Arithmetic Encoding of Markov Random Fields

This paper introduces methods for losslessly encoding a Markov random field (MRF) with arithmetic coding. The issues are how to choose the pixel scan order and how to produce coding distributions to accompany the pixels. For an MRF based on an acyclic graph, we choose a scan consistent with the graph and use Belief Propagation (BP) to efficiently compute the optimal coding distributions. For an MRF based on a cyclic graph, we use Local Conditioning (LC) to losslessly encode an appropriately chosen scan of a loop cutset, whose removal leaves an acyclic graph whose pixels can be encoded by the previous method. The results include BP-like formulas for LC in an undirected graph and a formula for the complexity of LC in a cyclic graph. As a first application of the methods, preliminary results of applying the method to an Ising model are given.