In a Bayesian tomographic maximum a posteriori (MAP) reconstruction, an estimate of the object f is computed by iteratively minimizing an objective function that typically comprises the sum of a log-likelihood (data consistency) term and prior (or penalty) term. The prior can be used to stabilize the solution and to also impose spatial properties on the solution. One such property, preservation of edges and locally monotonic regions, is captured by the well-known median root prior (MRP), an empirical method that has been applied to emission and transmission tomography. We propose an entirely new class of convex priors that depends on f and also on m, an auxiliary field in register with f. We specialize this class to our median prior (MP). The approximate action of the median prior is to draw, at each iteration, an object voxel toward its own local median. This action is similar to that of MRP and results in solutions that impose the same sorts of object properties as does MRP. Our MAP method is not empirical, since the problem is stated completely as the minimization of a joint (on f and m) objective. We propose an alternating algorithm to compute the joint MAP solution and apply this to emission tomography, showing that the reconstructions are qualitatively similar to those obtained using MRP.

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.