Two-dimensional discrete Markovian fields

A definition of discrete Markovian random fields is formulated analogously to a definition for the continuous case given by Lévy. This definition in the homogeneous Gaussian case leads to a difference equation that sets forth the state of the field in terms of its values on a band of minimum width
P
, where
P
is the order of the process. The state of the field at position
(i,j)
is given by the set of values of the nearest neighbors within distance
P
of the point
(i,j)
. Conversely, given a difference equation satisfying certain conditions relating to stability, there corresponds a homogeneous discrete Markov random field. This theory is applied to the problem of obtaining spectral estimates of a two-dimensional field, given observation over a limited aperture.