Analysis and application of a nonlocal Hessian

Jan Lellmann, Konstantinos Papafitsoros, Carola Schönlieb, Daniel Spector

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

In this work we introduce a formulation for a nonlocal Hessian that combines the ideas of higherorder and nonlocal regularization for image restoration, extending the idea of nonlocal gradients to higher-order derivatives. By intelligently choosing the weights, the model allows us to improve on the current state of the art higher-order method, total generalized variation, with respect to overall quality and preservation of jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights it can be shown to converge to classical second-order regularizers, and in fact it allows a novel characterization of higher-order Sobolev and BV spaces.

Original languageEnglish
Pages (from-to)2161-2202
Number of pages42
JournalSIAM Journal on Imaging Sciences
Volume8
Issue number4
DOIs
Publication statusPublished - 2015 Oct 6
Externally publishedYes

Keywords

  • Amoeba filters
  • Fast marching method
  • Nonlocal Hessian
  • Nonlocal total variation regularization
  • Variational methods

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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