Asymptotic analysis of higher-order perturbations of the Perona--Malik functional

TL;DR

This paper investigates the asymptotic behavior of higher-order perturbations of the Perona-Malik functional, revealing a Gamma-limit characterized by a free-discontinuity functional with explicit surface density, advancing understanding of image processing regularizations.

Contribution

It provides a rigorous Gamma-convergence analysis of higher-order singular perturbations of the Perona-Malik functional, identifying the limit as a free-discontinuity functional with explicit surface energy.

Findings

Gamma-limit characterized as a free-discontinuity functional

Surface density characterized via a one-dimensional optimal-profile problem

Extension of previous second-order results to higher-order perturbations

Abstract

The Gamma-limit of higher-order singular perturbations of the Perona-Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a k-th order regularization designed to balance bulk and interfacial effects. A compactness result is obtained, and the Gamma-limit is identified as a free-discontinuity functional on SBV, given by the sum of the Dirichlet energy and a surface term proportional to the jump amplitude to the power 1/k. The surface density is characterized through a one-dimensional optimal-profile problem with homogeneous boundary conditions on derivatives up to order k-1. As a consequence, the limit of the same energies at a different scaling is determined. That scaling had been previously studied in the second-order case to address the so-called staircasing phenomenon.