Simultaneous Patch-Group Sparse Coding with Dual-Weighted ℓp Minimization for Image Restoration

Sparse coding (SC) models have been proven as powerful tools applied in image restoration tasks, such as patch sparse coding (PSC) and group sparse coding (GSC). However, these two kinds of SC models have their respective drawbacks. PSC tends to generate visually annoying blocking artifacts, while GSC models usually produce over-smooth effects. Moreover, conventional ℓ1 minimization-based convex regularization was usually employed as a standard scheme for estimating sparse signals, but it cannot achieve an accurate sparse solution under many realistic situations. In this paper, we propose a novel approach for image restoration via simultaneous patch-group sparse coding (SPG-SC) with dual-weighted ℓp minimization. Specifically, in contrast to existing SC-based methods, the proposed SPG-SC conducts the local sparsity and nonlocal sparse representation simultaneously. A dual-weighted ℓp minimization-based non-convex regularization is proposed to improve the sparse representation capability of the proposed SPG-SC. To make the optimization tractable, a non-convex generalized iteration shrinkage algorithm based on the alternating direction method of multipliers (ADMM) framework is developed to solve the proposed SPG-SC model. Extensive experimental results on two image restoration tasks, including image inpainting and image deblurring, demonstrate that the proposed SPG-SC outperforms many state-of-the-art algorithms in terms of both objective and perceptual quality.

A unified approach for a 1D generalized total variation problem

We study a 1-dimensional discrete signal denoising problem that consists of minimizing a sum of separable convex fidelity terms and convex regularization terms, the latter penalize the differences of adjacent signal values. This problem generalizes the total variation regularization problem. We provide here a unified approach to solve the problem for general convex fidelity and regularization functions that is based on the Karush–Kuhn–Tucker optimality conditions. This approach is shown here to lead to a fast algorithm for the problem with general convex fidelity and regularization functions, and a faster algorithm if, in addition, the fidelity functions are differentiable and the regularization functions are strictly convex. Both algorithms achieve the best theoretical worst case complexity over existing algorithms for the classes of objective functions studied here. Also in practice, our C++ implementation of the method is considerably faster than popular C++ nonlinear optimization solvers for the problem.