Fast Solvers for Cahn--Hilliard Inpainting

Jessica Bosch; David Kay; Martin Stoll; Andrew J. Wathen

doi:10.1137/130921842

Fast Solvers for Cahn--Hilliard Inpainting

Jessica Bosch, David Kay, Martin Stoll, Andrew J. Wathen

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

36 Scopus citations

Abstract

The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results. © 2014 Society for Industrial and Applied Mathematics.

Original language	English (US)
Pages (from-to)	67-97
Number of pages	31
Journal	SIAM Journal on Imaging Sciences
Volume	7
Issue number	1
DOIs	https://doi.org/10.1137/130921842
State	Published - Jan 2 2014
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: This research was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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title = "Fast Solvers for Cahn--Hilliard Inpainting",

abstract = "The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results. {\textcopyright} 2014 Society for Industrial and Applied Mathematics.",

author = "Jessica Bosch and David Kay and Martin Stoll and Wathen, {Andrew J.}",

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AU - Bosch, Jessica

AU - Kay, David

AU - Stoll, Martin

AU - Wathen, Andrew J.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: This research was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2014/1/2

Y1 - 2014/1/2

N2 - The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results. © 2014 Society for Industrial and Applied Mathematics.

AB - The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results. © 2014 Society for Industrial and Applied Mathematics.

UR - http://hdl.handle.net/10754/598319

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DO - 10.1137/130921842

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EP - 97

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

IS - 1

ER -

Fast Solvers for Cahn--Hilliard Inpainting

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