  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.
