Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed $\ell_1/ell_2$ Regularization

Laurent Duval just sent me the following:

Dear Igor

Nuit Blanche is a nest of choice for many sparsities. The ones of concern here are those approximated by an $l_1/l_2$, or Taxicab/Euclidean norm ratio, which was already covered in some of your posts: 

http://nuit-blanche.blogspot.fr/2012/04/compressive-sensing-this-week.html

http://nuit-blanche.blogspot.fr/2012/03/old-and-new-algorithm-for-blind.html

http://nuit-blanche.blogspot.fr/2008/05/cs-kgg-explanation-of-l0-and-l1.html 

We propose in the following preprint a smoothed, parametrized penalty, termed SOOT for "Smoothed-One-Over-Two" norm ratio, with results on its theoretical convergence, and an algorithm based on proximal methods. It is applied to blind deconvolution, here for seismic data. We hope it could be of interest to your readership. 

Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed $\ell_1/ell_2$ Regularization, Audrey Repetti, Mai Quyen Pham, Laurent Duval, Emilie Chouzenoux, Jean-Christophe Pesquet

[Abstract]

The $\ell_1/ell_2$ 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 $\ell_1/ell_2$ function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such regularization penalties in current restoration methods.

In this paper, we propose a new penalty based on a smooth approximation to the $\ell_1/ell_2$ 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 $\ell_1/ell_2$ term, on an application to seismic data blind deconvolution.

[arXiv Link]

http://arxiv.org/abs/1407.5465

Amitiés

Laurent

******** Travail et autres activités/Work and misc. ********

SIVA (conferences): http://www.laurent-duval.eu/siva-conferences.html

WITS (wavelets): http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html

BLOG (variations): http://laurent-duval.blogspot.com

Thanks Laurent !

From the paper: "The code will be made available at http://www-syscom.univ-mlv.fr/ upon the paper acceptance"

All the blind deconvolution blog entries are at: 

http://nuit-blanche.blogspot.fr/search/label/BlindDeconvolution

Join the CompressiveSensing subreddit or the Google+ Community and post there !

Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.