Joint estimation of parameters and hyperparameters in a Bayesian approach of solving inverse problems

We propose a joint estimation of the parameters and hyperparameters (the parameters of the prior law) when a Bayesian approach with maximum entropy (ME) priors is used to solve the inverse problems which arise in signal and image reconstruction and restoration problems. In particular we propose two methods: one based on the expectation maximization (EM) algorithm who aims to find the marginalized MAP (MMAP) estimate and the second based on a joint MAP estimation (JMAP). We discuss and compare these methods and give some simulation results in image restoration to show the relative performances of the proposed methods