Wavelets in physics

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contributors......Page 14
Preface to the paperback edition......Page 19
References......Page 21
Acknowledgements......Page 26
0 A guided tour through the book......Page 27
1.1 What is wavelet analysis?......Page 35
1.2 The continuous WT......Page 38
1.3 The discrete WT: orthonormal bases of wavelets......Page 40
1.4 The wavelet transform in more than one dimension......Page 44
1.5 Outcome......Page 46
References......Page 47
2.1 Introduction......Page 49
2.2.1 Construction and main properties of the 2-D CWT......Page 50
2.2.2 Interpretation of the CWT as a singularity scanner......Page 52
2.2.3 Practical implementation: the various representations......Page 53
2.2.4.1 Isotropic wavelets......Page 55
2.2.4.2 Directional wavelets......Page 57
2.2.5.1 The scale and angle resolving power......Page 60
2.2.5.2 The reproducing kernel and the resolving power of the wavelet......Page 61
2.2.5.3 Calibration of a wavelet with benchmark signals......Page 62
2.2.5.4 Discretization of the CWT......Page 64
2.3.1.1 Contour detection, character recognition......Page 65
2.3.1.2 Analysis of 2-D fractals......Page 67
2.3.1.4 Analysis of astronomical images......Page 68
2.3.2.1 Application in fluid dynamics......Page 69
2.3.2.2 Detection of symmetries......Page 72
2.3.2.3 Geophysics: fault detection......Page 75
2.3.3 Local contrast: a nonlinear extension of the CWT......Page 76
2.4.1 A general set-up......Page 79
2.4.2.1 Definitions and main properties......Page 81
2.4.2.2 Examples: the 1-D and 2-D CWT......Page 83
2.4.2.3 Application: minimal uncertainty wavelets......Page 84
2.5.1 The three-dimensional case......Page 85
2.5.2 Wavelets on the 2-sphere......Page 87
2.5.3.1 Kinematical wavelets......Page 89
2.5.3.2 Relativistic wavelets......Page 90
2.6.1 Multiresolution analysis in 2-D and the 2-D DWT......Page 91
2.6.2.1 Biorthogonal wavelet bases......Page 92
2.6.2.3 More isotropic 2-D wavelets......Page 93
2.6.3 Physical applications of the DWT......Page 94
2.7 Outcome: why wavelets?......Page 96
References......Page 97
Abstract......Page 103
3.1 Introduction......Page 104
3.2.1 The world of astrophysical variable sources......Page 105
3.2.2 The application of the Fourier transform......Page 106
3.2.4 Regular and irregular variables......Page 107
3.2.5 The analysis of chaotic light curves......Page 108
3.2.6 Applications to solar time series......Page 109
3.3.1 Image compression......Page 110
3.3.2.2 Decision theory and significant coefficients......Page 112
3.3.2.3 The PDF of the wavelet coefficients......Page 113
3.3.2.4 Denoising by using the significant coefficients......Page 114
3.3.3 Multiscale adaptive deconvolution......Page 115
3.3.4 The restoration of aperture synthesis observations......Page 117
3.3.5 Applications to data fusion......Page 118
3.4.1 Astronomical surveys and vision models......Page 119
3.4.2.1 Object definition in the wavelet transform space......Page 120
3.4.2.3 The object identification......Page 121
3.4.2.4 The object image reconstruction......Page 122
3.4.3 Applications to the analysis of astrophysical sources......Page 123
3.4.4 Applications to galaxy counts......Page 125
3.4.5 Statistics on the large-scale structure of the Universe......Page 128
3.5 Conclusion......Page 132
A. The à trous algorithm......Page 133
B. The pyramidal algorithm......Page 134
Acknowledgements......Page 135
References......Page 136
4.1 Introduction......Page 143
4.2.1 Definitions......Page 147
4.2.2 Navier–Stokes equations......Page 150
4.2.3 Statistical theories of turbulence......Page 151
4.2.4 Coherent structures......Page 155
4.3.1 Introduction......Page 158
4.3.2 Detection and characterization of singularities......Page 161
4.3.3 Energy spectra......Page 163
4.3.4 Structure functions......Page 167
4.3.5 The singularity spectrum for multifractals......Page 169
4.3.6 Distinguishing between signals made up of isolated and dense singularities......Page 173
4.4.1 New diagnostics using wavelets......Page 174
4.4.2 Two-dimensional turbulence analysis......Page 177
4.4.3 Three-dimensional turbulence analysis......Page 185
4.5 Turbulence modelling......Page 186
4.5.1 Two-dimensional turbulence modelling......Page 187
4.5.2 Three-dimensional turbulence modelling......Page 193
4.5.3 Stochastic models......Page 194
4.6.1 Direct numerical simulations......Page 196
4.6.2 Wavelet-based numerical schemes......Page 197
4.6.3 Solving Navier–Stokes equations in wavelet bases......Page 200
4.6.3.1 The heat equation solution......Page 202
4.6.3.2 The Poisson equation......Page 203
4.6.3.3 The nonlinear term......Page 204
4.6.3.4 The boundary conditions......Page 205
4.6.4.1 Three vortex interaction......Page 206
4.6.4.2 Freely decaying turbulence......Page 208
4.6.4.3 Wavelet-forced turbulence......Page 209
4.7 Conclusion......Page 213
References......Page 216
5.1 Introduction......Page 227
5.3 Experimental details......Page 231
5.4.1 Methodology......Page 234
5.4.2 Estimation of the false-alarm rate......Page 235
5.4.3 Estimation of the probability of detection......Page 237
5.5.2 Variable Interval Time Average (VITA)......Page 238
5.5.3 Window Average Gradient (WAG)......Page 240
