Skip to main content
SpringerLink
Log in

Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decomposition

  • Conference paper
  • First Online:
Scale Space and Variational Methods in Computer Vision (SSVM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10302))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  4. Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM J. Imaging Sci. 7(4), 1937–1961 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  16. Haddad, A.: Texture separation \(BV-G\) and \(BV-L^1\) models. Multiscale Model. Simul. 6(1), 273–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duval, V., Aujol, J.F., Gousseau, Y.: The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  19. Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in \(R^{N}\). J. Differ. Equ. 184(2), 475–525 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Caselles, V., Chambolle, A., Novaga, M.: Uniqueness of the Cheeger set of a convex body. Pac. J. Math. 232(1), 77–90 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Twente, 7522 NB, Enschede, The Netherlands

    Leonie Zeune, Stephan A. van Gils & Christoph Brune

  2. Department of Medical Cell BioPhysics, University of Twente, 7522 NB, Enschede, The Netherlands

    Leonie Zeune & Leon W. M. M. Terstappen

Authors
  1. Leonie Zeune
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephan A. van Gils
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leon W. M. M. Terstappen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christoph Brune
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonie Zeune .

Editor information

Editors and Affiliations

  1. University of Copenhagen , Copenhagen, Denmark

    François Lauze

  2. Technical University of Denmark , Kongens Lyngby, Denmark

    Yiqiu Dong

  3. Technical University of Denmark , Kongens Lyngby, Denmark

    Anders Bjorholm Dahl

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Zeune, L., van Gils, S.A., Terstappen, L.W.M.M., Brune, C. (2017). Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decomposition. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-58771-4_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58770-7

  • Online ISBN: 978-3-319-58771-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation