Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization

TL;DR

This paper introduces a smooth approximation to the l1/l2 regularization for blind deconvolution, along with a proximal algorithm, improving the optimization process and demonstrating effectiveness on seismic data.

Contribution

It proposes a novel smooth penalty for l1/l2 regularization and develops a proximal algorithm with convergence guarantees for blind deconvolution.

Findings

Effective in seismic data blind deconvolution

Outperforms recent alternating optimization methods

Provides theoretical convergence results

Abstract

The l1/l2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the l1/l2 function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the l1/l2 function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact l1/l2 term, on an application to seismic data blind deconvolution.