The wavelet frame systems have been extensively studied due to their capability of sparsely approximating piecewise smooth functions, such as images, and the corresponding wavelet frame-based image restoration models are mostly based on the penalization of the ℓ ₁ norm of wavelet frame coefficients for sparsity enforcement. In this paper, we focus on the image inpainting problem based on the wavelet frame, propose a weighted sparse restoration model, and develop a corresponding efficient algorithm. The new algorithm combines the idea of iterative support detection method, first proposed by Wang and Yin for sparse signal reconstruction, and the split Bregman method for wavelet frame ℓ ₁ model of image inpainting, and more important, naturally makes use of the specific multilevel structure of the wavelet frame coefficients to enhance the recovery quality. This new algorithm can be considered as the incorporation of prior structural information of the wavelet frame coefficients into the traditional ℓ ₁ model. Our numerical experiments show that the proposed method is superior to the original split Bregman method for wavelet frame-based ℓ ₁ norm image inpainting model as well as some typical ℓ _p (0 ≤ p <; 1) norm-based nonconvex algorithms such as mean doubly augmented Lagrangian method, in terms of better preservation of sharp edges, due to their failing to make use of the structure of the wavelet frame coefficients.

Regularization by denoising (RED) framework has shown impressive performance for many imaging inverse problems, by leveraging the denoising method in defining an explicit regularization. In this letter, we propose a novel SLN-RED scheme for image restoration by exploiting the local and nonlocal denoisers simultaneously. Theoretically, we proves that for bounded denoisers, the SLN-RED under ADMM scheme with a continuation strategy converges to a fixed-point. Numerical experiments on deblurring and super-resolution tasks demonstrate promising performance of the proposed algorithm.