Lossless Reduced Cutset Coding of Markov Random Fields

This paper presents Reduced Cutset Coding, a new Arithmetic Coding (AC) based approach to lossless compression of Markov random fields. In recent work, the authors presented an efficient AC based approach to encoding acyclic MRFs and described a Local Conditioning (LC) based approach to encoding cyclic MRFs. In the present work, we introduce an algorithm for AC encoding of a cyclic MRF for which the complexity of the LC method of, or the acyclic MRF algorithm of combined with the Junction Tree (JT) algorithm, is too large. For encoding an MRF based on a cyclic graph G = (V,E), a cutset U ¿ V is selected such that the subgraph G_U induced by U, and each of the components of GU, are tractable to either LC or JT. Then, the cutset variables X_U are AC encoded with coding distributions based on a reduced MRF defined on G_U, and the remaining components X_VU of X_V are optimally AC encoded conditioned on X_U . The increase in rate over optimal encoding of X_V is the normalized divergence between the marginal distribution of X_U and the reduced MRF on G_U used for the AC encoding. We show this follows a Pythagorean decomposition and, additionally, that the optimal exponential parameter for the reduced MRF on G_U is the one that preserves the moments from the marginal distribution. We also show that the rate of encoding X_U with this moment-matching exponential parameter is equal to the entropy of the reduced MRF with this moment-matching parameter. We illustrate the concepts of our approach by encoding a typical image from an Ising model with a cutset consisting of evenly spaced rows. The performance on this image is similar to that of JBIG.