Skip to main content
SpringerLink
Log in

Correspondences between Wavelet Shrinkage and Nonlinear Diffusion

  • Conference paper
  • First Online:
Scale Space Methods in Computer Vision (Scale-Space 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2695))

Included in the following conference series:

Abstract

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet shrinkage denoising.

This joint research was supported by the project Relations between nonlinear filters in digital image processing within the DFG-Schwerpunktprogramm 1114: Mathematical methods for time series analysis and digital image processing. This is gratefully acknowledged.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Acar and C. R. Vogel. Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems, 10:1217–1229, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón. Minimizing total variation flow. Differential and Integral Equations, 14(3):321–360, March 2001.

    MATH  MathSciNet  Google Scholar 

  3. M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger. Robust anisotropic diffusion. IEEE Transactions on Image Processing, 7(3):421–432, March 1998.

    Article  Google Scholar 

  4. E. J. Candés and F. Guo. New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction. Signal Processing, 82(11):1519–1543, 2002.

    Article  MATH  Google Scholar 

  5. A. Chambolle, R. A. DeVore, N. Lee, and B. L. Lucier. Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Transactions on Image Processing, 7(3):319–335, March 1998.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Chambolle and B. L. Lucier. Interpreting translationally-invariant wavelet shrinkage as a new image smoothing scale space. IEEE Transactions on Image Processing, 10(7):993–1000, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. F. Chan and H. M. Zhou. Total variation improved wavelet thresholding in image compression. In Proc. Seventh International Conference on Image Processing, Vancouver, Canada, September 2000.

    Google Scholar 

  8. P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud. Two deterministic half-quadratic regularization algorithms for computed imaging. In Proc. 1994 IEEE International Conference on Image Processing, volume 2, pages 168–172, Austin, TX, November 1994. IEEE Computer Society Press.

    Article  Google Scholar 

  9. A. Cohen, R. DeVore, P. Petrushev, and H. Xu. Nonlinear approximation and the space BV (R 2). Americal Journal of Mathematics, 121:587–628, 1999.

    Article  MATH  Google Scholar 

  10. R. R. Coifman and D. Donoho. Translation invariant denoising. In A. Antoine and G. Oppenheim, editors, Wavelets in Statistics, pages 125–150. Springer, New York, 1995.

    Google Scholar 

  11. R. R. Coifman and A. Sowa. Combining the calculus of variations and wavelets for image enhancement. Applied and Computational Harmonic Analysis, 9(1):1–18, July 2000.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. R. Coifman and A. Sowa. New methods of controlled total variation reduction for digital functions. SIAM Journal on Numerical Analysis, 39(2):480–498, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. L. Donoho. De-noising by soft thresholding. IEEE Transactions on Information Theory, 41:613–627, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrica, 81(3):425–455, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Durand and J. Froment. Reconstruction of wavelet coefficients using total-variation minimization. Technical Report 2001-18, Centre de Mathématiques et de Leurs Applications, ENS de Cachan, France, 2001.

    Google Scholar 

  16. H.-Y. Gao. Wavelet shrinkage denoising using the non-negative garrote. Journal of Computational and Graphical Statistics, 7(4):469–488, 1998.

    Article  MathSciNet  Google Scholar 

  17. H.-Y. Gao and A. G. Bruce. WaveShrink with firm shrinkage. Statistica Sinica, 7:855–874, 1997.

    MATH  MathSciNet  Google Scholar 

  18. M. Holschneider, R. Kronland-Martinet, J. Morlet, and Ph. Tchamitchian. A real-time algorithm for signal analysis with the help of the wavelet transform. In J.M. Combes, A. Grossman, and Ph. Tchamitchian, editors, Wavelets: Time-Frequency Methods and Phase Space, pages 286–297. Springer-Verlag, 1987.

    Google Scholar 

  19. T. Iijima. Basic theory on normalization of pattern (in case of typical one-dimensional pattern). Bulletin of the Electrotechnical Laboratory, 26:368–388, 1962. In Japanese.

    Google Scholar 

  20. B. Kawohl and N. Kutev. Maximum and comparison principle for one-dimensional anisotropic diffusion. Mathematische Annalen, 311:107–123, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. L. Keeling and R. Stollberger. Nonlinear anisotropic diffusion filters for wide range edge sharpening. Inverse Problems, 18:175–190, January 2002.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Kijima. Markov Processes and Stochastic Modeling. Chapman and Hall, New York, 1997.

    Google Scholar 

  23. F. Malgouyres. Combining total variation and wavelet packet approaches for image deblurring. In Proc. First IEEE Workshop on Variational and Level Set Methods in Computer Vision, pages 57–64, Vancouver, Canada, July 2001. IEEE Computer Society Press.

    Google Scholar 

  24. F. Malgouyres. Mathematical analysis of a model which combines total variation and wavelet for image restoration. Inverse Problems, 2(1):1–10, 2002.

    Google Scholar 

  25. S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, San Diego, second edition, 1999.

    MATH  Google Scholar 

  26. Pavel Mrázek, Joachim Weickert, Gabriele Steidl, and Martin Welk. On iterations and scales of nonlinear filters. In O. Drbohlav, editor, Computer Vision Winter Workshop 2003, pages 61–66. Czech Pattern Recognition Society, 2003.

    Google Scholar 

  27. P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12:629–639, 1990.

    Article  Google Scholar 

  28. L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.

    Article  MATH  Google Scholar 

  29. G. Steidl and J. Weickert. Relations between soft wavelet shrinkage and total variation denoising. In L. Van Gool, editor, Pattern Recognition, volume 2449 of Lecture Notes in Computer Science, pages 198–205. Springer, Berlin, 2002.

    Chapter  Google Scholar 

  30. G. Steidl, J. Weickert, T. Brox, P. Mrázek, and M. Welk. On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs. Technical report, Series SPP-1114, Department of Mathematics, University of Bremen, Germany, 2003.

    Google Scholar 

  31. J. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.

    MATH  Google Scholar 

  32. J. Weickert and B. Benhamouda. A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox. In F. Solina, W. G. Kropatsch, R. Klette, and R. Bajcsy, editors, Advances in Computer Vision, pages 1–10. Springer, Wien, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building 27, 66123, Saarbrücken, Germany

    Pavel Mrázek & Joachim Weickert

  2. Faculty of Mathematics and Computer Science, D7, 27, University of Mannheim, 68131, Mannheim, Germany

    Gabriele Steidl

Authors
  1. Pavel Mrázek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joachim Weickert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gabriele Steidl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Imaging Sciences, 5th Floor Thomas Guy House, Guy’s Campus, London, SE1 9RT, UK

    Lewis D. Griffin

  2. IT University of Copenhagen, Glentevej 67-69, Copenhagen NV, Denmark

    Martin Lillholm

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Mrázek, P., Weickert, J., Steidl, G. (2003). Correspondences between Wavelet Shrinkage and Nonlinear Diffusion. In: Griffin, L.D., Lillholm, M. (eds) Scale Space Methods in Computer Vision. Scale-Space 2003. Lecture Notes in Computer Science, vol 2695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44935-3_8

Download citation

  • DOI: https://doi.org/10.1007/3-540-44935-3_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40368-5

  • Online ISBN: 978-3-540-44935-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation