APPROXIMATE MAXIMUM-LIKELIHOOD HYPERPARAMETER ESTIMATION FOR GIBBS-PRIORS

The parameters of the prior, the hyperparameters, play an important ro le in Bayesian image estimation, Of particular importance for the case of Gibbs priors is the global hyperparameter, beta, which multiplies the Hamiltonian, Here we consider maximum likelihood (ML) estimation o f beta from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degra dation or blurring process, Important applications include image resto ration and image reconstruction from projections. Exact ML estimation of beta from incomplete data is intractable for most image processing, Here we present an approximate ML estimator that is computed simultan eously with a maximum a posteriori (MAP) image estimate, The algorithm is based on a mean field approximation technique through which multid imensional Gibbs distributions are approximated by a separable functio n equal to a product of one-dimensional (1-D) densities, We show how t his approach can be used to simplify the ML estimation problem. We als o show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to opt imize the approximation for a restricted class of problems, We present the results of a Monte Carlo study that examines the bias and varianc e of this estimator when applied to image restoration.