Transformed sparsity-boosted low-rank model for image inpainting with non-convex γ-norm regularization and non-local prior

Low-rank prior has important applications in image restoration tasks, particularly in filling in missing information through low-rank matrix completion models. Although the truncated nuclear norm is a classic low-rank algorithm, practical solutions often rely on convex regularized nuclear norm to approximate the rank function, which limits its approximation ability and leads to blurry edges and loss of details. To improve restoration performance, we introduce a non-convex gamma-norm. Theoretical analysis shows that the gamma-norm approximates the rank function more accurately than the nuclear norm, leading to a novel non-convex low- rank approximation model. Furthermore, we enhance the model by introducing transform domain sparse regularization, aimed at capturing more local details and texture information, thereby improving inpainting quality. Addressing the limitations of traditional low-rank matrix restoration models in cases of entire row or column missing, we introduce a multi-pixel window strategy based on the new model, utilizing non-local similarity to search for similar blocks in the multi-pixel neighborhood of the target block to restore the entire column and eliminate residual column artifacts. Our method demonstrates excellent performance. We compare it with several state-of-the-art image restoration techniques across multiple tasks, including pixel restoration, text and scratch removal, column inpainting, and cloud removal. Experimental results prove that our method shows significant advantages in both visual quality and quantitative evaluation.