  Image deblurring is relevant 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 are
characterized by substituting the original problem with a minimization one
where the functional is composed of two terms, a data fidelity term and a
regularization term. In this paper we propose, in the classical $\ell^2-\ell^1$
minimization with the non-negativity constraint of the solution, 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 solve efficiently the proposed
minimization problem splitting the convolution operator and the graph Laplacian
by the alternating direction method of multipliers (ADMM). Some selected
numerical examples show the good performances of the proposed algorithm.
