On a Parabolic-Hyperbolic Filter for Multicolor Image Noise Reduction

TL;DR

This paper introduces a new PDE-based anisotropic filter for multicolor image noise reduction, generalizing previous models through hyperbolic relaxation and establishing well-posedness and long-term behavior.

Contribution

It presents a novel hyperbolic relaxation of existing PDE filters, with rigorous mathematical analysis of well-posedness and solution properties.

Findings

Proved well-posedness of the proposed PDE model.

Established existence and uniqueness of solutions.

Analyzed the long-time behavior of the filter.

Abstract

We propose a novel PDE-based anisotropic filter for noise reduction in multicolor images. It is a generalization of Nitzberg & Shiota's (1992) model being a hyperbolic relaxation of the well-known parabolic Perona & Malik's filter (1990). First, we consider a `spatial' molifier-type regularization of our PDE system and exploit the maximal $L^{2}$-regularity theory for non-autonomous forms to prove a well-posedness result both in weak and strong settings. Again, using the maximal $L^{2}$-regularity theory and Schauder's fixed point theorem, respective solutions for the original quasilinear problem are obtained and the uniqueness of solutions with a bounded gradient is proved. Finally, the long-time behavior of our model is studied.