Lossless Reduced Cutset Coding of Markov Random Fields

Abstract

This paper presents Reduced Cutset Coding, a new Arithmetic Coding (AC) based approach tolossless compression of Markov random fields. In recent work\cite{reye:09a}, the authors presented an efficient AC based approachto encoding acyclic MRFs and described a Local Conditioning (LC)based approach to encoding cyclic MRFs. In the present work, weintroduce an algorithm for AC encoding of a cyclic MRF for which thecomplexity of the LC method of \cite{reye:09a}, or the acyclicMRF algorithm of \cite{reye:09a} combined with the Junction Tree(JT) algorithm, is too large. For encoding an MRF based on acyclic graph $G=(V,E)$, a cutset $U\subset V$ is selected such thatthe subgraph $G_U$ induced by $U$, and each of the components of$G\setminus U$, are tractable to either LC or JT. Then, the cutsetvariables $X_U$ are AC encoded with coding distributions based on areduced MRF defined on $G_U$, and the remaining components$X_{V\setminus U}$ of $X_V$ are optimally AC encoded conditioned on$X_U$. The increase in rate over optimal encoding of $X_V$ is thenormalized divergence between the marginal distribution of $X_U$ and thereduced MRF on $G_U$ used for the AC encoding. We show this follows aPythagorean decomposition and, additionally, that the optimalexponential parameter for the reduced MRF on $G_U$ is the one thatpreserves the moments from the marginal distribution. We also showthat the rate of encoding $X_U$ with this moment-matchingexponential parameter is equal to the entropy of the reduced MRFwith this moment-matching parameter. We illustrate the concepts ofour approach by encoding a typical image from an Ising model with acutset consisting of evenly spaced rows. The performance on this image issimilar to that of JBIG.