Applied Mathematics & Optimization

, Volume 62, Issue 3, pp 323–339 | Cite as

Anisotropic Total Variation Filtering

  • Markus GrasmairEmail author
  • Frank Lenzen


Total variation regularization and anisotropic filtering have been established as standard methods for image denoising because of their ability to detect and keep prominent edges in the data. Both methods, however, introduce artifacts: In the case of anisotropic filtering, the preservation of edges comes at the cost of the creation of additional structures out of noise; total variation regularization, on the other hand, suffers from the stair-casing effect, which leads to gradual contrast changes in homogeneous objects, especially near curved edges and corners. In order to circumvent these drawbacks, we propose to combine the two regularization techniques. To that end we replace the isotropic TV semi-norm by an anisotropic term that mirrors the directional structure of either the noisy original data or the smoothed image. We provide a detailed existence theory for our regularization method by using the concept of relaxation. The numerical examples concluding the paper show that the proposed introduction of an anisotropy to TV regularization indeed leads to improved denoising: the stair-casing effect is reduced while at the same time the creation of artifacts is suppressed.


Image denoising Total variation Relaxation Anisotropic filtering Variational denoising 


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  1. 1.
    Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (2000) zbMATHGoogle Scholar
  3. 3.
    Aviles, P., Giga, Y.: Variational integrals on mappings of bounded variation and their lower semicontinuity. Arch. Ration. Mech. Anal. 115(3), 201–255 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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. Akademische Verlagsgesellschaft Aka GmbH, Berlin (2006) Google Scholar
  5. 5.
    Bouchitté, G., Fonseca, I., Mascarenhas, L.: A global method for relaxation. Arch. Ration. Mech. Anal. 145(1), 51–98 (1998) zbMATHCrossRefGoogle Scholar
  6. 6.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) zbMATHGoogle Scholar
  7. 7.
    Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chan, R.H., Setzer, S., Steidl, G.: Inpainting by flexible Haar-wavelet shrinkage. SIAM J. Imaging Sci. 1(3), 273–293 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, New York (2008) zbMATHGoogle Scholar
  10. 10.
    Dal Maso, G.: Integral representation on BV(Ω) of Γ-limits of variational integrals. Manuscripta Math. 30(4), 387–416 (1979/80) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Demengel, F., Temam, R.: Convex functions of a measure and applications. Indiana Univ. Math. J. 33(5), 673–709 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) zbMATHGoogle Scholar
  13. 13.
    Fonseca, I., Müller, S.: Relaxation of quasiconvex functionals in BV(Ω,R p) for integrands f(ξ,u, u). Arch. Ration. Mech. Anal. 123(1), 1–49 (1993) zbMATHCrossRefGoogle Scholar
  14. 14.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing, River Edge (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Grasmair, M.: Relaxation of nonlocal integrals with rational integrands. PhD thesis, University of Innsbruck, Austria, Innsbruck, June 2006 Google Scholar
  16. 16.
    Guidotti, P., Lambers, J.V.: Two new nonlinear nonlocal diffusions for noise reduction. J. Math. Imaging Vis. 33(1), 25–37 (2009) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990) CrossRefGoogle Scholar
  18. 18.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992) zbMATHCrossRefGoogle Scholar
  19. 19.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2009) zbMATHGoogle Scholar
  20. 20.
    Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vis. 12(1), 43–63 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998). European Consortium for Mathematics in Industry zbMATHGoogle Scholar
  22. 22.
    Yosida, K.: Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, vol. 123. Academic Press Inc., New York (1965) zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Computational Science CenterUniversity of ViennaViennaAustria
  2. 2.Heidelberg Collaboratory for Image ProcessingUniversity of HeidelbergHeidelbergGermany

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