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
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
3.4.3 Applications to the analysis of astrophysical
sources ............................................ 97
3.4.4 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 à 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
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
6.5.2 Bicoherence profile at the L/H transition on CCT .. 256
6.6 Conclusions .............................................. 260
References ............................................... 261
7 Transfers and fluxes of wind kinetic energy between
orthogonal wavelet components during atmospheric
blocking ................................................. 263
A. Fournier
7.1 Introduction ............................................. 263
7.2 Data and blocking description ............................ 264
7.3 Analysis ................................................. 265
7.3.1 Conventional statistics ........................... 266
7.3.2 Fundamental equations ............................. 266
7.3.3 Review of statistical equations ................... 267
7.3.4 Review of Fourier based energetics ................ 268
7.3.5 Basic concepts from the theory of wavelet
analysis .......................................... 270
7.3.6 Energetics in the domain of wavelet indices
(or any orthogonal basis) ......................... 273
7.3.7 Kinetic energy localized flux functions ........... 274
7.4 Results and interpretation ............................... 276
7.4.1 Time averaged statistics .......................... 276
7.4.2 Time dependent multiresolution analysis
at fixed (φ,ρ) .................................... 279
7.4.3 Kinetic energy transfer functions ................. 283
7.5 Concluding remarks ....................................... 295
References ............................................... 296
8 Wavelets in atomic physics and in solid state physics .... 299
J.-P. Antoine, Ph. Antoine and B. Piraux
8.1 Introduction ............................................. 299
8.2 Harmonic generation in atom-laser interaction ............ 301
8.2.1 The physical process .............................. 301
8.2.2 Calculation of the atomic dipole for a
one-electron atom ................................. 302
8.2.3 Time-frequency analysis of the dipole
acceleration: H(ls) ............................... 304
8.2.4 Extension to multi-electron atoms ................. 313
8.3 Calculation of multi-electronic wave functions ........... 314
8.3.1 The self-consistent Hartree-Fock method (HF) ...... 315
8.3.2 Beyond Hartree-Fock: inclusion of electron
correlations ...................................... 317
8.3.3 CWT realization of a 1-D HF equation .............. 317
8.4 Other applications in atomic physics ..................... 318
8.4.1 Combination of wavelets with moment methods ....... 318
8.4.2 Wavelets in plasma physics ........................ 319
8.5 Electronic structure calculations ........................ 320
8.5.1 Principle ......................................... 320
8.5.2 A non-orthogonal wavelet basis .................... 321
8.5.3 Orthogonal wavelet bases .......................... 324
8.5.4 Second generation wavelets ........................ 326
8.6 Wavelet-like orthonormal bases for the lowest
Landau level ............................................. 327
8.6.1 The Fractional Quantum Hall Effect setup .......... 328
8.6.2 The LLL basis problem ............................. 329
8.6.3 Wavelet-like bases ................................ 330
8.6.4 Further variations on the same theme .............. 333
8.7 Outcome: what have wavelet brought to us? ................ 334
References ............................................... 335
9 The thermodynamics of fractals revisited with wavelets ... 339
A. Arneodo, E. Bacry and J.F. Muzy
9.1 Introduction ............................................. 340
9.2 The multifractal formalism ............................... 343
9.2.1 Microcanonical description ........................ 343
9.2.2 Canonical description ............................. 346
9.3 Wavelets and multifractal formalism for fractal
functions ................................................ 348
9.3.1 The wavelet transform ............................. 348
9.3.2 Singularity detection and processing with
wavelets .......................................... 349
9.3.3 The wavelet transform modulus maxima method ....... 350
9.3.4 Phase transition in the multifractal spectra ...... 357
9.4 Multifractal analysis of fully developed turbulence
data ..................................................... 360
9.4.1 Wavelet analysis of local scaling properties
of a turbulent velocity signal .................... 361
9.4.2 Determination of the singularity spectrum of a
turbulent velocity signal with the WTMM method .... 363
9.5 Beyond multifractal analysis using wavelets .............. 366
9.5.1 Solving the inverse fractal problem from wavelet
analysis .......................................... 367
9.5.2 Wavelet transform and renormalization of the
transition to chaos ............................... 373
9.6 Uncovering a Fibonacci multiplicative process in
the arborescent fractal geometry of diffusion-limited
aggregates ............................................... 377
9.7 Conclusion ............................................... 384
References ............................................... 385
10 Wavelets in medicine and physiology ...................... 391
P.Ch. Ivanov, A.L. Goldberger, S. Havlin, C.-K. Peng,
M.G. Rosenblum and H.E. Stanley
10.1 Introduction ............................................. 391
10.2 Nonstationary physiological signals ...................... 394
10.3 Wavelet transform ........................................ 396
10.4 Hilbert transform ........................................ 397
10.5 Universal distribution of variations ..................... 400
10.6 Wavelets and scale invariance ............................ 405
10.7 A diagnostic for health vs. disease ...................... 407
10.8 Information in the Fourier phases ........................ 408
10.9 Concluding remarks ....................................... 412
References ............................................... 413
11 Wavelet dimension and time evolution ..................... 421
Ch.-A. Guérin and M. Holschneider
11.1 Introduction ............................................. 421
11.2 The lacunarity dimension ................................. 425
11.3 Quantum chaos ............................................ 429
11.4 The generalized wavelet dimensions ....................... 430
11.5 Time evolution and wavelet dimensions .................... 433
11.6 Appendix ................................................. 435
References ............................................... 446
Index .................................................... 449
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