Theoretical foundation of the weighted Laplace inpainting problem

Laurent Hoeltgen; Andreas Kleefeld; Isaac Harris; Michael Breuss

Applications of Mathematics (2019)

  • Volume: 64, Issue: 3, page 281-300
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Hoeltgen, Laurent, et al. "Theoretical foundation of the weighted Laplace inpainting problem." Applications of Mathematics 64.3 (2019): 281-300. <http://eudml.org/doc/294678>.

@article{Hoeltgen2019,
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.},
author = {Hoeltgen, Laurent, Kleefeld, Andreas, Harris, Isaac, Breuss, Michael},
journal = {Applications of Mathematics},
keywords = {image inpainting; image reconstruction; Laplace equation; Laplace interpolation; mixed boundary condition; partial differential equation; weighted Sobolev space},
language = {eng},
number = {3},
pages = {281-300},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Theoretical foundation of the weighted Laplace inpainting problem},
url = {http://eudml.org/doc/294678},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Hoeltgen, Laurent
AU - Kleefeld, Andreas
AU - Harris, Isaac
AU - Breuss, Michael
TI - Theoretical foundation of the weighted Laplace inpainting problem
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 281
EP - 300
AB - 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.
LA - eng
KW - image inpainting; image reconstruction; Laplace equation; Laplace interpolation; mixed boundary condition; partial differential equation; weighted Sobolev space
UR - http://eudml.org/doc/294678
ER -

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