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.
Repetti, A., Pham, M. Q., Duval, L., Chouzenoux, E., & Pesquet, J-C. (2015). Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ℓ1/ℓ2 Regularization. IEEE Signal Processing Letters, 22(5), 539-543. https://doi.org/10.1109/LSP.2014.2362861