This page intentionally left blank Wavelets in Physics Thisbooksurveystheapplicationoftherecentlydevelopedtechniqueofthewavelet transform to a wide range of physical fields, including astrophysics, turbulence, meteorology,plasmaphysics,atomicandsolidstatephysics,multifractalsoccurring in physics, biophysics (in medicine and physiology) and mathematical physics. The wavelet transorm can analyse scale-dependent characteristics of a signal (or image) locally, unlike the Fourier transform, and more flexibly than the windowed Fourier transform developed by Gabor 50 years ago. The continuous wavelet transform is used mostly for analysis, but the discrete wavelet transform allows very fast compression and transmission of data and speeds up numerical calculation, and is applied,forexample,inthesolutionofpartialdifferentialequationsinphysics.This book will be of interest to graduate students and researchers in many fields of physics, and to applied mathematicians and engineers interested in physical application. J. C. VANDEN BERG studied physics and mathematics at the University of Amsterdam. He graduated in high energy physics, doing some work on the automatization of the analysis of bubble chamber films exhibiting the paths of elementaryparticlesincollisionexperiments.Helatertookadegreeinphilosophyof science and logic at the same university, doing his masters thesis on quantum logic. He became a mathematics instructor at Wageningen University in 1973 and is now an Assistant Professor of Applied Mathematics at the Biometris group of Wageningen University and Research Center. After being interested in the foundations of quantum mechanics for many years, hemovedontonon-lineardynamics,especiallytheconceptofmultifractalsandthe difficulties of analysing them. In the writings of Alain Arne´odo on multifractals, he came acrossthewavelettransformforthefirsttime,taking hisfirsttechnicalcourse on the subject in 1991 at the CWI in Amsterdam. Soon after, discovering the pioneering works of Marie Farge in turbulence and Gerald Kaiser in electromagnetism,hebecameconvinced thatwaveletswereimportantforphysics at large. Gradually wavelets overshadowed all his other interests and have remained a mainfocuseversince.Thisbookisaresultofthatcontinuinginterestandhehopesit may stimulate others to explore the possibilities of the new tools wavelet analysis continues to deliver. Wavelets in Physics Edited by J.C. VAN DEN BERG WageningenUniversityandResearchCenter, Wageningen,TheNetherlands CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521593113 © Cambridge University Press 1999, 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 1999 ISBN-13 978-0-511-67489-1 eBook (NetLibrary) ISBN-13 978-0-521-59311-3 Hardback ISBN-13 978-0-521-53353-9 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents page List of contributors xii Preface to the paperback edition xvii J.C. van den Berg Preface to the first edition xxi J.C. van den Berg 0 A guided tour through the book 1 J.C. van den Berg 1 Wavelet analysis: a new tool in physics 9 J.-P. Antoine 1.1 What is wavelet analysis? 9 1.2 The continuous WT 12 1.3 The discrete WT: orthonormal bases of wavelets 14 1.4 The wavelet transform in more than one dimension 18 1.5 Outcome 20 References 21 2 The 2-D wavelet transform, physical applications and generalizations 23 J.-P. Antoine 2.1 Introduction 23 2.2 The continuous WT in two dimensions 24 2.2.1 Construction and main properties of the 2-D CWT 24 2.2.2 Interpretation of the CWT as a singularity scanner 26 2.2.3 Practical implementation: the various representations 27 v vi Contents 2.2.4 Choice of the analysing wavelet 29 2.2.5 Evaluation of the performances of the CWT 34 2.3 Physical applications of the 2-D CWT 39 2.3.1 Pointwise analysis 39 2.3.2 Applications of directional wavelets 43 2.3.3 Local contrast: a nonlinear extension of the CWT 50 2.4 Continuous wavelets as affine coherent states 53 2.4.1 A general set-up 53 2.4.2 Construction of coherent states from a square integrable group representation 55 2.5 Extensions of the CWT to other manifolds 59 2.5.1 The three-dimensional case 59 2.5.2 Wavelets on the 2-sphere 61 2.5.3 Wavelet transform in space-time 63 2.6 The discrete WT in two dimensions 65 2.6.1 Multiresolution analysis in 2-D and the 2-D DWT 65 2.6.2 Generalizations 66 2.6.3 Physical applications of the DWT 68 2.7 Outcome: why wavelets? 