Journal of Mathematical Sciences

, Volume 237, Issue 6, pp 804–809 | Cite as

Estimation of the Loss Function When Using Wavelet-Vaguelette Decomposition for Solving Ill-Posed Problems

  • A. A. KudryavtsevEmail author
  • O. V. Shestakov

The paper discusses the de-noising method in a model with an additive Gaussian noise, based on the wavelet-vaguelette decomposition and thresholding of the vaguelette coefficients. For soft and hard thresholding procedures, we derive the values of the thresholds and estimate asymptotically optimal orders of the loss function based on the probabilities of errors in calculating decomposition coefficients.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comp. Harm. An., 2, 101–126 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I.M. Johnstone, “Wavelet shrinkage for correlated data and inverse problems adaptivity results,” Stat. Sinica, 9, 51–83 (1999).MathSciNetzbMATHGoogle Scholar
  3. 3.
    A.V. Markin and O.V. Shestakov, “Consistency of risk estimation with thresholding of wavelet coefficients,” Moscow Univ. Comput. Math. Cybern., 34, No. 1, 22–30 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    O.V. Shestakov, “Asymptotic normality of adaptive wavelet thresholding risk estimation,” Dokl. Math., 86, No. 1, 556–558 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Sadasivan, S. Mukherjee, and C.S. Seelamantula, “An optimum shrinkage estimator based on minimum-probability-of-error criterion and application to signal denoising,” in: Proceedings of IEEE ICASSP (2014), pp. 4249–4253.Google Scholar
  6. 6.
    A.A. Kudryavtsev and O.V. Shestakov, “Asymptotic behavior of the threshold minimizing the average probability of error in calculation of wavelet coefficients,” Dokl. Math., 93, No. 3, 295–299 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A.A. Kudryavtsev and O.V. Shestakov, “Asymptotically optimal wavelet thresholding in the models with non-Gaussian noise distributions,” Dokl. Math., 94. No. 3, 615–619 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S.A Mallat, Wavelet Tour of Signal Processing, Academic Press, New York (1999).zbMATHGoogle Scholar
  9. 9.
    N. Lee, Wavelet-vaguelette decompositions and homogenous equations, Ph.D. Dissertation, Purdue University (1997).Google Scholar
  10. 10.
    F. Abramovich and B. W. Silverman, “Wavelet decomposition approaches to statistical inverse problems,” Biometrika, 85, No. 1, 115–129 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Jansen, Noise Reduction by Wavelet Thresholding, Springer, Berlin (2001).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Statistics, Faculty of Computational Mathematics and CyberneticsM. V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Institute of Informatics ProblemsFederal Research Center “Computer Science and Control” of the Russian Academy of SciencesMoscowRussia

Personalised recommendations