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Journal of Mathematical Imaging and Vision

, Volume 55, Issue 3, pp 401–427 | Cite as

A Second-Order TV-Type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data

  • Ronny Bergmann
  • Andreas Weinmann
Article

Abstract

In this paper, we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space-valued data. These kinds of data appear when dealing with nonlinear color spaces such as HSV, and they can be obtained by changing the space domain of, e.g., an optical flow field to polar coordinates. For such nonlinear data spaces, we develop algorithms for the solution of the corresponding second-order total variation-type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and apply the algorithms to concrete problems.

Keywords

Higher order total variation minimization Vector-valued TV Cyclic data Combined denoising and inpainting Cyclic proximal point algorithm 

Mathematics Subject Classification

65K05 65K10 68U10 94A08 

Notes

Acknowledgments

This research was started when RB visited the Helmholtz-Zentrum München in summer 2014. We thank Gabriele Steidl for valuable discussions and both reviewers for their valuable comments and suggestions. The authors acknowledge funding by the DFG project BE 5888/2-1 & WE 5886/3-1. AW is supported by the Helmholtz Association within the young investigator group VH-NG-526. AW also acknowledges the support by the DFG scientific network “Mathematical Methods in Magnetic Particle Imaging.”

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Departement of MathematicsTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Department of MathematicsTU MünchenMunichGermany
  3. 3.Department of Mathematics and Natural Sciences, Darmstadt University of Applied SciencesDarmstadtGermany
  4. 4.Fast Algorithms for Biomedical Imaging Group, Helmholtz Center MunichNeuherbergGermany

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