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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> is the order of the process. The state of the field at position <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i,j)</tex> is given by the set of values of the nearest neighbors within distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> of the point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i,j)</tex> . 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.