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Truncated convex models (TCM) are a special case of pairwise random fieldsthat have been widely used in computer vision. However, by restricting theorder of the potentials to be at most two, they fail to capture useful imagestatistics. We propose a natural generalization of TCM to high-order randomfields, which we call truncated max-of-convex models (TMCM). The energyfunction of TMCM consistsof two types of potentials: (i) unary potential, whichhas no restriction on its form; and (ii) clique potential, which is the sum ofthe m largest truncated convex distances over all label pairs in a clique. Theuse of a convex distance function encourages smoothness, while truncationallows for discontinuities in the labeling. By using m > 1, TMCM providesrobustness towards errors in the definition of the cliques. In order tominimize the energy function of a TMCM over all possible labelings, we designan efficient st-MINCUT based range expansion algorithm. We prove the accuracyof our algorithm by establishing strong multiplicative bounds for severalspecial cases of interest. Using synthetic and standard real data sets, wedemonstrate the benefit of our high-order TMCM over pairwise TCM, as well asthe benefit of our range expansion algorithm over other st-MINCUT basedapproaches.

Truncated Max-of-Convex Models