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Variational methods have become an important kind of methods in signal andimage restoration - a typical inverse problem. One important minimization modelconsists of the squared $\ell_2$ data fidelity (corresponding to Gaussiannoise) and a regularization term constructed by a potential function composedof first order difference operators. It is well known that total variation (TV)regularization, although achieved great successes, suffers from a contrastreduction effect. Using a typical signal, we show that, actually all convexregularizers and most nonconvex regularizers have this effect. With thismotivation, we present a general truncated regularization framework. Thepotential function is a truncation of existing nonsmooth potential functionsand thus flat on $(\tau,+\infty)$ for some positive $\tau$. Some analysis in 1Dtheoretically demonstrate the good contrast-preserving ability of theframework. We also give optimization algorithms with convergence verificationin 2D, where global minimizers of each subproblem (either convex or nonconvenx)are calculated. Experiments numerically show the advantages of the framework.

A General Truncated Regularization Framework for Contrast-Preserving Variational Signal and Image Restoration: Motivation and Implementation