Approximate maximum likelihood hyperparameter estimation for Gibbs priors

We describe an approximate ML estimator for the hyperparameters of a Gibbs prior which can be 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