A two-term penalty function for inverse problems with sparsity constrains

Abstract

Inverse problems with sparsity constrains, such Basis Pursuit denoising (BPDN) and Convolutional BPDN (CBPDN), usually use the l 1 -norm as the penalty function; however such choice leads to a solution that is biased towards zero. Recently, several works have proposed and assessed the properties of other non-standard penalty functions (most of them non-convex), which avoid the above mentioned drawback and at the same time are intended to induce sparsity more strongly than the l 1 -norm. In this paper we propose a two-term penalty function consisting of a synthesis between the ?i-norm and the penalty function associated with the Non-Negative Garrote (NNG) thresholding rule. Although the proposed two-term penalty function is non-convex, the total cost function for the BPDN / CBPDN problems is still convex. The performance of the proposed two-term penalty function is compared with other reported choices for practical denoising, deconvolution and convolutional sparse coding (CSC) problems within the BPDN / CBPDN frameworks. Our experimental results show that the proposed two-term penalty function is particularly effective (better reconstruction with sparser solutions) for the CSC problem while attaining competitive performance for the denoising and deconvolution problems.