Stationary Markov random fields on a finite rectangular lattice

Abstract

This paper provides a complete characterization of stationary Markov random fields on a finite rectangular (nontoroidal) lattice in the basic case of a second-order neighborhood system. Equivalently, it characterizes stationary Markov fields on 2 whose restrictions to finite rectangular subsets are still Markovian (i.e., even on the boundaries). Until now, Pickard random fields formed the only known class of such fields. First, we derive a necessary and sufficient condition for Markov random fields on a finite lattice to be stationary. It is shown that their joint distribution factors in terms of the marginal distribution on a generic (2 2 2) cell which must fulfill some consistency constraints. Second, we solve the consistency constraints and provide a complete characterization of such measures in three cases. Symmetric measures and Gaussian measures are shown to necessarily belong to the Pickard class, whereas binary measures belong either to the Pickard class, or to a new nontrivial class which is further studied. In particular, the corresponding fields admit a simple parameterization and may be simulated in a simple, although nonunilateral manner.

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“…The class of binary Markov random fields defined as stationary extension of the distribution on 2 x 2 elements was completely characterized in [6].…”

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“…It is also known that if , then . Thus (11) Consequently, QMRF of size 2 is a MRF of size 4 with no abnormality. Theorem 2 is only proven for QMRF of size 2; however, it is simple to extend it to any order of QMRF.…”

“…Therefore, QMRF with the neighboring size of 2 is PRF. Note that PRF is the only known class of nontrivial stationary MRF [6], [11].…”

“…Pickard shows [6] that the constructed random field is MMRF and consequently MRF. It is farther proven that PRF is the only acknowledged class of nontrivial stationary Markov field on finite rectangular lattices [11].…”

“…In spite of such deficiency, the local conditional probabilities are computable without needing to compute the partition function. Derivation of the global closed-form pdf of the MRF from the local conditional probabilities is only possible under the unilateral (causal) assumption [9]- [11]. The pioneering work on unilateral subclass of MRF by Abend is noteworthy, which is called Markov Mesh in the literature and later on denoted as Markov mesh random field (MMRF) [4].…”