Wavelets The Key to Intermittent Information

, by Silverman, B. W.; Vassilicos, J. C.

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ISBN: 9780198507161 | 019850716X
Cover: Hardcover
Copyright: 8/24/2000

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In recent years there has been an explosion of interest in wavelets, in a wide range of fields in science and engineering and beyond. This book brings together contributions from researchers from disparate fields, both in order to demonstrate to a wide readership the current breadth of workin wavelets, and to encourage cross-fertilization of ideas. It demonstrates the genuinely interdisplinary nature of wavelet research and applications. Particular areas covered include turbulence, statistics, time series analysis, signal and image processing, the physiology of vision, astronomy,economics and acoustics. Some of the work uses standard wavelet approaches and in other cases new methodology is developed. The papers were originally presented at a Royal Society Discussion Meeting, to a large and enthusiastic audience of specialists and non-specialists.

Contributors

Wavelets on irregular point sets

(20)

Ingrid Daubechies

Igor Guskov

Peter Schroder

Wim Sweldens

Introduction

(6)

One-dimensional subdivision

(3)

Two-dimensional subdivision

(1)

Overview

(1)

The one-dimensional case

(4)

Multilevel grids

(1)

Subdivision schemes

(1)

Smoothness results

(2)

Wavelets

(1)

The two-dimensional case

(6)

Polygonal simplification

(1)

Functional setting: multivariate commutation formula

(1)

Constructing a subdivision scheme

(2)

Functions on surfaces

(1)

Burt-Adelson pyramid

(1)

Applications

(1)

Conclusion

(1)

Acknowledgements

(1)

References

(3)

Revealing a lognormal cascading process in turbulent velocity statistics with wavelet analysis

(28)

A. Arneodo

S. Manneville

J. F. Muzy

S. G. Roux

Introduction

(4)

Experimental evidence for lognormal statistics in high-Reynolds-number turbulent flows

(7)

A method for determining the kernel G

(2)

Experimental determination of the kernel G

(5)

Experimental evidence for a non-scale-invariant lognormal cascading process in high-Reynolds-number turbulent flows

(6)

Space-scale correlation functions from wavelet analysis

(1)

Analysis of random cascades using space-scale correlation functions

(4)

Analysis of velocity data using space-scale correlation functions

(1)

The multifractal description of intermittency revisited with wavelets

(6)

WTMM probability density functions

(2)

ζq scaling exponents

(4)

Conclusions and perspectives

(2)

Acknowledgements

(1)

References

(3)

Wavelets for the study of intermittency and its topology

(22)

F. Nicolleau

J. C. Vassilicos

Introduction

(1)

Intermittency and eddy capacity

(2)

Validity of F(r) ∼ rDE--1

(11)

Application to high resolution turbulent velocity signals

(5)

Conclusion

(2)

Acknowledgements

(1)

DE and D'K

(1)

References

(1)

Wavelets in statistics: beyond the standard assumptions

(18)

Bernard W. Silverman

Introduction

(2)

The standard assumptions

(1)

Prior information and modelling uncertainty

(1)

Correlated and non-stationary noise

(1)

Level-dependent thresholding

(1)

Adaptivity results

(1)

The non-decimated wavelet transform

(3)

The transform and the average basis reconstruction

(1)

Using the NDWT for curve estimation

(1)

A neurophysiological example

(2)

Dealing with irregular data

(2)

Coefficient-dependent thresholding

(1)

Finding the variances of the wavelet coefficients

(1)

Robust estimation

(1)

Deformable templates

(3)

Images collected in palaeopathology

(2)

Models for deformations

(1)

Gaining information from the wavelet model

(1)

Discussion

(3)

Acknowledgements

(1)

The discrete wavelet transform

(1)

References

(3)

Wavelets and the theory of non-parametric function estimation

(22)

Iain M. Johnstone

Introduction

(1)

A simple model for sparsity

(7)

The 'signal in Gaussian white-noise' model

(2)

Optimality in the white-noise model

(8)

Linear estimators

100

(2)

Wavelet bases and thresholding

102

(5)

Concluding remarks

107

(1)

Acknowledgements

108

(1)

References

108

(3)

Ridgelets: a key to higher-dimensional intermittency?

