#### Technical Report USC-SIPI-285

“Approximate Maximum Likelihood Hyperparameter Estimation for Gibbs Priors”

by Zhenyu Zhou and Richard M. Leahy

June 1995

The parameters of the prior, the hyperparameters, play a critical role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, _, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of _ from incomplete data, i.e. problems in which the image, which is drawn from the Gibbs distribution, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of _ from incomplete data is intractable for most image processing. Here we present an approximate ML estimator which 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 densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman 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.

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