Image Restoration by Second‐Order Total Generalized Variation and Wavelet Frame Regularization

Abstract

It has been proved that total generalized variation (TGV) can better preserve edges while suppressing staircase effect. In this paper, we propose an effective hybrid regularization model based on second-order TGV and wavelet frame. The proposed model inherits the advantages of TGV regularization and wavelet frame regularization, can eliminate staircase effect while protecting the sharp edge, and simultaneously has good capability of sparsely estimating the piecewise smooth functions. The alternative direction method of multiplier (ADMM) is employed to solve the new model. Numerical results show that our proposed model can preserve more details and get higher image visual quality than some current state-of-the-art methods.

Combined Non-convex Second-order Total Variation with Overlapping Group Sparsity for Full Waveform Inversion

“…Li et al [12] developed an effective sparsity-promoting TV regularization method for FWI that incorporates nonlocal similarity in the model, which can be seen as a generalization of TV regularization with adaptive dictionary learning. Recently, some researchers found that the second-order formula for the total generalized variation can generate superior results compared with the classical first-order TV [13][14][15]. Gao et al [16] introduced a total generalized p-variation regularization scheme into acoustic and elastic-waveform inversion, and proposed an efficient iterative algorithm to implement the defined regularization scheme by using the split-Bregman strategy.…”

Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability

“…On the other hand, the classical maximal operator S Ω was originally introduced by Chen and Lin [21] who proved that if Ω ∈ C 1 (S −1 ), then S Ω is of type ( , ) for any > 2 /(2 −1) and the range of is best possible. Subsequently, the mapping properties of S Ω have been discussed extensively by many authors (see [22][23][24][25][26], for example). Particularly, Al-Salman [23] proved the following result.…”