Calculus of Variations and Geometric Measure Theory

J. Lellmann - K. Papafitsoros - C. Schönlieb - D. Spector

Analysis and Application of a Non-Local Hessian

created by spector on 03 Nov 2014
modified on 12 Oct 2015


Accepted Paper

Inserted: 3 nov 2014
Last Updated: 12 oct 2015

Journal: SIAM J. Imaging Sci.
Pages: 31
Year: 2014
Doi: arXiv:1410.8825


In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to 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 allows a novel characterization of higher-order Sobolev and BV spaces.