SOOT l1/l2 norm ratio sparse blind deconvolution

Sparse blind deconvolution in Gaussian noise with a non-convex regularized l1/l2 norm ratio penalty

Blind deconvolution refers to deconvolution of a signal without exact knowledge of the impulse response function used in the convolution. This is usually achieved by adding appropriate assumptions on the input and/or the impulse response to restore the output. We take into account here the sparsity or parsimony of the input signal. It is generally measured with the l0 cost function, and often addressed with a l1 norm penalty. The l1/l2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works. 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. The SOOT toolbox (for Smooth One-Over-Two norm ratio) implements the method published in the paper "Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization", Audrey Repetti, Mai Quyen Pham, Laurent Duval, Emilie Chouzenoux, Jean-Christophe Pesquet in IEEE Signal Processing Letters, May 2015, http://dx.doi.org/10.1109/LSP.2014.2362861
The script demo_SOOT_toolbox.m runs the demonstration of the method.