Analysis and application of a nonlocal Hessian

Daniel Spector National Chiao Tung University Jan Lellman University of Cambridge Konstantinos Papafitsoros University of Cambridge Carola Schönlieb University of Cambridge

Functional Analysis mathscidoc:1703.12002

SIAM J. Imaging Sci., 8, (4), 2161–2202., 2015
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.
Non-local Hessian, Non-local Total Variation Regularization, Variational Methods, Fast Marching Method, Amoeba Filters
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  title={Analysis and application of a nonlocal Hessian},
  author={Daniel Spector, Jan Lellman, Konstantinos Papafitsoros, and Carola Schönlieb},
  booktitle={ SIAM J. Imaging Sci.},
Daniel Spector, Jan Lellman, Konstantinos Papafitsoros, and Carola Schönlieb. Analysis and application of a nonlocal Hessian. 2015. Vol. 8. In SIAM J. Imaging Sci.. pp.2161–2202..
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