Half-quadratic regularization, preconditioning and applications

The article addresses 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 an 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. The optimal solution is then found using an alternating minimization (AM) scheme. We focus specifically on Huber regularization. We exhibit the possibility of getting very fast calculations while preserving the edges in the solution. Preliminary numerical results are reported to illustrate the effectiveness of our method