Maximum-likelihood parameter estimation of the harmonic, evanescent, and purely indeterministic components of discrete homogeneous random fields

This paper presents a maximum-likelihood solution to the general problem of tting a parametric model to observations from a single realization of a 2-D homogeneous random eld with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the eld is represented as a sum of mutually orthogonal components of three types: purelyindeterministic, harmonic, and evanescent. The suggested algorithm involves a two-stage procedure. In the rst stage we obtain a suboptimal initial estimate for the parameters of the spectral support of the evanescent and harmonic components. In the second stage we re ne these initial estimates by iterative maximization of the conditional likelihood of the observed data, which is expressed as a function of only the parameters of the spectral supports of the evanescent and harmonic components. The solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the other unknown parameters of the eld to linear least squares. The CramerRao lower bound on the accuracy of jointly estimating the parameters of the di erent components is derived, and it is shown that the bounds on the purely-indeterministic and deterministic components are decoupled. Numerical evaluation of the bounds provides some insight into the e ects of various parameters on the achievable estimation accuracy. The performance of the Maximum-Likelihood algorithm is illustrated by Monte-Carlo simulations and is compared with the Cramer-Rao bound.