Resolving scaling ambiguities with the ℓ1/ℓ2 norm in a blind deconvolution problem with feedback

Abstract

Compared to more mundane blind deconvolution problems, blind deconvolution in seismic applications involves a feedback mechanism related to the free surface. The presence of this feedback mechanism gives us an unique opportunity to remove ambiguities that have plagued blind deconvolution for a long time. While beneficial, this feedback by itself is insufficient to remove the ambiguities even with`1 constraints. However, when paired with an`1/`2 constraint the feedback allows us to resolve the scaling ambiguity under relatively mild assumptions. Inspired by lifting approaches, we propose to split the sparse signal into positive and negative components and apply an`1/`2 constraint to the difference, thereby obtaining a constraint that is easy to implement. Numerical experiments demonstrate robustness to the initialization as well as to noise in the data.

Accelerated Schemes for the $L_1/L_2$ Minimization

“…Note that w(•) can not be explicitly determined from (27). By applying a proximal gradient method (PGM) [45], [46], [47] on the model (26), we obtain the following scheme…”

“…Since the gradient of the L 2 norm is ∇ x 2 = x x 2 , Lemma 2 implies that the gradient of Euclidean norm is Lipschitz-continuous in the domain {x | Ax = b}. The next lemma is about the Lipschitz property for the implicit function w(•) that satisfies (27).…”

“…Esser et al [25], [14] focused on nonnegative signals and established the equivalence between L 1 /L 2 and L 0 . The ratio model was later formulated as a nonlinear constraint that was solved by a lifted approach [26], [27]. Some applications of L 1 /L 2 include blind deconvolution [28], [29] and sparse filtering [30], [31].…”

A Noise-Robust Method with Smoothed \ell_1/\ell_2 Regularization for Sparse Moving-Source Mapping

“…However, in [11], Benichoux et al showed that use of the norm ℓ 1 suffers from scaling and shift ambiguities due to the nonlinear relation between the blurring kernel and the signal, as also discussed in [12,13]. Felix et al extended this result for the case of ℓ p , (p < 1)-norm in [14]. In particular, both of these reports showed that using the ℓ 1 /ℓ 2 function can overcome this difficulty.…”