Joint-MAP Bayesian tomographic reconstruction with a gamma-mixture prior

We address the problem of Bayesian image reconstruction with a prior that captures the notion of a clustered intensity histogram. The problem is formulated in the framework of a joint-MAP (maximum a posteriori) estimation with the prior PDF modeled as a mixture-of-gammas density. This prior PDF has appealing properties, including positivity enforcement. The joint MAP optimization is carried out as an iterative alternating descent wherein a regularized likelihood estimate is followed by a mixture decomposition of the histogram of the current tomographic image estimate. The mixture decomposition step estimates the hyperparameters of the prior PDF. The objective functions associated with the joint MAP estimation are complicated and difficult to optimize, but we show how they may be transformed to allow for much easier optimization while preserving the fixed point of the iterations. We demonstrate the method in the context of medical emission and transmission tomography.