A class of forward-backward regularizations of the Perona-Malik equation with variable exponent

TL;DR

This paper introduces a new class of regularizations for the Perona-Malik equation with variable exponents, demonstrating their mathematical properties and potential in image processing applications.

Contribution

It develops a variational framework for forward-backward regularizations of the Perona-Malik equation with variable exponents, including existence results for Young measure solutions.

Findings

Existence of Young measure solutions established

Theoretical results align with numerical observations

Potential applications in image processing

Abstract

This paper investigates a novel class of regularizations of the Perona-Malik equation with variable exponents, of forward-backward parabolic type, which possess a variational structure and have potential applications in image processing. The existence of Young measure solutions to the Neumann initial-boundary value problem for the proposed equation is established via Sobolev approximation and the vanishing viscosity limit. The proofs rely on Rothe's method, variational principles, and Young measure theory. The theoretical results confirm numerical observations concerning the generic behavior of solutions with suitably chosen variable exponents.