Efficient belief propagation for higher-order cliques using linear constraint nodes

Authors:
Brian Potetz;Tai Sing Lee
Affiliations:
Department of Computer Science & Center for the Neural Basis of Cognition Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA;Department of Computer Science & Center for the Neural Basis of Cognition Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA
Venue:
Computer Vision and Image Understanding
Year:
2008

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Abstract

Belief propagation over pairwise-connected Markov random fields has become a widely used approach, and has been successfully applied to several important computer vision problems. However, pairwise interactions are often insufficient to capture the full statistics of the problem. Higher-order interactions are sometimes required. Unfortunately, the complexity of belief propagation is exponential in the size of the largest clique. In this paper, we introduce a new technique to compute belief propagation messages in time linear with respect to clique size for a large class of potential functions over real-valued variables. We discuss how this technique can be generalized to still wider classes of potential functions at varying levels of efficiency. Also, we develop a form of nonparametric belief representation specifically designed to address issues common to networks with higher-order cliques and also to the use of guaranteed-convergent forms of belief propagation. To illustrate these techniques, we perform efficient inference in graphical models where the spatial prior of natural images is captured by 2x2 cliques. This approach shows significant improvement over the commonly used pairwise-connected models, and may benefit a variety of applications using belief propagation to infer images or range images, including stereo, shape-from-shading, image-based rendering, segmentation, and matting.