Analysis and application of a nonlocal Hessian

Jan Lellmann, Konstantinos Papafitsoros, Carola Schönlieb, Daniel Eli Spector

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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
Issue number4
StatePublished - 6 Oct 2015


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


Dive into the research topics of 'Analysis and application of a nonlocal Hessian'. Together they form a unique fingerprint.

Cite this