A Smoothed l 0 -Norm and l 1 -Norm Regularization Algorithm for Computed Tomography

Abstract

The nonmonotone alternating direction algorithm (NADA) was recently proposed for effectively solving a class of equality-constrained nonsmooth optimization problems and applied to the total variation minimization in image reconstruction, but the reconstructed images suffer from the artifacts. Though by the $l_{\mathrm{0}}$-norm regularization the edge can be effectively retained, the problem is NP hard. The smoothed $l_{\mathrm{0}}$-norm approximates the $l_{\mathrm{0}}$-norm as a limit of smooth convex functions and provides a smooth measure of sparsity in applications. The smoothed $l_{\mathrm{0}}$-norm regularization has been an attractive research topic in sparse image and signal recovery. In this paper, we present a combined smoothed $l_{\mathrm{0}}$-norm and $l_{\mathrm{1}}$-norm regularization algorithm using the NADA for image reconstruction in computed tomography. We resolve the computation challenge resulting from the smoothed $l_{\mathrm{0}}$-norm minimization. The numerical experiments demonstrate that the proposed algorithm improves the quality of the reconstructed images with the same cost of CPU time and reduces the computation time significantly while maintaining the same image quality compared with the $l_{\mathrm{1}}$-norm regularization in absence of the smoothed $l_{\mathrm{0}}$-norm.