On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing

TL;DR

This paper develops a new mathematical framework for analyzing time-dependent PDEs with critical variable exponents, demonstrating existence of solutions and potential applications in image restoration.

Contribution

It introduces the bounded vectorial partial variation space and its variable exponent version, proving existence of solutions for critical vectorial p(t,x)-Laplacian flows, advancing the mathematical theory.

Findings

Existence of weak solutions for critical vectorial p(t,x)-Laplacian flows.

Semigroup solutions for non-time-dependent critical flows.

Potential applications in color image restoration.

Abstract

Variable exponent spaces have found interesting applications in real world problems. Recently, there have been considerable interest in utilizing variational and evolution problems based on variable exponents for imaging applications. The main classes of partial differential equations (PDEs) related to the variable exponents involve the $p(\cdot)$-Laplacian. In imaging applications, the variable exponent can approach the critical value $1$, and this poses unique challenges in proving existence of solutions, which have not been mastered earlier. In this work, we develop some additional functional framework to study the time-dependent parabolic flows with critical variable exponents. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We prove the existence of weak solutions of critical vectorial $p(t,x)$-Laplacian flow in our variable exponent space. For non-time-dependent variable exponent based critical vectorial $p(x)$-Laplacian flow we obtain a semigroup solution. The results are new even in the scalar case. This is a theory-oriented draft, and the full paper will provide detailed experimental results on color image restoration using various example for the variable exponents and compare them traditional PDE based image processing procedures. Our results will indicate the applicability of variable exponent Laplacian flows in image processing in general and image restoration in particular.