111

(18)

Emmanuel J. Candes

David L. Donoho

Introduction

111

(4)

Wavelets and point singularities

112

(1)

Singularities along lines

112

(1)

Ridgelet analysis

113

(1)

Ridgelets and ridge functions

114

(1)

The continuous ridgelet transform

115

(2)

Relation to Radon transform

116

(1)

An example

116

(1)

Discrete ridgelet transform: frames

117

(2)

Orthonormal ridgelets in dimension 2

119

(2)

Ridgelet synthesis of linear singularities

121

(1)

Using dual frames

121

(1)

Using orthonormal ridgelets

121

(1)

Ridgelet analysis of ridge functions

122

(1)

Using dual ridgelets

122

(1)

Using orthonormal ridgelets

122

(1)

Ridge spaces

123

(3)

Ridgelets and curves

126

(1)

Discussion

126

(1)

References

127

(2)

Wavelets in time-series analysis

129

(18)

Guy P. Nason

Rainer von Sachs

Introduction

129

(2)

Estimation of `global' scale behaviour

131

(6)

Long-memory processes, Allan and wavelet variance

132

(3)

Wavelet spectral analysis of 1 / f processes

135

(2)

Synthesis of long-memory processes using wavelets

137

(1)

Wavelet processes: a particular time-scale model

137

(6)

Local stationarity

137

(2)

Locally stationary wavelet processes

139

(4)

Acknowledgements

143

(2)

Discrete non-decimated wavelets

144

(1)

References

145

(2)

Wavelets, vision and the statistics of natural scenes

147

(18)

D. J. Field

Introduction

147

(1)

The mammalian visual system

148

(3)

Wavelet-like transforms

150

(1)

Image transforms and the statistics of natural scenes

151

(1)

The goal of visual coding

152

(6)

Principal components and the amplitude spectra of natural scenes

153

(2)

Discovering sparse structure

155

(2)

Sparse structure in natural scenes

157

(1)

Neural networks and independent coding

158

(3)

Nonlinear decorrelation

160

(1)

Overview

161

(1)

Acknowledgements

162

(1)

References

162

(3)

Image processing with complex wavelets

165

(22)

Nick Kingsbury

Introduction

165

(1)

The wavelet tree for multi-dimensional signals

166

(2)

Common wavelets with perfect reconstruction

168

(3)

Some problems with common wavelets

171

(2)

Shift dependence

171

(1)

Poor directional selectivity

172

(1)

Properties of complex wavelets

173

(2)

Extension to multiple dimensions

174

(1)

The dual-tree complex wavelet transform

175

(3)

Shift-invariant filter design

178

(5)

Applications of the complex wavelet transform

183

(1)

References

184

(3)

Application of wavelets to filtering of noisy data

187

(12)

Ue-Li Pen

Introduction

187

(1)

Classical filters

188

(1)

Thresholding

188

(1)

Wiener filtering

189

(1)

Non-Gaussian filtering

189

(4)

Estimating prior Θ

191

(1)

Maximizing non-Gaussianity using wavelets

192

(1)

Examples

193

(4)

Discussion

197

(1)

Conclusions

197

(1)

Acknowledgements

197

(1)

References

198

(1)

Approximation and compression of piecewise smooth functions

199

(22)

Paolo Prandoni

Martin Vetterli

Introduction

199

(2)

The compression problem

201

(2)

RD upper bounds for a piecewise polynomial function

203

(7)

Polynomial approximation

204

(3)

Wavelet-based approximation

207

(1)

Commentary

208

(2)

RD optimal approximation

210

(7)

Joint segmentation and allocation

211

(2)

Efficient solution

213

(2)

Implementation and results

215

(2)

Conclusions

217

(2)

Local Legendre expansion

218

(1)

Estimate of the series truncation error

218

(1)

References

219

(2)

The contribution of wavelets to the analysis of economic and financial data

221

(16)

James B. Ramsey

Introduction

221

(2)

A selective review of the literature

223

(8)

Non-stationarity and complex functions

223

(3)

Time-scale decompositions

226

(3)

Forecasting

229

(1)

Some miscellaneous papers

229

(2)

The research potential

231

(3)

Exploratory analysis: time-scale versus frequency

231

(1)

Density estimation and local inhomogeneity

232

(1)

Time-scale decomposition

232

(1)

Aspects for forecasting

233

(1)

Conclusion

234

(1)

References

234

(3)

Harmonic wavelets in vibrations and acoustics

237

(1)

D. E. Newland

Introduction

237

(1)

Harmonic wavelet theory

238

(1)

Numerical algorithm

238

(3)

Phase interpretation

241

(1)

Bending wave transmission in a beam

241

(3)

Response of an acoustic waveguide

244

(2)

Underground train vibration

246

(3)

Geotechnical centrifuge testing

249

(1)

Experimental data

250

(1)

Power spectral densities

250

(2)

Cross-spectral densities

252

(2)

Simulated results

254

(1)

Acknowledgements

254

(1)

References

255

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Wavelets The Key to Intermittent Information

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Table of Contents

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