Performance comparison of convex-nonquadratic priors for Bayesian tomographic reconstruction

Abstract

Bayesian methods have formed the core of the development of reconstruction algorithms for emission tomography. In particular, there has been considerable interest in edge-preserving prior models, which are associated with smoothing penalty functions that are nonquadratic functions of nearby pixel differences. In spite of several advantages of nonconvex prior models, their use in routine applications has been hindered by several factors, such as the computational expense due to the nonconvexity of penalty functions and the difficulty in the selection of hyperparameters. We note here that, by choosing a penalty function which is nonquadratic but is still convex, both the problem of nonconvexity involved in some nonquadratic priors and the edge-oversmoothing problem of conventional quadratic priors may be avoided. In this paper, we use a class of two-dimensional smoothing splines with low (first) and high (second) spatial derivatives applied to convex-nonquadratic (CNQ) penalty functions. To compare quantitative performance of our new priors, we use the quantitation of bias/variance and total squared error over noise trials. Our numerical results show that a linear combination of first- and second-order spatial derivatives applied to CNQ penalty functions improves reconstructions in terms of total squared error.