Spline-based sparse tomographic reconstruction with Besov priors

Abstract

Tomographic reconstruction from limited X-ray data is an ill-posed inverse problem. A common Bayesian approach is to search for the maximum a posteriori (MAP) estimate of the unknowns that integrates the prior knowledge, about the nature of biomedical images, into the reconstruction process. Recent results on the Bayesian inversion have shown the advantages of Besov priors for the convergence of the estimates as the discretization of the image is refined. We present a spline framework for sparse tomographic reconstruction that leverages higher-order basis functions for image discretization while incorporating Besov space priors to obtain the MAP estimate. Our method leverages tensor-product B-splines and box splines, as higher order basis functions for image discretization, that are shown to improve accuracy compared to the standard, first-order, pixel-basis. Our experiments show that the synergy produced from higher order B-splines for image discretization together with the discretization-invariant Besov priors leads to significant improvements in tomographic reconstruction. The advantages of the proposed Bayesian inversion framework are examined for image reconstruction from limited number of projections in a few-view setting.