Proximity algorithms for image models: denoising

Abstract

This paper introduces a novel framework for the study of the total-variation model for image denoising. In the model, the denoised image is the proximity operator of the total-variation evaluated at a given noisy image. The total-variation can be viewed as the composition of a convex function (the ℓ 1 norm for the anisotropic total-variation or the ℓ 2 norm for the isotropic total-variation) with a linear transformation (the first-order difference operator). These two facts lead us to investigate the proximity operator of the composition of a convex function with a linear transformation. Under the assumption that the proximity operator of a given convex function (e.g., the ℓ 1 norm or the ℓ 2 norm) can be readily obtained, we propose a fixed-point algorithm for computing the proximity operator of the composition of the convex function with a linear transformation. We then specialize this fixed-point methodology to the total-variation denoising models. The resulting algorithms are compared with the Goldstein–Osher split-Bregman denoising algorithm. An important advantage of the fixed-point framework leads us to a convenient analysis for convergence of the proposed algorithms as well as a platform for us to develop efficient numerical algorithms via various fixed-point iterations. Our numerical experience indicates that the methods proposed here perform favorably.