In recent years the Markov Random Field (MRF) has become the de facto probabilistic model for low-level vision applications.
However, in a maximum a posteriori (MAP) framework, MRFs inherently encourage delta function marginal statistics.
By contrast, many low-level vision problems have heavy tailed marginal statistics, making the MRF model unsuitable.
In this paper we introduce a more general Marginal Probability Field (MPF), of which the MRF is a special, linear case,
and show that convex energy MPFs can be used to encourage arbitrary marginal statistics.
We introduce a flexible, extensible framework for effectively optimizing the resulting NP-hard MAP problem,
based around dual-decomposition and a modified min-cost flow algorithm, and which achieves global optimality in some instances.
We use a range of applications, including image denoising and texture synthesis, to demonstrate the benefits of this class of MPF over MRFs.