Graph Laplacian for image deblurring

Image deblurring is a relevant problem in many fields of science and engineering. To solve this problem, many different approaches have been proposed, and, among the various methods, variational ones are extremely popular. These approaches substitute the original problem with a minimization problem where the functional is composed of two terms, a data fidelity term and a regularization term. In this paper we propose, in the classical non-negative constrained '2-'1 minimization framework, the use of the graph Laplacian as regularization operator. Firstly, we describe how to construct the graph Laplacian from the observed noisy and blurred image. Once the graph Laplacian has been built, we efficiently solve the proposed minimization problem by splitting the convolution operator and the graph Laplacian by the Alternating Direction Multiplier Method (ADMM). Some selected numerical examples show the good performances of the proposed algorithm.