On the Statistics of Best Bases Criteria

  • H. Krim
  • J.-C. Pesquet
Part of the Lecture Notes in Statistics book series (LNS, volume 103)

Abstract

Wavelet packets are a useful extension of wavelets providing an adaptive time- scale analysis. In using noisy observations of a signal of interest, the criteria for best bases representation are random variables. The search may thus be very sensitive to noise. In this paper, we characterize the asymptotic statistics of the criteria to gain insight which can in turn, be used to improve on the performance of the analysis. By way of a well-known information-theoretic principle, namely the Minimum Description Length, we provide an alternative approach to Minimax methods for deriving various attributes of nonlinear wavelet packet estimates.

Keywords

Entropy Covariance Radar Shrinkage Pyramid 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BB+]
    Basseville, M., Benveniste, A., Chou, K. C., Golden, S. A., Nikoukhah, R., Willsky, A. S.: Modeling and estimation of multiresolution stochastic processes. IEEE Trans. Informat. Theory.38(1992) 766–784CrossRefGoogle Scholar
  2. [CH1]
    Cambanis, S., Houdré., C.: On the continuous wavelet transform of second order random processes, preprint (1993)Google Scholar
  3. [CW1]
    Coifman, R. R., Wickerhauser, M. V.: Entropy-based algorithms for best basis selection. IEEE Trans. Informat. Theory. 38 (1992) 713–718MATHCrossRefGoogle Scholar
  4. [Da1]
    Daubechies I.: Ten Lectures on Wavelets. CBMS-NSF, SIAM, Philadelphia, USA (1992)MATHGoogle Scholar
  5. [DJ1]
    Donoho, D. L., Johnstone, I. M.: Ideal spatial adaptation by wavelet shrinkage. Bio-metrika (to appear)Google Scholar
  6. [DJ2]
    Donoho, D. L., Johnstone, I. M.: Ideal denoising in an orthogonal basis chosen from a library of bases. Note CRAS Paris (to appear)Google Scholar
  7. [Fl1]
    Flandrin, P.: On the spectrum of fractional Brownian motion. IEEE Trans. Informat. Theory.35(1989) 197–199MathSciNetCrossRefGoogle Scholar
  8. [Ib1]
    Ibragimov, I. A .: Some limit theorems for stationary processes. Th. Prob. Appl. .VII(1962) 349–382MATHCrossRefGoogle Scholar
  9. [KPW]
    Krim, H., Pesquet, J.-C., Willsky, A. S.: Robust multiscale representation of processes and optimal signal reconstruction. Proceedings of IEEE Symposium on Time-Frequency and Time-Scale Analysis, Philadelphia, USA. (1994) 1–4Google Scholar
  10. [LP+]
    Lumeau, B., Pesquet, J.-C., Bercher, J.-F., Louveau, L.: Optimisation of bias-variance tradeoff in non parametric spectral analysis by decomposition into wavelet packets. Progress in wavelet analysis and applications, Editions Frontieres. (1993) 285–290Google Scholar
  11. [Mo1]
    Moulin, P.: A wavelet regularization method for diffuse radar target imaging and speckle noise reduction. Journ. Math. Imaging and Vision. 3 (1993) 123–134MATHCrossRefGoogle Scholar
  12. [PC1]
    Pesquet, J.-C., Combettes, P. L.: Wavelet synthesis by alternating projections, submitted to IEEE Trans, on S.P.Google Scholar
  13. [Pr1]
    Price, R.: A useful theorem for nonlinear devices having gaussian inputs. IRE Trans. Informat. Theory.4(1958) 69–72MATHCrossRefGoogle Scholar
  14. [Ri1]
    Rissanen, J.: Modeling by shortest data description. Automatica.14(1978) 465–471MATHCrossRefGoogle Scholar
  15. [Sa1]
    Saito, N.: Local feature extraction and its applications using a library of bases. PhD thesis, Yale University (1994)Google Scholar
  16. [Un1]
    Unser, M.: On the optimality of ideal filters for pyramid and wavelet signal approximation. IEEE Trans. Signal Processing.41(1993) 3591–3596MATHCrossRefGoogle Scholar
  17. [Wi1]
    Wickerhauser, M. V.: INRIA lectures on wavelet packet algorithms. Ondelettes et paquets d’ondelettes, Roquencourt, France. (1991) 31–99Google Scholar
  18. [Wo1]
    Wornell., G. W.: A Karhunen-Loève-like expansion for 1/fprocesses via wavelets. IEEE Trans. Informat. Theory.36(1990) 859–861CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York 1995

Authors and Affiliations

  • H. Krim
    • 1
  • J.-C. Pesquet
    • 2
  1. 1.Stochastic Systems Group, LIDSMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Laboratoire des Signaux et SystèmesCNRS/UPS, GDR TdSI, ESEGif sur Yvette CédexFrance

Personalised recommendations