This paper presents a maximum-likelihood solution to the general problem of fitting a parametric model to observations from a single realization of a two-dimensional (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 suggested algorithm involves a two-stage procedure. In the first 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 refine 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 field to a linear least squares. The Cramer-Rao lower bound on the accuracy of jointly estimating the parameters of the different 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 effects 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.

Abstruct-We present a solution to the problem of modeling, parameter estimation, and synthesis of natural textures.The texture field is assumed to be a realization of a regular homogeneous random field, which can have a mixed spectral distribution.On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of a purely indeterministic component, a harmonic component, and a countable number of evanescent fields.We present a maximum-likelihood solution to the joint parameter estimation problem of these components from a single observed realization of the texture field.The proposed solution is a twostage algorithm.In the first stage, we obtain an estimate for the number of harmonic and evanescent components in the field, and a suboptimal initial estiqate for the parameters of their spectral supports.In the second stage, we refine these initial estimates by iterative maximization of the likelihood function of the observed data.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 then provides a complete solution of the field-model parameter estimation problem.The Wold-based model and the resulting analysis and synthesis algorithms are seen applicable to a wide variety of texture types found in natural images.The support or shape of the analyzed texture patch may be arbitrary.Our model is very efficient in terms of the number of parameters required to represent and faithfully reconstruct the original texture.