Best Basis Representations with Prior Statistical Models
Wavelet packets and local trigonometric bases provide an efficient framework and fast algorithms to obtain a “best representation” of a deterministic signal. Applying these deterministic search techniques to stochastic signals may, however, lead to statistically unreliable results. In this chapter, we revisit this problem and introduce prior models on the underlying signal in noise. We propose several techniques to derive the prior parameters and develop a Bayesian-based approach to the best basis problem. As illustrated by applications to signal denoising, this leads to reduced estimation errors while preserving the classical tree search algorithm.
KeywordsProbability Density Function Reconstruction Error Wavelet Packet Prior Model Decomposition Tree
Unable to display preview. Download preview PDF.
- [BD95]J.B. Buckheit and D.L. Donoho. Wavelab and Reproducible Research. Stanford University, 1995.Google Scholar
- [CI95]F. Champagnat and J. Idier. An alternative to standard maximum likelihood for Gaussian mixtures. In Proc. IEEE Conf. Acoust., Speech, Signal Processing, pages 2020-2023, Detroit, USA, May 9–12 1995.Google Scholar
- [Hub67]P. Huber. The behaviour of maximum-likelihood estimates under nonstandard conditions. In Proc. Berkeley Symp. on Math. Stat. and Prob., volume 1, pages 73–101, 1967.Google Scholar
- [KMDW95]H. Krim, S. Mallat, D. Donoho, and A. Willsky. Best basis algorithm for signal enhancement. In Proc. IEEE Conf. Acoust., Speech, Signal Processing, Detroit, MI, May 1995.Google Scholar
- [KP95]H. Krim and J.-C. Pesquet. On the statistics of best bases criteria. In A. Antoniadis, editor, Wavelets and statistics. Lecture Notes in Statistics, Springer Verlag, 1995.Google Scholar
- [Lep98]D. Leporini. Modélisation Statistique et Paquets d’Ondelettes: Application au Débruitage de Signaux Transitoires d’Acoustique Sous-Marine. PhD thesis, Université Paris XI, Sept. 1998.Google Scholar
- [NH98]R. M. Neal and G. E. Hinton. A new view of the EM algorithm that justifies incremental and other variants. In M.I. Jordan, editor, Learning in Graphical Models. Kluwer, 1998.Google Scholar
- [PKLH96]J.-C. Pesquet, H. Krim, D. Leporini, and E. Hamman. Bayesian Approach to Best Basis Selection. In Proc. IEEE Conf. Acoust., Speech, Signal Processing, pages 2634-2638, Atlanta, USA, May 7-9 1996.Google Scholar
- [Sai94]N. Saito. Local feature extraction and its applications using a library of bases. PhD thesis, Yale University, Dec. 1994.Google Scholar