Block Pickard Models for Two-Dimensional Constraints

Abstract

In Pickard random fields (PRF), the probabilities of finite configurations and the entropy of the field can be calculated explicitly, but only very simple structures can be incorporated into such a field. Given two Markov chains describing a boundary, an algorithm is presented which determines whether a PRF consistent with the distribution on the boundary and a 2-D constraint 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. A PRF is defined for the domino tiling constraint represented by a quaternary alphabet. PRF models are also presented for higher order constraints, including the no isolated bits (n.i.b.) constraint, and a minimum distance 3 constraint by defining super symbols on blocks of binary symbols.