Approximate maximum likelihood hyperparameter estimation for Gibbs priors

TLDR: An approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate and the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration are presented.

Abstract: The parameters of the prior, the hyperparameters, play an important role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, /spl beta/, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of /spl beta/ from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of /spl beta/ from incomplete data is intractable for most image processing. Here we present an approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate. The algorithm is based on a mean field approximation technique through which multidimensional Gibbs distributions are approximated by a separable function equal to a product of one-dimensional (1-D) densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to optimize the approximation for a restricted class of problems. We present the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration.