Efficient Belief Propagation for Vision Using Linear Constraint Nodes

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 demonstrate this technique in two applications. First, we perform efficient inference in graphical models where the spatial prior of natural images is captured by 2 times 2 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. Finally, we apply these techniques to shape-from-shading and demonstrate significant improvement over previous methods, both in quality and in flexibility.