A total variation recursive space-variant filter for image denoising

Total Variation (TV) regularization is a widely used convex but non-smooth regularizer in image restoration and reconstruction. Many algorithms involve solving a denoising problem as an intermediate step or in each iteration. Most existing solvers were proposed in the context of a specific application. In this paper, we propose a denoising method which can be used as a proximal mapping (denoising operator) for noises other than additive and Gaussian. We formulate the Maximum A-Posteriori (MAP) estimation in terms of a spatially adaptive and recursive filtering operation on the Maximum Likelihood (ML) estimate. The only dependence on the model is the ML estimate and the second order derivative, which are computed at the beginning and remain fixed throughout the iterative process. The proposed method generalizes the MAP estimation with a quadratic regularizer using an infinite impulse response filter, to the case with TV regularization. Due to the fact that TV is non-smooth and has spatial dependencies, the resulting filter after reweighted least squares formulation of the TV term, is recursive and spatially variant. The proposed method is an instance of the Majorization–Minimization (MM) algorithms, for which convergence conditions are defined and can be shown to be satisfied by the proposed method. The method can also be extended to image inpainting and higher order TV in an intuitively straight-forward manner.