Applications of Mathematics, Vol. 64, No. 3, pp. 281-300, 2019

Theoretical foundation of the weighted Laplace inpainting problem

Laurent Hoeltgen, Andreas Kleefeld, Isaac Harris, Michael Breuss

Received July 31, 2018.   Published online April 24, 2019.

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.
Keywords:  image inpainting; image reconstruction; Laplace equation; Laplace interpolation; mixed boundary condition; partial differential equation; weighted Sobolev space
Classification MSC:  35J15, 35J70, 46E35, 94A08

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Affiliations:   Laurent Hoeltgen, Institute for Mathematics, Brandenburg Technical University, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany, e-mail:; Andreas Kleefeld, Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany, e-mail:; Isaac Harris, Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A., e-mail:; Michael Breuss, Institute for Mathematics, Brandenburg Technical University, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany, e-mail:

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