Texture Enhancing Based on Variational Image Decomposition

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 5)


In this paper we consider the Augmented Lagrangian Method for image decomposition. We propose a method which decomposes an image into texture, which is characterized to have finite l 1 curvelet coefficients, a cartoon part, which has finite total variation norm, and noise and oscillating patterns, which have finite G-norm. In the second part of the paper we utilize the equivalence of the Augmented Lagrangian Method and the iterative Bregman distance regularization to show that the dual variables can be used for enhancing of particular components. We concentrate on the enhancing feature for the texture and propose two different variants of the Augmented Lagrangian Method for decomposition and12.5pc]The first author has been considered as corresponding author. Please check. enhancing.


Image decomposition Image enhancement Anisotropic diffusion texture Curvelets Total variation 


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  1. 1.
    Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)CrossRefGoogle Scholar
  2. 2.
    Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Candès, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5, 861–899 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cidade, G.A.G., Anteneodo, C., Roberty, N.C., Neto, A.J.S.: A generalized approach for atomic force microscopy image restoration with Bregman distances as Tikhonov regularization terms. Inverse Probl. Sci. Eng. 8, 1068–2767 (2000)CrossRefGoogle Scholar
  5. 5.
    Duval, V., Aujol, J.-F., Vese, L.A.: Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm. J. Math. Imaging Vis. 37, 232–248 (2010).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)MATHGoogle Scholar
  7. 7.
    Garnett, J., Le, T., Meyer, Y., Vese, L.: Image decompositions using bounded variation and generalized homogeneous Besov spaces. Appl. Comput. Harmon. Anal. 23(1), 25–56 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ma, J., Plonka, G.: Combined curvelet shrinkage and nonlinear anisotropic diffusion. IEEE Trans. Image Process. 16, 2198–2206 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, volume 22 of University Lecture Series. American Mathematical Society, Providence, RI (2001)Google Scholar
  10. 10.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H  − 1-norm. Multiscale Model. Simul. 1(3), 349–370 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)MATHCrossRefGoogle Scholar
  13. 13.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging, volume 167 of Applied Mathematical Sciences. Springer, New York (2009)Google Scholar
  14. 14.
    Starck, J.L., Elad, M., Donoho, D.L.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14, 1570–1582 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003) Special issue in honor of the sixtieth birthday of Stanley OsherGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Florian Frühauf
    • 1
  • Carsten Pontow
    • 2
  • Otmar Scherzer
    • 2
    • 3
  1. 1.RinnAustria
  2. 2.Computational Science CenterUniversity of ViennaViennaAustria
  3. 3.Radon Institute of Computational and MathematicsAustrian Academy of SciencesLinzAustria

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