Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ${\ell _1}/{\ell _2}$ Regularization

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.