Analysis and Application of a Nonlocal Hessian

Jan Lellmann; Konstantinos Papafitsoros; Carola Schoenlieb; Daniel Spector

doi:10.1137/140993818

Analysis and Application of a Nonlocal Hessian

Jan Lellmann, Konstantinos Papafitsoros, Carola Schoenlieb, Daniel Spector

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

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 language	English (US)
Pages (from-to)	2161-2202
Number of pages	42
Journal	SIAM Journal on Imaging Sciences
Volume	8
Issue number	4
DOIs	https://doi.org/10.1137/140993818
State	Published - 2015
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-10-08
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: This project was supported by King Abdullah University of Science and Technology (KAUST) award KUK-I1-007-43.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1137/140993818

Cite this

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title = "Analysis and Application of a Nonlocal Hessian",

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.",

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note = "KAUST Repository Item: Exported on 2021-10-08 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: This project was supported by King Abdullah University of Science and Technology (KAUST) award KUK-I1-007-43. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1137/140993818",

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AU - Lellmann, Jan

AU - Papafitsoros, Konstantinos

AU - Schoenlieb, Carola

AU - Spector, Daniel

N1 - KAUST Repository Item: Exported on 2021-10-08 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: This project was supported by King Abdullah University of Science and Technology (KAUST) award KUK-I1-007-43. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

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JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

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Analysis and Application of a Nonlocal Hessian

Abstract

Bibliographical note

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