Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ℓ 1 /ℓ 2 Regularization

Audrey Repetti (1) , Mai Quyen Pham (1) , Laurent Duval (2) , Emilie Chouzenoux (1) , Jean-Christophe Pesquet (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 IFPEN - IFP Energies nouvelles

The ℓ 1 /ℓ 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 ℓ 1 /ℓ 2 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 ℓ 1 /ℓ 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 ℓ 1 /ℓ 2 term, on an application to seismic data blind deconvolution. Index Terms Smoothed ℓ 1 /ℓ 2 regularization, norm ratio, sparsity, blind deconvolution, nonconvex optimization, preconditioned forward-backward algorithm, seismic data processing.

Mots clés: concave programming; deconvolution; geophysical signal processing; smoothing methods; norm ratio; Blind deconvolution; sparsity; Optimization; nonconvex minimization; nonsmooth minimization

Domaines: Traitement du signal et de l'image [eess.SP]; Bio-informatique [q-bio.QM]

Audrey Repetti, Mai Quyen Pham, Laurent Duval, Emilie Chouzenoux, Jean-Christophe Pesquet. Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ℓ 1 /ℓ 2 Regularization. IEEE Signal Processing Letters, 2015, 22 (5), pp.539-543. ⟨10.1109/LSP.2014.2362861⟩. ⟨hal-01328398⟩