70 References 71 3 Wavelets and astrophysical applications 77 A. Bijaoui 3.1 Introduction 78 3.2 Time–frequency analysis of astronomical sources 79 3.2.1 The world of astrophysical variable sources 79 3.2.2 The application of the Fourier transform 80 3.2.3 From Gabor’s to the wavelet transform 81 3.2.4 Regular and irregular variables 81 3.2.5 The analysis of chaotic light curves 82 3.2.6 Applications to solar time series 83 3.3 Applications to image processing 84 3.3.1 Image compression 84 3.3.2 Denoising astronomical images 86 3.3.3 Multiscale adaptive deconvolution 89 3.3.4 The restoration of aperture synthesis observations 91 3.3.5 Applications to data fusion 92 3.4 Multiscale vision 93 3.4.1 Astronomical surveys and vision models 93 3.4.2 A multiscale vision model for astronomical images 94 Contents vii 3.4.3 Applications to the analysis of astrophysical sources 97 3.3.4 Applications to galaxy counts 99 3.4.5 Statistics on the large-scale structure of the Universe 102 3.5 Conclusion 106 Appendices to Chapter 3 107 A. The a` trous algorithm 107 B. The pyramidal algorithm 108 C. The denoising algorithm 109 D. The deconvolution algorithm 109 References 110 4 Turbulence analysis, modelling and computing using wavelets 117 M. Farge, N.K.-R. Kevlahan, V. Perrier and K. Schneider 4.1 Introduction 117 4.2 Open questions in turbulence 121 4.2.1 Definitions 121 4.2.2 Navier–Stokes equations 124 4.2.3 Statistical theories of turbulence 125 4.2.4 Coherent structures 129 4.3 Fractals and singularities 132 4.3.1 Introduction 132 4.3.2 Detection and characterization of singularities 135 4.3.3 Energy spectra 137 4.3.4 Structure functions 141 4.3.5 The singularity spectrum for multifractals 143 4.3.6 Distinguishing between signals made up of isolated and dense singularities 147 4.4 Turbulence analysis 148 4.4.1 New diagnostics using wavelets 148 4.4.2 Two-dimensional turbulence analysis 150 4.4.3 Three-dimensional turbulence analysis 158 4.5 Turbulence modelling 160 4.5.1 Two-dimensional turbulence modelling 160 4.5.2 Three-dimensional turbulence modelling 165 4.5.3 Stochastic models 168 4.6 Turbulence computation 170 4.6.1 Direct numerical simulations 170 4.6.2 Wavelet-based numerical schemes 171 4.6.3 Solving Navier–Stokes equations in wavelet bases 172 4.6.4 Numerical results 179 viii Contents 4.7 Conclusion 185 References 190 5 Wavelets and detection of coherent structures in fluid turbulence 201 L. Hudgins and J.H. Kaspersen 5.1 Introduction 201 5.2 Advantages of wavelets 205 5.3 Experimental details 205 5.4 Approach 208 5.4.1 Methodology 208 5.4.2 Estimation of the false-alarm rate 209 5.4.3 Estimation of the probability of detection 211 5.5 Conventional coherent structure detectors 212 5.5.1 Quadrant analysis (Q2) 212 5.5.2 Variable Interval Time Average (VITA) 212 5.5.3 Window Average Gradient (WAG) 214 5.6 Wavelet-based coherent structure detectors 215 5.6.1 Typical wavelet method (psi) 215 5.6.2 Wavelet quadature method (Quad) 216 5.7 Results 219 5.8 Conclusions 225 References 225 6 Wavelets, non-linearity and turbulence in fusion plasmas 227 B.Ph. van Milligen 6.1 Introduction 227 6.2 Linear spectral analysis tools 228 6.2.1 Wavelet analysis 228 6.2.2 Wavelet spectra and coherence 231 6.2.3 Joint wavelet phase-frequency spectra 233 6.3 Non-linear spectral analysis tools 234 6.3.1 Wavelet bispectra and bicoherence 234 6.3.2 Interpretation of the bicoherence 237 6.4 Analysis of computer-generated data 240 6.4.1 Coupled van der Pol oscillators 242 6.4.2 A large eddy simulation model for two-fluid plasma turbulence 245 6.4.3 A long wavelength plasma drift wave model 249 6.5 Analysis of plasma edge turbulence from Langmuir probe data 255 6.5.1 Radial coherence observed on the TJ-IU torsatron 255