Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed <formula formulatype="inline"><tex Notation="TeX">${\ell _1}/{\ell _2}$</tex></formula> Regularization

Abstract

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.

Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

“…In addition, we show that the optimization-based deconvolution approach is robust to dense and sparse noise. Recent works in this direction focus on randomized measurements [2,47] and alternating optimization [57]. Quantifying the discretization error incurred by solving`1-norm minimization on a fine grid instead of solving the continuous TV norm minimization problem, in the spirit of [31].…”

“…Corruptions Per Spike Studying blind deconvolution, i.e., joint estimation of the convolution kernel and the signal, for deterministic kernels. Recent works in this direction focus on randomized measurements [2,47] and alternating optimization [57]. Throughout the appendices, we assume there is some compact interval I R containing the support of the true measure (given in (1.1)).…”

A fast algorithm for sparse multichannel blind deconvolution

“…In Kazemi and Sacchi (2014), the authors use a Bayesian approach (Kormylo and Mendel, 1983;Cheng et al, 1996;Rosec et al, 2003;Repetti et al, 2015) to estimate the reflectivity. Bayesian estimation usually aims at estimating the parameters s of a statistical model by solving the following problem:…”

“…The hybrid l 1 ∕l 2 -norm function achieves this end automatically. In fact, if one thinks of the loss function as the log likelihood of a random variable, the corresponding distribution has a peak similar to the Gaussian distribution, but a tail resembles the Laplace distribution (Huber and Ronchetti, 2009;Repetti et al, 2015).…”

“…The hybrid l 1 ∕l 2 -norm function can also be seen as a smoothed version of the l 1 norm (Bube and Langan, 1997;Li et al, 2010;Zhang and Claerbout, 2011;Repetti et al, 2015), and hence it can be used as a measure of sparsity of a signal (Donoho, 2006). To illustrate such a relationship, consider the function,…”