In this talk, we address a wide class of image deconvolution or reconstruction situations where a sought image is recovered from degraded observed image. The sought solution is defined to be the minimizer of an objective function combining a data-fidelity term and a edge-preserving, convex regularization term. Our objective is to speed up the calculation of the solution in a wide range of situations. We propose a method applying pertinent preconditioning to an adapted half-quadratic equivalent form of the objective function. We focus specifically on Huber regularization. We exhibit the possibility get very fast calculations while preserving the edges in the solution.