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

TL;DR

The paper addresses ill-posed forward-backward diffusion modeled by a Perona-Malik–type equation with a spatially varying growth exponent $p(x)$. It proposes a Sobolev-type regularization and a vanishing viscosity scheme, proving the existence of weak solutions to the regularized problem and establishing uniform energy bounds via a Rothe-time discretization. Using Young measure theory, it passes to the limit as the regularization vanishes, obtaining a Young measure solution $(u,\nu)$ to the original Neumann problem with the gradient representation $\nabla u=\langle\nu,\mathrm{id}\rangle$ and flux convergence $\langle\nu,\vec{q}(x,\cdot)\rangle$, thereby providing a rigorous well-posedness framework for variable-exponent forward-backward diffusion. The results connect variational structure, nonconvex diffusion, and measure-valued limits, explaining observed microstructures and offering a robust mathematical foundation for related image-processing models.

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.

Figures (3)

• Figure 1: Manifestation of the staircasing phenomenon widely observed in discretizations of the Perona-Malik equation, where the graphs of $u(\cdot,t)$ and $u_x(\cdot,t)$ are plotted at time $t = 2$.
• Figure 2: Microramping phenomenon observed for the regularized equation \ref{['P1']} with constant exponent $p$ and $\delta = 0.001$.
• Figure 3: Gentler microramping phenomenon observed for the new regularized equation \ref{['P1']} with $\delta = 0.001$.

Theorems & Definitions (17)

• Theorem 1: Existence of Young measure solutions
• Lemma 1
• Lemma 2
• Lemma 3: Hölder's inequality
• Lemma 4
• Lemma 5: Poincaré inequality
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9