Relations between Soft Wavelet Shrinkage and Total Variation Denoising

  • Gabriele Steidl
  • Joachim Weickert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2449)


Soft wavelet shrinkage and total variation (TV) denoising are two frequently used techniques for denoising signals and images, while preserving their discontinuities. In this paper we show that — under specific circumstances — both methods are equivalent. First we prove that 1-D Haar wavelet shrinkage on a single scale is equivalent to a single step of TV diffusion or regularisation of two-pixel pairs. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularisation of the Laplacian pyramid of the signal.


Wavelet Packet Threshold Parameter Haar Wavelet Wavelet Method Wavelet Shrinkage 
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  1. 1.
    F. Andreu, V. Caselles, J. I. Diaz, and J. M. Mazón. Qualitative properties of the total variation flow. Journal of Functional Analysis, 2002. To appear.Google Scholar
  2. 2.
    G. Bellettini, V. Caselles, and M. Novaga. The total variation flow in R N. Journal of Differential Equations, 2002. To appear.Google Scholar
  3. 3.
    P. J. Burt and E. H. Adelson. The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, 31:532–540, 1983.CrossRefGoogle Scholar
  4. 4.
    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, Mar. 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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, Sept. 2000.Google Scholar
  6. 6.
    D. Donoho. De-noising by soft thresholding. IEEE Transactions on Information Theory, 41:613–627, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, San Diego, 1998.zbMATHGoogle Scholar
  10. 10.
    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.CrossRefGoogle Scholar
  11. 11.
    L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.zbMATHCrossRefGoogle Scholar
  12. 12.
    O. Scherzer and J. Weickert. Relations between regularization and diffusion filtering. Journal of Mathematical Imaging and Vision, 12(1):43–63, Feb. 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, 1996.Google Scholar
  14. 14.
    J. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Gabriele Steidl
    • 1
  • Joachim Weickert
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of MannheimMannheimGermany
  2. 2.Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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