Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decomposition

  • Leonie ZeuneEmail author
  • Stephan A. van Gils
  • Leon W. M. M. Terstappen
  • Christoph Brune
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


This paper focuses on multi-scale approaches for variational methods and corresponding gradient flows. Recently, for convex regularization functionals such as total variation, new theory and algorithms for nonlinear eigenvalue problems via nonlinear spectral decompositions have been developed. Those methods open new directions for advanced image filtering. However, for an effective use in image segmentation and shape decomposition, a clear interpretation of the spectral response regarding size and intensity scales is needed but lacking in current approaches. In this context, \(L^1\) data fidelities are particularly helpful due to their interesting multi-scale properties such as contrast invariance. Hence, the novelty of this work is the combination of \(L^1\)-based multi-scale methods with nonlinear spectral decompositions. We compare \(L^1\) with \(L^2\) scale-space methods in view of spectral image representation and decomposition. We show that the contrast invariant multi-scale behavior of \(L^1-TV\) promotes sparsity in the spectral response providing more informative decompositions. We provide a numerical method and analyze synthetic and biomedical images at which decomposition leads to improved segmentation.


\(L^{1}\)-TV Denoising Scale-spaces Nonlinear spectral decomposition Multiscale segmentation Eigenfunctions Calibrable sets 



LZ, LT, CB acknowledge support by the EUFP7 program #305341 CTCTrap and the IMI EU program #115749 CANCER-ID. CB acknowledges support for his TT position by the NWO via Veni grant 613.009.032 within the NDNS+ cluster.


  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brune, C., Sawatzky, A., Burger, M.: Primal and dual Bregman methods with application to optical nanoscopy. Int. J. Comput. Vis. 92(2), 211–229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gilboa, G.: A spectral approach to total variation. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds.) SSVM 2013. LNCS, vol. 7893, pp. 36–47. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38267-3_4 CrossRefGoogle Scholar
  4. 4.
    Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM J. Imaging Sci. 7(4), 1937–1961 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burger, M., Eckardt, L., Gilboa, G., Moeller, M.: Spectral representations of one-homogeneous functionals. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 16–27. Springer, Cham (2015). doi: 10.1007/978-3-319-18461-6_2 Google Scholar
  6. 6.
    Burger, M., Gilboa, G., Moeller, M., Eckardt, L., Cremers, D.: Spectral decompositions using one-homogeneous functionals. SIAM J. Imaging Sci. 9(3), 1374–1408 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gilboa, G., Moeller, M., Burger, M.: Nonlinear spectral analysis via one-homogeneous functionals: overview and future prospects. J. Math. Imaging Vis. 56(2), 300–319 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Aujol, J.F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition-modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Yin, W., Goldfarb, D., Osher, S.: The total variation regularized \(L^1\) model for multiscale decomposition. Multiscale Model. Simul. 6(1), 190–211 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dong, Y., Hintermüller, M., Neri, M.: An efficient primal-dual method for \(L^1\)TV image restoration. SIAM J. Imaging Sci. 2(4), 1168–1189 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wu, C., Zhang, J., Tai, X.C.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yuan, J., Shi, J., Tai, X.C.: A convex and exact approach to discrete constrained TV- \(L^1\) image approximation. East Asian J. Appl. Math. 1(02), 172–186 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zach, C., Pock, T., Bischof, H.: A duality based approach for realtime TV-L 1 optical flow. In: Hamprecht, F.A., Schnörr, C., Jähne, B. (eds.) DAGM 2007. LNCS, vol. 4713, pp. 214–223. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74936-3_22 CrossRefGoogle Scholar
  16. 16.
    Haddad, A.: Texture separation \(BV-G\) and \(BV-L^1\) models. Multiscale Model. Simul. 6(1), 273–286 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Duval, V., Aujol, J.F., Gousseau, Y.: The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zeune, L., van Dalum, G., Terstappen, L.W., van Gils, S.A., Brune, C.: Multiscale segmentation via Bregman distances and nonlinear spectral analysis. SIAM J. Imaging Sci. 10(1), 111–146 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in \(R^{N}\). J. Differ. Equ. 184(2), 475–525 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Caselles, V., Chambolle, A., Novaga, M.: Uniqueness of the Cheeger set of a convex body. Pac. J. Math. 232(1), 77–90 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chambolle, A., Duval, V., Peyré, G., Poon, C.: Geometric properties of solutions to the total variation denoising problem. arXiv preprint arXiv:1602.00087 (2016)
  22. 22.
    Darbon, J.: Total variation minimization with \(L^1\) data fidelity as a contrast invariant filter. In: 4th International Symposium on Image and Signal Processing and Analysis (ISpPA 2005), pp. 221–226 (2005)Google Scholar
  23. 23.
    Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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
  25. 25.
    Hoover, A., Kouznetsova, V., Goldbaum, M.: Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response. IEEE Trans. Med. Imaging 19(3), 203–210 (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Leonie Zeune
    • 1
    • 2
    Email author
  • Stephan A. van Gils
    • 1
  • Leon W. M. M. Terstappen
    • 2
  • Christoph Brune
    • 1
  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  2. 2.Department of Medical Cell BioPhysicsUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations