Variational Image Denoising with Adaptive Constraint Sets

  • Frank Lenzen
  • Florian Becker
  • Jan Lellmann
  • Stefania Petra
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)


We propose a generalization of the total variation (TV) minimization method proposed by Rudin, Osher and Fatemi. This generalization allows for adaptive regularization, which depends on the minimizer itself. Existence theory is provided in the framework of quasi-variational inequalities. We demonstrate the usability of our approach by considering applications for image and movie denoising.


solution dependent adaptivity quasi-variational inequalities spatio-temporal TV anisotropic TV image denoising 


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  1. 1.
    Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: Vision, Modeling, and Visualization Proceedings, pp. 293–300 (2006)Google Scholar
  2. 2.
    Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization (2003)Google Scholar
  3. 3.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1–2), 89–97 (2004)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chan, D., Pang, T.S.: The generalized quasi-variational inequality problem. Math. Operat. Res. 7(2), 211–222 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dong, Y., Hintermüller, M.: Multi-scale total variation with automated regularization parameter selection for color image restoration. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 271–281. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Eberly, D.: Distance from a point to an ellipse in 2D. Technical report (2002)Google Scholar
  8. 8.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Grasmair, M.: Locally adaptive total variation regularization. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 331–342. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Grasmair, M., Lenzen, F.: Anisotropic Total Variation Filtering. Appl. Math. Optim. 62(3), 323–339 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005) (electronic)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Rellich, F.: Störungstheorie der Spektralzerlegung, I. Math. Ann (1936)Google Scholar
  13. 13.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Setzer, S., Steidl, G., Teuber, T.: Restoration of images with rotated shapes. Numerical Algorithms 48, 49–66 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Steidl, G., Teuber, T.: Anisotropic smoothing using double orientations. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 477–489. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Wilkinson, J.H.: The algebraic eigenvalue problem. In: Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Frank Lenzen
    • 1
    • 2
  • Florian Becker
    • 1
  • Jan Lellmann
    • 1
  • Stefania Petra
    • 1
  • Christoph Schnörr
    • 1
  1. 1.HCI & IPA, Heidelberg UniversityHeidelbergGermany
  2. 2.Intel Visual Computing InstituteSaarland UniversitySaarbrückenGermany

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