5.6.1 Typical wavelet method (psi)......Page 241
5.6.2 Wavelet quadrature method (Quad)......Page 242
5.7 Results......Page 245
References......Page 251
6.1 Introduction......Page 253
6.2.1 Wavelet analysis......Page 254
6.2.2 Wavelet spectra and coherence......Page 257
6.2.3 Joint wavelet phase-frequency spectra......Page 259
6.3.1 Wavelet bispectra and bicoherence......Page 260
6.3.2 Interpretation of the bicoherence......Page 263
6.4 Analysis of computer-generated data......Page 266
6.4.1 Coupled van der Pol oscillators......Page 268
6.4.2 A large eddy simulation model for two-fluid plasma turbulence......Page 271
6.4.3 A long wavelength plasma drift wave model......Page 275
6.5.1 Radial coherence observed on the TJ-IU torsatron......Page 281
6.5.2 Bicoherence profile at the L/H transition on CCT......Page 282
6.6. Conclusions......Page 286
References......Page 287
7.1 Introduction......Page 289
7.2 Data and blocking description......Page 290
7.3 Analysis......Page 291
7.3.2 Fundamental equations......Page 292
7.3.3 Review of statistical equations......Page 293
7.3.4 Review of Fourier based energetics......Page 294
7.3.5 Basic concepts from the theory of wavelet analysis......Page 296
7.3.6 Energetics in the domain of wavelet indices (or any orthogonal basis)......Page 299
7.3.7 Kinetic energy localized flux functions......Page 300
7.4.1 Time averaged statistics......Page 302
7.4.2 Time dependent multiresolution analysis at fixed (phi,p)......Page 305
7.4.3 Kinetic energy transfer functions......Page 309
7.5 Concluding remarks......Page 321
References......Page 322
8.1 Introduction......Page 325
8.2.1 The physical process......Page 327
8.2.2 Calculation of the atomic dipole for a one-electron atom......Page 328
8.2.3 Time–frequency analysis of the dipole acceleration: H(1s)......Page 330
8.2.3.1 Time dependence of harmonic emission in H(1s)......Page 331
8.2.3.2 Harmonic emission in H(2s)......Page 335
8.2.3.4 Harmonic spectrum at fixed time......Page 336
8.2.3.5 Which time–frequency method?......Page 337
8.2.4 Extension to multi-electron atoms......Page 339
8.3 Calculation of multi-electronic wave functions......Page 340
8.3.1 The self-consistent Hartree–Fock method (HF)......Page 341
8.3.3 CWT realization of a 1-D HF equation......Page 343
8.4.1 Combination of wavelets with moment methods......Page 344
8.4.2 Wavelets in plasma physics......Page 345
8.5.1 Principle......Page 346
8.5.2 A non-orthogonal wavelet basis......Page 347
8.5.3.1 Diagonalizing the LDA Hamiltonian in a Daubechies basis......Page 350
8.5.3.2 Molecular dynamics algorithm in a Daubechies basis......Page 351
8.5.4 Second generation wavelets......Page 352
8.6 Wavelet-like orthonormal bases for the lowest Landau level......Page 353
8.6.1 The Fractional Quantum Hall Effect setup......Page 354
8.6.2 The LLL basis problem......Page 355
8.6.3.1 The Haar basis......Page 356
8.6.3.2 The Littlewood–Paley and other wavelet bases......Page 357
8.6.4 Further variations on the same theme......Page 359
8.7 Outcome: what have wavelets brought to us?......Page 360
References......Page 361
Abstract......Page 365
9.1 Introduction......Page 366
9.2.1.1 The f(alpha) singularity spectrum......Page 369
9.2.1.2 The generalized fractal dimensions......Page 371
9.2.2 Canonical description......Page 372
9.3.1 The wavelet transform......Page 374
9.3.2 Singularity detection and processing with wavelets......Page 375
9.3.3.1 Determination of the singularity spectrum of fractal functions from wavelet analysis......Page 376
9.3.3.2 Application of the WTMM method to recursive fractal functions......Page 380
9.3.4 Phase transition in the multifractal spectra......Page 383
9.4 Multifractal analysis of fully developed turbulence data......Page 386
9.4.1 Wavelet analysis of local scaling properties of a turbulent velocity signal......Page 388
9.4.2 Determination of the singularity spectrum of a turbulent velocity signal with the WTMM method......Page 390
9.5 Beyond multifractal analysis using wavelets......Page 392
9.5.1 Solving the inverse fractal problem from wavelet analysis......Page 393
9.5.1.1 Linear cookie-cutters......Page 394
9.5.1.2 Nonlinear cookie-cutters......Page 398
9.5.2 Wavelet transform and renormalization of the transition to chaos......Page 399
9.6 Uncovering a Fibonacci multiplicative process in the arborescent fractal geometry of diffusion-limited aggregates......Page 403
9.7 Conclusion......Page 410
References......Page 411
10.1 Introduction......Page 417
10.2 Nonstationary physiological signals......Page 420
10.3 Wavelet transform......Page 422
10.4 Hilbert transform......Page 423
10.5 Universal distribution of variations......Page 426
10.6 Wavelets and scale invariance......Page 431
10.7 A diagnostic for health vs. disease......Page 433
10.8 Information in the Fourier phases......Page 434
10.9 Concluding remarks......Page 438
References......Page 439
11.1 Introduction......Page 447
11.2 The lacunarity dimension......Page 451
11.3 Quantum chaos......Page 455
11.4 The generalized wavelet dimensions......Page 456
11.5 Time evolution and wavelet dimensions......Page 459
11.6 Appendix......Page 461
References......Page 472
Index......Page 475