Efficient Message Representations for Belief Propagation

Belief propagation (BP) has been successfully used to approximate the solutions of various Markov random field (MRF) formulated energy minimization problems. However, large MRFs require a significant amount of memory to store the intermediate belief messages. We observe that these messages have redundant information due to the imposed smoothness prior. In this paper, we study the feasibility of applying compression techniques to the messages in the min-sum/max-product BP algorithm with 1D labels to improve the memory efficiency and reduce the read/write bandwidth. We articulate properties that an efficient message representation should satisfy. We investigate two common compression schemes, predictive coding and linear transform coding (PCA), and then propose a novel envelope point transform (EPT) method. Predictive coding is efficient and supports linear operations directly in the compressed domain, but it is only compatible with the L₁ smoothness function. PCA has the disadvantage that it does not guarantee the preservation of the minimal label. EPT is not limited to L₁ smoothness cost and allows a flexible quality vs. compression ratio tradeoff compared with predictive coding. Experiments on dense stereo reconstruction have shown that the predictive scheme and EPT can achieve 8times or more compression without significant loss of depth accuracy.