Stationary Markov Random Fields on a Finite Rectangular Lattice

—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 n...

• Type: Journal
• Year: 1998
• Where: TIT
• Authors: Fréderic Champagnat, Jérôme Idier, Yves Goussard
• Keywords: Complete Characterization, Markov Random, Stationary Markov, TIT 1998