Two-dimensional linear prediction models--part I: Spectral factorization and realization

In this paper we present several results for three different canonical forms of linear prediction on a plane. These filters have causal, semicausal, and noncausal prediction geometries. Starting from their properties we consider the problem of realization of these filters from a given power spectral density function (SDF). Since it is not possible in general to obtain rational spectral factors of a two-dimensional SDF, we propose algorithms for obtaining rational approximations which are stable and converge to their limit (irrational) factors as the order of approximation is increased. It is also shown that the normal equations associated with the minimum variance two-dimensional prediction filters give a useful algorithm for obtaining rational approximations which are stable and converge to their unique limit filters. This result allows design of finite-order stable filters by solving a finite number of equations while realizing the given SDF arbitrarily closely.