Denoising of Image Gradients and Constrained Total Generalized Variation

  • Birgit KomanderEmail author
  • Dirk A. Lorenz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


We derive a denoising method that uses higher order derivative information. Our method is motivated by work on denoising of normal vectors to the image which then are used for a better denoising of the image itself. We propose to denoise image gradients instead of image normals, since this leads to a convex optimization problem. We show how the denoising of the image gradient and the image itself can be done simultaneously in one optimization problem. It turns out that the resulting problem is similar to total generalized variation denoising, thus shedding more light on the motivation of the total generalized variation penalties. Our approach, however, works with constraints, rather than penalty functionals. As a consequence, there is a natural way to choose one of the parameters of the problems and we motivate a choice rule for the second involved parameter.



This material was based upon work supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


  1. 1.
    Lysaker, M., Osher, S., Tai, X.C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. Img. Proc. 13(10), 1345–1357 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Komander, B., Lorenz, D., Fischer, M., Petz, M., Tutsch, R.: Data fusion of surface normals and point coordinates for deflectometric measurements. J. Sens. Sens. Syst. 3, 281–290 (2014)CrossRefGoogle Scholar
  3. 3.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brinkmann, E.-M., Burger, M., Grah, J.: Regularization with sparse vector fields: from image compression to TV-type reconstruction. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 191–202. Springer, Cham (2015). doi: 10.1007/978-3-319-18461-6_16 Google Scholar
  5. 5.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65(2), 480–491 (2011)CrossRefGoogle Scholar
  6. 6.
    Lorenz, D.A., Worliczek, N.: Necessary conditions for variational regularization schemes. Inverse Prob. 29, 075016 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.TU BraunschweigBraunschweigGermany

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