A doubly sparse and low-patch-rank prior model for image restoration - He, Hongjin; Zhao, Lulu (Applied mathematical modelling, 2022)
•A unified doubly sparse and low-patch-rank prior model including two complementary sparse terms and one nuclear norm term.•A new low-patch-rank minimization model without total variation regularization.•An implementable three-block alternating minimization algorithm with global convergence and O(1/k) convergence rate.•An extra sparse term under discrete cosine transform is able to improve the performance of the model on image restoration. Image restoration is a core problem in computer vision and image processing. In this paper, we introduce a unified low-patch-rank minimization model, which possesses one nuclear norm regularization term promoting the low-patch-rankness, and two sparse regularization terms including the classical total variation (TV) norm and a general sparse term under certain transform such as discrete cosine transform. By setting balancing parameters, our unified model reduces to the classical TV-regularized low-patch-rank minimization model and yields a new non-TV-regularized low-patch-rank prior image restoration model. Due to the multi-block structure of the model, we introduce a three-block alternating minimization algorithm to find approximate solutions of the proposed models. A series of computational results on image inpainting and deblurring further show that our approaches are reliable to recover high-quality images from degraded ones.