Cartoon-Texture-Noise Decomposition with Transport Norms

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

Abstract

We investigate the problem of decomposing an image into three parts, namely a cartoon part, a texture part and a noise part. We argue that norms originating in the theory of optimal transport should have the ability to distinguish certain noise types from textures. Hence, we present a brief introduction to optimal transport metrics and show their relation to previously proposed texture norms. We propose different variational models and investigate their performance.

Keywords

Image decomposition Texture Optimal transport Oscillating patterns 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefMATHGoogle Scholar
  2. 2.
    Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22(University Lecture Series). AMS (2001)Google Scholar
  3. 3.
    Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing 19(1–3), December 2003Google Scholar
  4. 4.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(\mathit{H}^{-1}\) norm. SIAM Multiscale Modeling and Simulation 1(3), 349–370 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40, 120–145 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. Journal of Mathematical Imaging and Vision (2014)Google Scholar
  7. 7.
    Lellmann, J., Lorenz, D.A., Schönlieb, C.B., Valkonen, T.: Imaging with Kantorovich-Rubinstein discrepancy. SIAM Journal on Imaging Sciences, July 2014 (to appear). arXiv
  8. 8.
    Kaipio, J., Somersalo, E.: Statistical and computational inverse problems. Springer-Verlag, New York (2005)MATHGoogle Scholar
  9. 9.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM Journal on Applied Mathematics 65(5), 1817–1837 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Clason, C., Jin, B., Kunisch, K.: A semismooth Newton method for \(L^1\) data fitting with automatic choice of regularization parameters and noise calibration. SIAM Journal on Imaging Sciences 3(2), 199–231 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Clason, C.: \(L^\infty \) fitting for inverse problems with uniform noise. Inverse Problems 28(10), 104007 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM Journal on Imaging Sciences 3(3), 492–526 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais. De l’Imprimerie Royale (1781)Google Scholar
  14. 14.
    Kantorovič, L.V.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)MathSciNetGoogle Scholar
  15. 15.
    Vasershtein, L.N.: Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii 5(3), 64–72 (1969)MathSciNetGoogle Scholar
  16. 16.
    Kantorovič, L.V., Rubinšteĭn, G.Š.: On a functional space and certain extremum problems. Doklady Akademii Nauk SSSR 115, 1058–1061 (1957)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Analysis und AlgebraTU BraunschweigBraunschweigGermany

Personalised recommendations