Sparse Representation via Unfolding ℓ 0 -Regularized Dictionary Learning for Image Restoration

Sparse representation via dedicated dictionary learning facilitates image restoration by simultaneously preserving significant representation and removing trivial perturbations. However, existing dictionary learning under strict ℓ0-norm based sparsity constraint lacks of convergence guarantees and suffers from the dilemma between block artifacts and degraded quality in restoring fine details, and could hamper the consequent deep unfolded networks (DUNs). To address these problems, in this paper, we propose the first ℓ0-regularized dictionary learning with a theoretical guarantee of global convergence that achieves globally consistent sparse approximation for image restoration. Specifically, we first formulate an ℓ0-norm based optimization for dictionary learning that simultaneously achieves sparse approximation of local patches and ensures global consistency between the recovered and observed images. Consequently, we develop ℓ0-regularized global-local dictionary learning (ℓ0-GLDL) that alternately achieves sparse code pursuit, local dictionary update, and global image reconstruction based on alternating proximal linearized method. We demonstrate in theory that ℓ0-GLDL converges globally to a critical point. Furthermore, we design a deep unfolded network by unrolling ℓ0 -GLDL into diverse subblocks to strictly realize alternating optimization for stable image restoration. Experimental results demonstrate that the proposed methods achieve state-of-the-art performance and outperform existing deep learning based methods in image restoration tasks such as image denoising, image deblurring, and image inpainting.