A Novel Fast Tensor-Based Preconditioner for Image Restoration.

Abstract

Image restoration is one of the main parts of image processing. Mathematically, this problem can be modeled as a large-scale structured ill-posed linear system. Ill-posedness of this problem results in low-convergence rate of iterative solvers. For speeding up the convergence, preconditioning usually is used. Despite the existing preconditioners for image restoration, which are constructed based on approximations of the blurring matrix, in this paper, we propose a novel preconditioner with a different viewpoint. Here, we show that image restoration problem can be modeled as a tensor contractive linear equation. This modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator. Due to the particular structure of the blurring tensor for zero boundaries, we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner. Experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well-known preconditioners.