Existence theory for the EED inpainting problem
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- by M. Bildhauer, M. Cárdenas, M. Fuchs and J. Weickert
- St. Petersburg Math. J. 32, 481-497
- DOI: https://doi.org/10.1090/spmj/1657
- Published electronically: May 11, 2021
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Abstract:
An existence theory is developed for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoration of missing data in an image as the steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. The existence of a weak solution is established by applying the Leray–Schauder fixed point theorem, and it is shown that the set of all possible weak solutions is bounded. Moreover, it is demonstrated that under certain conditions the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.References
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Bibliographic Information
- M. Bildhauer
- Affiliation: Department of Mathematics, Saarland University, Faculty Math. and Computer Sci., P.O. Box 15 11 50, 66041 Saarbrücken, Germany
- Email: bibi@math.uni-sb.de
- M. Cárdenas
- Affiliation: Math. Image Analysis Group, Saarland University, Faculty Math. and Computer Sci., Campus E1.7, 66041 Saarbrücken, Germany
- Email: cardenas@mia.uni-saarland.de
- M. Fuchs
- Affiliation: Department of Mathematics, Saarland University, Faculty Math. and Computer Sci., P.O. Box 15 11 50, 66041 Saarbrücken, Germany
- Email: fuchs@math.uni-sb.de
- J. Weickert
- Affiliation: Math. Image Analysis Group, Saarland University, Faculty Math. and Computer Sci., Campus E1.7, 66041 Saarbrücken, Germany
- Email: weickert@mia.uni-saarland.de
- Received by editor(s): June 5, 2019
- Published electronically: May 11, 2021
- Additional Notes: The research of the second and fourth authors has received funding by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 741215, ERC Advanced Grant INCOVID).
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32, 481-497
- MSC (2020): Primary 94A08; Secondary 68U10
- DOI: https://doi.org/10.1090/spmj/1657
- MathSciNet review: 4099096
Dedicated: Dedicated to Professor Nina Ural’tseva on the occasion of her $85$th birthday.