MRF inference by k-fan decomposition and tight Lagrangian relaxation
ECCV'10 Proceedings of the 11th European conference on computer vision conference on Computer vision: Part III
Collective Inference for Extraction MRFs Coupled with Symmetric Clique Potentials
The Journal of Machine Learning Research
Joint shape segmentation with linear programming
Proceedings of the 2011 SIGGRAPH Asia Conference
An optimization approach for extracting and encoding consistent maps in a shape collection
ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH Asia 2012
Closed-Form relaxation for MRF-MAP tissue classification using discrete laplace equations
MICCAI'12 Proceedings of the 15th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II
Computer Vision and Image Understanding
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The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), and Wainwright et al. (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by Kumar et al. (2006) for a general label set. We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP, the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCP-MS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field. Using some examples we show that the new SOCP relaxations strictly dominate the previous approaches.