Validation of new Gibbs priors for Bayesian tomographic reconstruction using numerical studies and physically acquired data

The variety of Bayesian MAP approaches to emission tomography proposed in recent years can both stabilize reconstructions and lead to improved bias and variance. In our previous work [1, 2], we showed that the thin-plate (TP) prior, which is less sensitive to variations in first spatial derivatives than the conventional membrane (MM) prior, yields improved reconstructions in the sense of low bias. In spite of the several advantages of such quadratic smoothing priors, they are still less than ideal due to their limitations in edge preservation. In this paper we use a convex but nonquadratic (CNQ) potential function, which provides a degree of edge preservation. As in the case of quadratic priors, a class of two-dimensional smoothing splines with first and second partial derivatives are applied to the new potential function. In order to reduce difficulties such as oversmoothing for MM and edge overshooting for TP, we also generalize the prior energy definition to that of a linear combination of MM and TP using a control parameter [3], and observe its transition between the two extreme cases. To validate advantages of our new priors, we first perform extensive numerical studies using a digital phantom to compare the bias/variance behavior of CNQ priors with that of quadratic priors. We also use physically acquired PET emission and transmission data from phantoms to observe the efficacies of our new priors. Our numerical studies and results using physical phantoms show that a combination of first and second partial derivatives applied to the CNQ potential yields improved quantitative results in terms of scalar metrics of image quality computed from independent noise trials and good qualitative results for both emission and transmission images.