A Model for the Two-Dimensional No Isolated Bits Constraint

Abstract

A stationary model is presented for the two-dimensional (2-D) no isolated bits (n.i.b.) constraint over an extended alphabet defined by the elements within 1 by 2 blocks. This block-wise model is based on a set of sufficient conditions for a Pickard random field (PRF) over an m-ary alphabet. Iterative techniques are applied as part of determining the model parameters. Given two Markov chains describing a boundary, an algorithm is presented which determines whether a certain PRF consistent with the boundary exists. Iterative scaling is used as part of the algorithm, which also determines the conditional probabilities yielding the maximum entropy for the given boundary description if a solution exists. Optimizing over the parameters for a class of boundaries with certain symmetry properties, an entropy of 0.9156 is achieved for the n.i.b. constraint, providing a lower bound. An algorithm for iterative search for a PRF solution starting from a set of conditional probabilities is also presented