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Theoretical foundation of the weighted laplace inpainting problem

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Abstract

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.

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Authors and Affiliations

  1. Institute for Mathematics, Brandenburg Technical University, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany

    Laurent Hoeltgen & Michael Breuss

  2. Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425, Jülich, Germany

    Andreas Kleefeld

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Isaac Harris

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  1. Laurent Hoeltgen
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  2. Andreas Kleefeld
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Correspondence to Laurent Hoeltgen, Andreas Kleefeld, Isaac Harris or Michael Breuss.

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Hoeltgen, L., Kleefeld, A., Harris, I. et al. Theoretical foundation of the weighted laplace inpainting problem. Appl Math 64, 281–300 (2019). https://doi.org/10.21136/AM.2019.0206-18

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