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Front. Appl. Math. Stat. | doi: 10.3389/fams.2019.00027

Modeling variational inpainting methods with splines

 Nada Sissouno1*, Florian Bossmann2 and Tomas Sauer3
  • 1Department of Mathematics, Technical University of Munich, Germany
  • 2Department of Mathematics, Harbin Institute of Technology, China
  • 3Department of Digital Image Processing, Faculty of Computer Science and Mathematics, University of Passau, Germany

Mathematical methods of image inpainting often involve the discretization of a given continuous model. Typically, this is done by a pointwise discretization. We present a method that avoids this by modeling known variational approaches using a finite dimensional spline space. The way we build our algorithm we are able to prove that the basis of the spline space is stable. Further, due to the compact supports and structure of the basis the model involves a sparse system matrix allowing a fast and memory efficient implementation. Besides the analysis of the resulting model, we present a numerical implementation based on the alternating method of multipliers. We compare the results numerically with classical TV inpainting and give examples of applications such as text removal, restoration and noise removal.

Keywords: spline interpolation, Inpainting, Discrete approximation, optimization, variational methods

Received: 22 Dec 2018; Accepted: 17 May 2019.

Edited by:

Volker Michel, University of Siegen, Germany

Reviewed by:

Gerlind Plonka, University of Göttingen, Germany
Matthias Augustin, Saarland University, Germany  

Copyright: © 2019 Sissouno, Bossmann and Sauer. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Nada Sissouno, Department of Mathematics, Technical University of Munich, Munich, Germany, sissouno@ma.tum.de