Abstract
In this chapter, we discuss two important statistical applications of wavelets: (a) signal estimation and (b) signal detection. Several popular wavelet-based signal estimation/de-noising schemes are reviewed, followed by the introduction of a new signal estimator that attempts to estimate and preserve some moments of the underlying original signal. A relationship between signal estimation and data compression is also discussed. Analytical and experimental results are presented to explain the performance of these signal estimation schemes.
The second part of this chapter deals with hypothesis testing-based signal detection. Wavelet decomposition level-dependent binary hypothesis tests are first presented, followed by global detection procedures that combine these decisions to obtain the final decision. Likelihood ratio-based statistical detectors are discussed toward this goal. The combinatorial explosion of the global detector design results in the investigation of two specific global detection fusion rules: OR and AND. Some mathematical approximations are exploited that aid in trading off complexity for the global detector performance. Theorems and their proofs relating to this issue are presented.
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References
I. Daubechies, Ten Lectures on Wavelets SIAM CBMS-NSF Regional Conference Series in Applied Mathematics 1992.
S. G. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation IEEE Trans. on PAMI vol. 11, no. 7, July 1989, pp. 674–693.
C. S. Burrus, R. A. Gopinath, and H. Guo Introduction to wavelets and wavelet transforms: a primer Englewood Cliffs, NJ: Prentice-Hall, 1998.
D. L. Donoho, Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data Proc. Symposia Appl. Math. vol. 47, 1993, pp. 173–205.
D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage Biometrika vol. 81, 1994, pp. 425–455.
D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage J. American Stat. Assoc. vol. 90, 1995, pp. 1200–1224.
D. L. Donoho and I. M. Johnstone, Minimax estimation via wavelet shrinkage, Available on-line at http://www-stat.stanford.edu/~vdonoho/Reports/index.html
N. S. Jayant and P. Noll Digital Coding of Waveforms Principles and Applications to Speech and Video Englewood Cliffs, NJ: Prentice-Hall, 1984.
Y. Wu and S. Tai, Efficient BTC image compression technique IEEE Trans. on Consumer Electronics vol. 44, May 1998, pp. 317–324.
E. J. Delp and O. R. Mitchell, Image compression using block truncation codingIEEE Trans. on Communicationsvol. 27, Sept. 1979, pp. 1335–1342.
T. B. Nguyen and B. J. Oommen, Moment-preserving piecewise linear approximations of signals and systems IEEE Trans. on Pattern Analysis and Machine Intelligence vol. 19, Jan. 1997, pp. 84–91.
S. G. Chang, B. Yu, and M. Vetterli, Image denoising via lossy compression and wavelet thresholding Proc. of International Conf on Image Processing vol. 1, 1997, pp. 604–607.
M. Hansen and B. Yu, Assessing MDL in wavelet denoising and compression Information Theory and Networking Workshop 1999, p. 34.
G. Chang, B. Yu, and M. Vetterli, Bridging compression to wavelet thresholding as a de-noising method,“ Proceedings of 1997 Conference on Information Sciences and Systems Baltimore, MD, 1997.
S.M.Berman Sojourns and Extremes of Stochastic Processes Wadsworth, Reading, MA, 1989.
C. Stein, Estimation of the mean of a multivariate normal distribution Ann. Statistics vol. 9, 1981, pp. 1135–1151.
A. K. Jain Fundamentals of digital image processing Englewood Cliffs, NJ: Prentice-Hall, 1989
J. A. Bucklew and N. C. Gallagher, A note on optimum quantization IEEE Trans. on Information Theory vol. 24, May 1979, pp. 365–366.
P. Billingsley Probability and Measure, New York: John Wiley, 1979.
S. Mallat and S. Zhong, Characterization of signals from multiscale edges IEEE Trans. Patt. Recog. and Mach. Intell. vol. 14, July 1992, pp. 710–732.
E. J. Delp and O. R. Mitchell, Moment preserving quantization IEEE Trans. on Communications vol. 39, no. 11, Nov. 1991, pp. 1549–1558.
G. Szego Orthogonal Polynomials vol. 23, American Mathematical Society, Providence, RI, 1975.
D. L. Donoho, De-noising by soft-thresholding IEEE Trans. on Information Theory vol. 41, no. 3, May 1995, pp. 613–627.
R. A. DeVore and B. J. Lucier, Fast wavelet techniques for near optimal processing IEEE Military Communications Conf 1992, pp. 48.3.1–48.3.7.
Source: Dr. Adrain Maudsley, MRS Unit, VA Medical Center, San Francisco, CA.
MATLAB software package:http://www-stat.stanford.edurwavelab/
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Chandramouli, R., Ramachandran, K.M. (2003). Wavelets for Statistical Estimation and Detection. In: Debnath, L. (eds) Wavelets and Signal Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0025-3_5
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DOI: https://doi.org/10.1007/978-1-4612-0025-3_5
Publisher Name: Birkhäuser, Boston, MA
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