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

TL;DR

This paper introduces a general truncated regularization framework for variational signal and image restoration that better preserves contrast compared to traditional methods, supported by theoretical analysis and practical algorithms.

Contribution

It proposes a novel truncated regularization approach that enhances contrast preservation in variational restoration, with proven theoretical properties and effective optimization algorithms.

Findings

The framework effectively preserves contrast in 1D signals.

Numerical experiments demonstrate improved restoration quality.

Algorithms with convergence guarantees are developed for 2D problems.

Abstract

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 $\ell_2$ 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 $(\tau,+\infty)$ for some positive $\tau$. Some analysis in 1D theoretically demonstrate the good contrast-preserving ability of the framework. We also give optimization algorithms with convergence verification in 2D, where global minimizers of each subproblem (either convex or nonconvenx) are calculated. Experiments numerically show the advantages of the framework.