Non-local Functionals Related to the Total Variation and Connections with Image Processing

Haïm Brezis, Hoai Minh Nguyen

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

We present new results concerning the approximation of the total variation, ∫ Ω| ∇ u| , of a function u by non-local, non-convex functionals of the form Λδ(u)=∫Ω∫Ωδφ(|u(x)-u(y)|/δ)|x-y|d+1dxdy,as δ→ 0 , where Ω is a domain in Rd and φ: [0 , + ∞) → [0 , + ∞) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi’s concept of Γ -convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.

Original languageEnglish (US)
Article number9
JournalAnnals of PDE
Volume4
Issue number1
DOIs
StatePublished - Jun 1 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics
  • Geometry and Topology
  • Mathematical Physics
  • General Physics and Astronomy

Keywords

  • Bounded variation
  • Non-convex functional
  • Non-local functional
  • Sobolev spaces
  • Total variation
  • Γ -Convergence

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