Skip to main content
SpringerLink
Log in

Nonlocal Inpainting of Manifold-Valued Data on Finite Weighted Graphs

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

Included in the following conference series:

Abstract

Recently, there has been a strong ambition to translate models and algorithms from traditional image processing to non-Euclidean domains, e.g., to manifold-valued data. While the task of denoising has been extensively studied in the last years, there was rarely an attempt to perform image inpainting on manifold-valued data. In this paper we present a nonlocal inpainting method for manifold-valued data given on a finite weighted graph. We introduce a new graph infinity-Laplace operator based on the idea of discrete minimizing Lipschitz extensions, which we use to formulate the inpainting problem as PDE on the graph. Furthermore, we derive an explicit numerical solving scheme, which we evaluate on two classes of synthetic manifold-valued images.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    open source, http://www.mathematik.uni-kl.de/imagepro/members/bergmann/mvirt/.

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton and Oxford (2008)

    Book  MATH  Google Scholar 

  2. Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A second order non-smooth variational model for restoring manifold-valued images. SIAM J. Sci. Comput. 38(1), A567–A597 (2016)

    Article  MATH  Google Scholar 

  3. Basser, P., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)

    Article  Google Scholar 

  4. Bergmann, R., Persch, J., Steidl, G.: A parallel Douglas-Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric hadamard manifolds. SIAM J. Imaging Sci. 9(4), 901–937 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bergmann, R., Tenbrinck, D.: A graph framework for manifold-valued data (2017). arXiv preprint https://arxiv.org/abs/1702.05293

  6. Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: Proceedings of the IEEE CVPR 2005, vol. 2, pp. 60–65 (2005)

    Google Scholar 

  7. Caselles, V., Morel, J.M., Sbert, C.: An axiomatic approach to image interpolation. Trans. Image Process. 7(3), 376–386 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Crandall, M., Evans, L., Gariepy, R.: Optimal lipschitz extensions and the infinity laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)

    MATH  MathSciNet  Google Scholar 

  9. Cremers, D., Strekalovskiy, E.: Total cyclic variation and generalizations. J. Math. Imaging Vis. 47(3), 258–277 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elmoataz, A., Desquesnes, X., Lezoray, O.: Non-local morphological PDEs and p-Laplacian equation on graphs with applications in image processing and machine learning. IEEE J. Sel. Topics Signal Process. 6(7), 764–779 (2012)

    Article  MATH  Google Scholar 

  11. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  12. Lellmann, J., Strekalovskiy, E., Koetter, S., Cremers, D.: Total variation regularization for functions with values in a manifold. In: Proceedings of the IEEE ICCV 2013, pp. 2944–2951 (2013)

    Google Scholar 

  13. Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, Chichester (2000)

    MATH  Google Scholar 

  14. Massonnet, D., Feigl, K.L.: Radar interferometry and its application to changes in the earth’s surface. Rev. Geophys. 36(4), 441–500 (1998)

    Article  Google Scholar 

  15. Moakher, M., Batchelor, P.G.: Symmetric positive-definite matrices: From geometry to applications and visualization. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 285–298. Springer, Heidelberg (2006)

    Google Scholar 

  16. Oberman, A.M.: A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7(4), 2226–2257 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Kaiserslautern, 67663, Kaiserslautern, Germany

    Ronny Bergmann

  2. Institute for Computational and Applied Mathematics, Westfälische Wilhelms-Universität Münster, 48149, Münster, Germany

    Daniel Tenbrinck

Authors
  1. Ronny Bergmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Tenbrinck
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ronny Bergmann .

Editor information

Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Bergmann, R., Tenbrinck, D. (2017). Nonlocal Inpainting of Manifold-Valued Data on Finite Weighted Graphs. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_70

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68445-1_70

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68444-4

  • Online ISBN: 978-3-319-68445-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation