Bayesian image reconstruction for transmission tomography using deterministic annealing

We previously introduced a new, effective Bayesian reconstruction method for transmission tomographic reconstruction that is useful in attenuation correction in single-photon-emission computed tomography (SPECT) and positron-emission tomography (PET). The Bayesian reconstruction method uses a novel object model (prior) in the form of a mixture of gamma distributions. The prior models the object as comprising voxels whose values (attenuation coefficients) cluster into a few classes. This model is particularly applicable to transmission tomography since the attenuation map is usually well-clustered and the approximate values of attenuation coefficients in each anatomical region are known. The reconstruction is implemented as a maximum a posteriori (MAP) estimate obtained by iterative maximization of an associated objective function. As with many complex model-based estimations, the objective is nonconcave, and different initial conditions lead to different reconstructions corresponding to different local maxima. To make it more practical, it is important to avoid such dependence on initial conditions. We propose and test a deterministic annealing (DA) procedure for the optimization. Deterministic annealing is designed to seek approximate global maxima to the objective, and thus robustify the problem to initial conditions. We present the Bayesian reconstructions with and without DA and demonstrate the independence of initial conditions when using DA. In addition, we empirically show that DA reconstructions are stable with respect to small measurement changes.