AdaTL₁: an adaptive non-convex sparse solver with applications to CT reconstruction and image denoising

Abstract

Sparsity-promoting optimization techniques have, in recent years, gained prominence due to the availability of efficient solvers and their capability to efficiently use inherent duplication present in underlying data. In clinical Tomography, a faithful reconstruction of the cross-section of an object or image from a reduced set of X-ray attenuation samples attains importance. Though sparsity methods in Computed Tomography (CT) have already been well attended to, there is still a need for newer ideas that accommodate the highly coherent nature of the sub-sampled Radon matrix and incorporate adaptive information into the recovery process. To this end, we propose an adaptive transformed $L_1$ minimization (referred to as {\it Ada$TL_1$}), which being a non-convex problem is a generalization of transformed $L_1$ minimization. We discuss the recovery guarantees of {\it Ada$TL_1$} and propose its solver. Further, we compare and contrast its performance against the known methods in the reconstruction of CT as well as the denoising of images.