Maximum likelihood parameter estimation of textures using a Wold-decomposition based model

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 two-stage algorithm. In the first stage, we obtain an estimate for the number of harmonic and evanescent components in the field, and a suboptimal initial estimate 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. 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 a 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 applicable to a wide variety of texture types found in natural images.