Simplifying Energy Optimization using Partial Enumeration

Energies with higher order non-submodular interactions have been shown to be very useful in vision due to their high modeling power. Optimization of such energies, however, is generally NP-hard. One naive approach that works for small problem instances is exhaustive search, that is, enumeration of all possible labelings of the underlying graph. We propose a general approach for minimizing complex high-order energies on large graphs that is based on enumeration of labelings of certain small subsets. We show how to benefit from such partial enumerations by equivalently reformulating the optimization problem on a new graph. Each node of this graph represents a subset of pixels with feasible labels (node states) corresponding to enumeration of this subset. If pixel subsets are chosen as an overlapping partition of the image then our partial enumeration approach simplifies the original high-order energy optimization to a pairwise interaction problem that can be efficiently solved by methods like TRW-S. We demonstrate that our approach often yields the exact global minimum (zero duality gap) for some high-order (curvature) and non-submodular (deconvolution) energies known to be difficult.