Sparsity reconstruction using nonconvex TGpV-shearlet regularization and constrained projection

In many sparsity-based image processing problems, compared with the convex ℓ₁ norm approximation of the nonconvex ℓ₀ quasi-norm, one can often preserve the structures better by taking full advantage of the nonconvex ℓ_p quasi-norm (0≤p<1). In this paper, we propose a nonconvex ℓ_p quasi-norm approximation in the total generalized variation (TGV)-shearlet regularization for image reconstruction. By introducing some auxiliary variables, the nonconvex nonsmooth objective function can be solved by an efficient alternating direction method of multipliers with convergence analysis. Especially, we use a generalized iterated shrinkage operator to deal with the ℓ_p quasi-norm subproblem, which is easy to implement. Extensive experimental results show clearly that the proposed nonconvex sparsity approximation outperforms some state-of-the-art algorithms in both the visual and quantitative measures for different sampling ratios and noise levels.