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A Well-Posedness Framework for Inpainting Based on Coherence Transport

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Abstract

Image inpainting is the process of touching up damaged or unwanted portions of a picture and is an important task in image processing. For this purpose Bornemann and März (J Math Imaging Vision, 28:259–278, 2007) introduced a very efficient method, called image inpainting based on coherence transport, that fills the missing region by advecting the image information along integral curves of a coherence vector field from the boundary toward the interior of the hole. The mathematical model behind this method is a first-order functional advection partial differential equation posed on a compact domain with all inflow boundaries. We show that this problem is well posed under certain conditions.

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Acknowledgments

The author would like to thank Folkmar Bornemann and Colin Macdonald for their advice and the inspiring discussions. This work was supported in part by the Graduiertenkolleg Angewandte Algorithmische Mathematik (GKAAM) funded by the Deutsche Forschungsgemeinschaft (DFG) at the Technische Universität München (TUM), and by Award KUK-C1-013-04 granted by King Abdullah University of Science and Technology (KAUST).

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  1. Oxford Centre for Collaborative Applied Mathematics, University of Oxford, Radcliffe Observatory Quarter, Andrew Wiles Building, Woodstock Road, Oxford , OX2 6GG, UK

    Thomas März

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  1. Thomas März
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Correspondence to Thomas März.

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Communicated by Peter Olver.

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März, T. A Well-Posedness Framework for Inpainting Based on Coherence Transport. Found Comput Math 15, 973–1033 (2015). https://doi.org/10.1007/s10208-014-9199-7

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