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

Andrea Braides, Irene Fonseca

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.

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

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.
Paper Structure (12 sections, 19 theorems, 93 equations, 2 figures)

This paper contains 12 sections, 19 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of the convergence results
  3. Reduction to dimension one
  4. Preliminaries
  5. Estimates for sets with large derivatives below the threshold $c_\varepsilon/\varepsilon$
  6. Estimate of sets below threshold without points of small derivatives
  7. Fine analysis of small sets below threshold
  8. Energy densities
  9. A lower bound with (large) linear growth at $0$.
  10. Asymptotic optimal lower bound
  11. One-dimensional $\Gamma$-limit and compactness
  12. Surface scaling of the Perona--Malik energies

Key Result

Theorem 2.1

Let $\Omega$ be a bounded open set in $\mathbb R^d$ and let be defined on $H^k(\Omega)$, where $\|\cdot\|$ denotes the operator norm of a $k$-th order tensor. Then for every sequence $u_\varepsilon$ such that $F_\varepsilon(u_\varepsilon)\le S<+\infty$ there exist a subsequence and constants $C_\varepsilon$ such that $u_\varepsilon+C_\varepsilon$ converge in with the constant $m_k$ given by

Figures (2)

  • Figure 1: A function $u_\varepsilon$ and the related $[\tau,\sigma]$ subdivision
  • Figure 2: A function $u_\varepsilon$ and the related $[\tau^*,\sigma^*]$ subdivision

Theorems & Definitions (41)

  • Theorem 2.1
  • Lemma 3.1
  • Proposition 3.2
  • Remark 3.3: Estimates by localized interpolations
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Proposition 3.7: Asymptotics for the relative measure of intervals below threshold
  • ...and 31 more