIFFERENTIABLE COST FUNCTIONALS , 1973 .

[15]  Richard Szeliski,et al.  A Comparative Study of Energy Minimization Methods for Markov Random Fields with Smoothness-Based Priors , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[17]  Nikos Komodakis,et al.  Beyond Loose LP-Relaxations: Optimizing MRFs by Repairing Cycles , 2008, ECCV.

[18]  Tommi S. Jaakkola,et al.  New Outer Bounds on the Marginal Polytope , 2007, NIPS.

[19]  Brendan J. Frey,et al.  A Revolution: Belief Propagation in Graphs with Cycles , 1997, NIPS.

[20]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[21]  Dmitry M. Malioutov,et al.  Lagrangian Relaxation for MAP Estimation in Graphical Models , 2007, ArXiv.

[22]  Michael J. Black,et al.  Fields of Experts: a framework for learning image priors , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[23]  Yair Weiss,et al.  Linear Programming Relaxations and Belief Propagation - An Empirical Study , 2006, J. Mach. Learn. Res..

[24]  Tom Heskes,et al.  Convexity Arguments for Efficient Minimization of the Bethe and Kikuchi Free Energies , 2006, J. Artif. Intell. Res..

[25]  Tomás Werner,et al.  High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF) , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[26]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[27]  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.

[28]  Daniel Tarlow,et al.  Using Combinatorial Optimization within Max-Product Belief Propagation , 2006, NIPS.

[29]  Benjamin B. Kimia,et al.  Euler Spiral for Shape Completion , 2003, International Journal of Computer Vision.

[30]  William T. Freeman,et al.  Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology , 1999, Neural Computation.

[31]  Tommi S. Jaakkola,et al.  Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations , 2007, NIPS.

[32]  Andrew Zisserman,et al.  OBJ CUT , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[33]  Martin J. Wainwright,et al.  Message-passing for graph-structured linear programs: proximal projections, convergence and rounding schemes , 2008, ICML '08.

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

[35]  T. Heskes Stable Fixed Points of Loopy Belief Propagation Are Minima of the Bethe Free Energy , 2002 .

[36]  Vladimir Kolmogorov,et al.  Visual correspondence using energy minimization and mutual information , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[37]  Vladimir Kolmogorov,et al.  An Analysis of Convex Relaxations for MAP Estimation , 2007, NIPS.

[38]  Tom Heskes,et al.  Stable Fixed Points of Loopy Belief Propagation Are Local Minima of the Bethe Free Energy , 2002, NIPS.

[39]  Brendan J. Frey,et al.  Graphical Models for Machine Learning and Digital Communication , 1998 .

[40]  L. Williams,et al.  Contents , 2020, Ophthalmology (Rochester, Minn.).

[41]  Tommi S. Jaakkola,et al.  Tightening LP Relaxations for MAP using Message Passing , 2008, UAI.

[42]  Nikos Komodakis,et al.  Image Completion Using Global Optimization , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[43]  Vladimir Kolmogorov,et al.  Optimizing Binary MRFs via Extended Roof Duality , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

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

[45]  Yair Weiss,et al.  MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies , 2007, UAI.

[46]  Tom Heskes,et al.  Fractional Belief Propagation , 2002, NIPS.

[47]  Ian McGraw,et al.  Residual Belief Propagation: Informed Scheduling for Asynchronous Message Passing , 2006, UAI.

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

[49]  Daniel P. Huttenlocher,et al.  Pictorial Structures for Object Recognition , 2004, International Journal of Computer Vision.

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

[51]  Michael Patriksson,et al.  Ergodic, primal convergence in dual subgradient schemes for convex programming , 1999, Mathematical programming.

[52]  Gurmeet Singh,et al.  MRF's forMRI's: Bayesian Reconstruction of MR Images via Graph Cuts , 2006, CVPR.

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

[54]  Hanif D. Sherali,et al.  Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs , 1996, Oper. Res. Lett..

[55]  Vladimir Kolmogorov,et al.  Feature Correspondence Via Graph Matching: Models and Global Optimization , 2008, ECCV.

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