A general truncated regularization framework for contrast-preserving variational signal and image restoration: Motivation and implementation

Variational methods have become an important kind of methods in signal and image restoration—a typical inverse problem. One important minimization model consists of the squared ?_2 data fidelity(corresponding to Gaussian noise) and a regularization term constructed by a potential function composed of first order difference operators. It is well known that total variation(TV) regularization, although achieved great successes,suffers from a contrast reduction effect. Using a typical signal, we show that, actually all convex regularizers and most nonconvex regularizers have this effect. With this motivation, we present a general truncated regularization framework. The potential function is a truncation of existing nonsmooth potential functions and thus flat on(τ, +∞) for some positive τ. Some analysis in 1 D theoretically demonstrate the good contrast-preserving ability of the framework. We also give optimization algorithms with convergence verification in 2 D, where global minimizers of each subproblem(either convex or nonconvex) are calculated. Experiments numerically show the advantages of the framework.