Maximum-likelihood parameter estimation of discrete homogeneous random fields with mixed spectral distributions

This paper presents a maximum-likelihood solution to the general problem of fitting a parametric model to observations from a single realization of a real valued, 2-D, homogeneous random field with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. The proposed algorithm provides a complete solution to the joint estimation problem of the random field components. By introducing appropriate parameter transformations, the highly nonlinear least-squares problem that results from the maximization of the likelihood function is transformed into a separable least-squares problem. In this new problem, the solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the transformed parameters of the field to linear least squares. Solution of the transformation equations provides a complete solution of the field model parameter estimation problem.