A comparison of sparse Bayesian regularization methods on computed tomography reconstruction

Abstract

Design of regularization term is an important part of solution of an ill-posed linear inverse problem. Another important issue is selection of tuning parameters of the regularization term. We address this problem using Bayesian approach which treats tuning parameters as unknowns and estimates them from the data. Specifically, we study a regularization model known as Automatic Relevance Determination (ARD) and several methods of its solution. The first approach is the conventional Variational Bayes method using the symmetrical factorization of the posterior of the vector of unknowns and the vector of tuning parameters. The second approach is based on the idea of marginalization over the vector of unknowns or the vector of tuning parameters, while the complementary vector is estimated using maximum likelihood. The resulting algorithm is thus an optimization task with non-convex objective function, which is solved using standard gradient methods. The proposed algorithms are tested on real tomographic X-ray data and the comparison with conventional regularization techniques (Tikhonov and Lasso) is performed. The algorithm using marginalization over the tuning parameter is found to be closest to the ground truth with acceptable computational cost. MATLAB®implementation of the reconstruction algorithms is freely available for download.