A Convex Discrete-Continuous Approach for Markov Random Fields

We propose an extension of the well-known LP relaxation for Markov random fields to explicitly allow continuous label spaces. Unlike conventional continuous formulations of labelling problems which assume that the unary and pairwise potentials are convex, our formulation allows them to be general piecewise convex functions with continuous domains. Furthermore, we present the extension of the widely used efficient scheme for handling L 1 smoothness priors over discrete ordered label sets to continuous label spaces. We provide a theoretical analysis of the proposed model, and empirically demonstrate that labelling problems with huge or continuous label spaces can benefit from our discrete-continuous representation.