Signal, Image and Video Processing

, Volume 5, Issue 1, pp 11–28 | Cite as

Wavelet shrinkage: unification of basic thresholding functions and thresholds

  • Abdourrahmane M. AttoEmail author
  • Dominique Pastor
  • Grégoire Mercier
Original Paper


This work addresses the unification of some basic functions and thresholds used in non-parametric estimation of signals by shrinkage in the wavelet domain. The soft and hard thresholding functions are presented as degenerate smooth sigmoid-based shrinkage functions. The shrinkage achieved by this new family of sigmoid-based functions is then shown to be equivalent to a regularization of wavelet coefficients associated with a class of penalty functions. Some sigmoid-based penalty functions are calculated, and their properties are discussed. The unification also concerns the universal and the minimax thresholds used to calibrate standard soft and hard thresholding functions: these thresholds pertain to a wide class of thresholds, called the detection thresholds. These thresholds depend on two parameters describing the sparsity degree for the wavelet representation of a signal. It is also shown that the non-degenerate sigmoid shrinkage adjusted with the new detection thresholds is as performant as the best up-to-date parametric and computationally expensive method. This justifies the relevance of sigmoid shrinkage for noise reduction in large databases or large size images.


Non-parametric estimation Wavelets Shrinkage function Penalty function Detection thresholds 


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  1. 1.
    Donoho D.L., Johnstone I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bruce A.G., Gao H.Y.: Understanding waveshrink: variance and bias estimation. Biometrika 83(4), 727–745 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Donoho D.L., Johnstone I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Atto, A.M., Pastor, D., Mercier, G.: Detection threshold for non-parametric estimation. Signal, Image and Video Processing, vol. 2(3). Springer, Heidelberg (2008)Google Scholar
  5. 5.
    Simoncelli, E.P., Adelson, E.H.: Noise removal via bayesian wavelet coring. IEEE Int. Conf. Image Proc. (ICIP) 379–382 (1996)Google Scholar
  6. 6.
    Do M.N., Vetterli M.: Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance. IEEE Trans. Image Process. 11(2), 146–158 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Şendur L., Selesnick I.V.: Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process. 11, 2744–2756 (2002)Google Scholar
  8. 8.
    Portilla J., Strela V., Wainwright M.J., Simoncelli E.P.: Image denoising using scale mixtures of gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Johnstone I.M., Silverman B.W.: Empirical bayes selection of wavelet thresholds. Ann. Stat. 33(4), 1700–1752 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    ter Braak C.J.F.: Bayesian sigmoid shrinkage with improper variance priors and an application to wavelet denoising. Comput. Stat. Data Anal. 51(2), 1232–1242 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gao H.Y.: Waveshrink shrinkage denoising using the non-negative garrote. J. Comput. Graph. Stat. 7(4), 469–488 (1998)CrossRefGoogle Scholar
  12. 12.
    Antoniadis A., Fan J.: Regularization of wavelet approximations. J. Am. Stat. Assoc. 96(455), 939–955 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Luisier F., Blu T., Unser M.: A new sure approach to image denoising: interscale orthonormal wavelet thresholding. IEEE Trans. Image Process. 16(3), 593–606 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Atto, A.M., Pastor, D., Mercier, G.: Smooth sigmoid wavelet shrinkage for non-parametric estimation. IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP, Las Vegas, Nevada, USA, 30 March–4 April (2008)Google Scholar
  15. 15.
    Pastor, D., Atto, A.M.: Sparsity from binary hypothesis testing and application to non-parametric estimation. European Signal Processing Conference, EUSIPCO, Lausanne, Switzerland, August 25–29 (2008)Google Scholar
  16. 16.
    Benedetto, J.J., Frasier, M.W.: Wavelets : Mathematics and applications. CRC Press, Boca Raton (1994), chap. 9: Wavelets, probability, and statistics: Some bridges, by Christian Houdré, pp. 365–398Google Scholar
  17. 17.
    Zhang J., Walter G.: A wavelet-based KL-like expansion for wide-sense stationary random processes. IEEE Trans. Signal Process. 42(7), 1737–1745 (1994)CrossRefGoogle Scholar
  18. 18.
    Leporini D., Pesquet J.-C.: High-order wavelet packets and cumulant field analysis. IEEE Trans. Inf. Theory 45(3), 863–877 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Atto A.M., Pastor D., Isar A.: On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process. Signal Process. 87(10), 2320–2335 (2007)zbMATHCrossRefGoogle Scholar
  20. 20.
    Atto, A.M., Pastor, D.: Limit distributions for wavelet packet coefficients of band-limited stationary random processes. European Signal Processing Conference, EUSIPCO, Lausanne, Switzerland, 25–28 August (2008)Google Scholar
  21. 21.
    Flandrin P.: Wavelet analysis and synthesis of fractional brownian motion. IEEE Trans. Inf. Theory 38(2), 910–917 (1992)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Tewfik A.H., Kim M.: Correlation structure of the discrete wavelet coefficients of fractional brownian motion. IEEE Trans. Inf. Theory 38(2), 904–909 (1992)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Dijkerman R.W., Mazumdar R.R.: On the correlation structure of the wavelet coefficients of fractional brownian motion. IEEE Trans. Inf. Theory 40(5), 1609–1612 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kato T., Masry E.: On the spectral density of the wavelet transform of fractional brownian motion. J. Time Ser. Anal. 20(50), 559–563 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Craigmile P.F., Percival D.B.: Asymptotic decorrelation of between-scale wavelet coefficients. IEEE Trans. Inf. Theory 51(3), 1039–1048 (2005)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Antoniadis A.: Wavelet methods in statistics: some recent developments and their applications. Stat. Surveys 1, 16–55 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Berman S.M.: Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, USA (1992)zbMATHGoogle Scholar
  28. 28.
    Mallat S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, New York (1999)zbMATHGoogle Scholar
  29. 29.
    Coifman, R.R., Donoho, D.L.: Translation invariant de-noising. Lect. Notes Stat. (103), 125–150 (1995)Google Scholar
  30. 30.
    Wang Z., Bovik A.C., Sheikh H.R., Simoncelli E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  31. 31.
    Sveinsson J.R., Benediktsson J.A.: Speckle reduction and enhancement of sar images in the wavelet domain. Geosci. Remote Sens. Symp. IGARSS 1, 63–66 (1996)CrossRefGoogle Scholar
  32. 32.
    Xie H., Pierce L.E., Ulaby F.T.: Sar speckle reduction using wavelet denoising and markov random field modeling. IEEE Trans. Geosci. Remote Sens. 40(10), 2196–2212 (2002)CrossRefGoogle Scholar
  33. 33.
    Argenti F., Bianchi T., Alparone L.: Multiresolution map despeckling of sar images based on locally adaptive generalized gaussian pdf modeling. IEEE Trans. Image Process. 15(11), 3385–3399 (2006)CrossRefGoogle Scholar
  34. 34.
    Buades A., Coll B., Morel J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Johnstone I.M., Silverman B.W.: Wavelet threshold estimators for data with correlated noise. J. Royal Stat. Soc. Ser. B 59(2), 319–351 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Müller J.W.: Possible advantages of a robust evaluation of comparisons. J. Res. Natl. Inst. Stand. Technol. 105(4), 551–555 (2000)Google Scholar

Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  • Abdourrahmane M. Atto
    • 1
    Email author
  • Dominique Pastor
    • 1
  • Grégoire Mercier
    • 1
  1. 1.TELECOM BretagneBrestFrance

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