Truncated Max-of-Convex Models

Abstract: Truncated convex models (TCM) are a special case of pairwise random fields that have been widely used in computer vision. However, by restricting the order of the potentials to be at most two, they fail to capture useful image statistics. We propose a natural generalization of TCM to high-order random fields, which we call truncated max-of-convex models (TMCM). The energy function of TMCM consistsof two types of potentials: (i) unary potential, which has no restriction on its form; and (ii) clique potential, which is the sum of the m largest truncated convex distances over all label pairs in a clique. The use of a convex distance function encourages smoothness, while truncation allows for discontinuities in the labeling. By using m > 1, TMCM provides robustness towards errors in the definition of the cliques. In order to minimize the energy function of a TMCM over all possible labelings, we design an efficient st-MINCUT based range expansion algorithm. We prove the accuracy of our algorithm by establishing strong multiplicative bounds for several special cases of interest. Using synthetic and standard real data sets, we demonstrate the benefit of our high-order TMCM over pairwise TCM, as well as the benefit of our range expansion algorithm over other st-MINCUT based approaches.