A gradient system based on anisotropic monochrome image processing with orientation auto-adjustment

TL;DR

This paper analyzes a pseudo-parabolic PDE system for anisotropic monochrome image processing, introducing an energy functional with orientation auto-adjustment, and proves its stability and well-posedness for improved computational efficiency.

Contribution

It presents a simplified energy functional with reduced derivatives, along with rigorous proofs of stability and well-posedness for the associated PDE system.

Findings

Proves well-posedness of the PDE system

Establishes stability of the image denoising process

Provides a mathematically rigorous framework for anisotropic image processing

Abstract

This paper is devoted to the mathematical analysis of a system of pseudo-parabolic partial differential equations governed by an energy functional, associated with anisotropic monochrome image processing. The energy functional is based on the one proposed by [Berkels et al. Cartoon extraction based on anisotropic image classification, SFB 611, 2006], which incorporates an orientation auto-adjustment mechanism, and our energy is a simplified version which reduces the order of derivatives to improve computational efficiency. The aim of this paper is to establish a stable minimization process that addresses some instability of the algorithm caused by the reduction of derivative order. As a part of the study, we here prove Main Theorems concerned with the well-posedness of the pseudo-parabolic system, and provide a mathematically rigorous guarantee for the stability of the image denoising process derived from our system.