Subspace Correction Methods for a Class of Non-smooth and Non-additive Convex Variational Problems in Image Processing

The minimization of a functional composed of a non-smooth and non-additive regularization term and a combined L1 and L2 data-fidelity term is proposed. It is shown analytically and numerically that the new model has noticeable advantages over popular models in image processing tasks. For the numerical minimization of the new objective, subspace correction methods are introduced which guarantee the convergence and monotone decay of the associated energy along the iterates. Moreover, an estimate of the distance between the outcome of the subspace correction method and the global minimizer of the non-smooth objective is derived. This estimate and numerical experiments for image denoising, inpainting, and deblurring show that in practice the proposed subspace correction methods indeed converge to the global solution of the underlying minimization problem.