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

摘要

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 regularization function(potential function) composed of first order difference operators. As contrastsare important features in signals and images, we study, in this paper, thepossibility of contrast-preserving restoration by variational methods. Wepresent both the motivation and implementation of a general truncatedregularization framework. In particular, we show that, in both 1D and 2D, anyconvex or smooth regularization based variational model is impossible or withlow probabilities to preserve edge contrasts. It is better to use thosenonsmooth potential functions flat on $(\tau,+\infty)$ for some positive$\tau$, which are nonconvex. These discussions naturally yield a generalregularization framework based on truncation. Some analysis in 1D theoreticallydemonstrate its good contrast-preserving ability. We also give optimizationalgorithms with convergence verification in 2D, where global minimizers of eachsubproblem (either convex or nonconvenx) are calculated. Experimentsnumerically show the advantages of the framework.