Wavelets in Physics

by Edited by J. C. van den Berg

• ISBN13: 9780521593113
• ISBN10: 0521593115
• Format: Hardcover
• Copyright: 1999-09-13
• Publisher: Cambridge University Press

Summary

This book surveys the application of the recently developed technique of the wavelet transform to a wide range of physical fields, including astrophysics, turbulence, meteorology, plasma physics, atomic and solid state physics, multifractals occurring in physics, biophysics (in medicine and physiology) and mathematical physics. The wavelet transform can analyze scale-dependent characteristics of a signal (or image) locally, unlike the Fourier transform, and more flexibly than the windowed Fourier transform developed by Gabor fifty 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, for example, in the solution of partial differential equations in physics. 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.

Table of Contents

List of contributors	xiii
Preface	xix
J.C. van den Berg (ed.)
A guided tour through the book	1	(8)
J.C. van den Berg
Wavelet analysis: a new tool in physics	9	(14)
J.-P. Antoine
What is wavelet analysis?	9	(3)
The continuous WT	12	(2)
The discrete WT: orthonormal bases of wavelets	14	(4)
The wavelet transform in more than one dimension	18	(2)
Outcome	20	(3)
References	21	(2)
The 2-D wavelet transform, physical applications and generalizations	23	(54)
J.-P. Antoine
Introduction	23	(1)
The continuous WT in two dimensions	24	(15)
Construction and main properties of the 2-D CWT	24	(2)
Interpretation of the CWT as a singularity scanner	26	(1)
Practical implementation: the various representations	27	(2)
Choice of the analysing wavelet	29	(5)
Evaluation of the performances of the CWT	34	(5)
Physical applications of the 2-D CWT	39	(14)
Pointwise analysis	39	(4)
Applications of directional wavelets	43	(7)
Local contrast: a nonlinear extension of the CWT	50	(3)
Continuous wavelets as affine coherent states	53	(6)
A general set-up	53	(2)
Construction of coherent states from a square integrable group representation	55	(4)
Extensions of the CWT to other manifolds	59	(6)
The three-dimensional case	59	(2)
Wavelets on the 2-sphere	61	(2)
Wavelet transform in space-time	63	(2)
The discrete WT in two dimensions	65	(5)
Multiresolution analysis in 2-D and the 2-D DWT	65	(1)
Generalizations	66	(2)
Physical applications of the DWT	68	(2)
Outcome: why wavelets?	70	(7)
References	71	(6)
Wavelets and astrophysical applications	77	(40)
A. Bijaoui
Introduction	78	(1)
Time-frequency analysis of astronomical sources	79	(5)
The world of astrophysical variable sources	79	(1)
The application of the Fourier transform	80	(1)
From Gabor's to the wavelet transform	81	(1)
Regular and irregular variables	81	(1)
The analysis of chaotic light curves	82	(1)
Applications to solar time series	83	(1)
Applications to image processing	84	(9)
Image compression	84	(2)
Denoising astronomical images	86	(3)
Multiscale adaptive deconvolution	89	(2)
The restoration of aperture synthesis observations	91	(1)
Applications to data fusion	92	(1)
Multiscale vision	93	(13)
Astronomical surveys and vision models	93	(1)
A multiscale vision model for astronomical images	94	(3)
Applications to the analysis of astrophysical sources	97	(2)
Applications to galaxy counts	99	(3)
Statistics on the large-scale structure of the Universe	102	(4)
Conclusion	106	(11)
Appendices to Chapter 3	107	(1)
The a trous algorithm	107	(1)
The pyramidal algorithm	108	(1)
The denoising algorithm	109	(1)
The deconvolution algorithm	109	(1)
References	110	(7)
Turbulence analysis, modelling and computing using wavelets	117	(84)
M. Farge
N.K.-R. Kevlahan
V. Perrier
K. Schneider
Introduction	117	(4)
Open questions in turbulence	121	(11)
Definitions	121	(3)
Navier-Stokes equations	124	(1)
Statistical theories of turbulence	125	(4)
Coherent structures	129	(3)
Fractals and singularities	132	(16)
Introduction	132	(3)
Detection and characterization of singularities	135	(2)
Energy spectra	137	(4)
Structure functions	141	(2)
The singularity spectrum for multifractals	143	(4)
Distinguishing between signals made up of isolated and dense singularities	147	(1)
Turbulence analysis	148	(12)
New diagnostics using wavelets	148	(2)
Two-dimensional turbulence analysis	150	(8)
Three-dimensional turbulence analysis	158	(2)
Turbulence modelling	160	(10)
Two-dimensional turbulence modelling	160	(5)
Three-dimensional turbulence modelling	165	(3)
Stochastic models	168	(2)
Turbulence computation	170	(15)
Direct numerical simulations	170	(1)
Wavelet-based numerical schemes	171	(1)
Solving Navier-Stokes equations in wavelet bases	172	(7)
Numerical results	179	(6)
Conclusion	185	(16)
References	190	(11)
Wavelets and detection of coherent structures in fluid turbulence	201	(26)
L. Hudgins
J.H. Kaspersen
Introduction	201	(4)
Advantages of wavelets	205	(1)
Experimental details	205	(3)
Approach	208	(4)
Methodology	208	(1)
Estimation of the false-alarm rate	209	(2)
Estimation of the probability of detection	211	(1)
Conventional coherent structure detectors	212	(3)
Quadrant analysis (Q2)	212	(1)
Variable Interval Time Average (VITA)	212	(2)
Window Average Gradient (WAG)	214	(1)
Wavelet-based coherent structure detectors	215	(4)
Typical wavelet method (psi)	215	(1)
Wavelet quadature method (Quad)	216	(3)
Results	219	(6)
Conclusions	225	(2)
References	225	(2)
Wavelets, non-linearity and turbulence in fusion plasmas	227	(36)
B.Ph. van Milligen
Introduction	227	(1)
Linear spectral analysis tools	228	(6)
Wavelet analysis	228	(3)
Wavelet spectra and coherence	231	(2)
Joint wavelet phase-frequency spectra	233	(1)
Non-linear spectral analysis tools	234	(6)
Wavelet bispectra and bicoherence	234	(3)
Interpretation of the bicoherence	237	(3)
Analysis of computer-generated data	240	(15)
Coupled van der Pol oscillators	242	(3)
A large eddy simulation model for two-fluid plasma turbulence	245	(4)
A long wavelength plasma drift wave model	249	(6)
Analysis of plasma edge turbulence from Langmuir probe data	255	(5)
Radial coherence observed on the TJ-IU torsatron	255	(1)
Bicoherence profile at the L/H transition on CCT	256	(4)
Conclusions	260	(3)
References	261	(2)
Transfers and fluxes of wind kinetic energy between orthogonal wavelet components during atmospheric blocking	263	(36)
A. Fournier
Introduction	263	(1)
Data and blocking description	264	(1)
Analysis	265	(11)
Conventional statistics	266	(1)
Fundamental equations	266	(1)
Review of statistical equations	267	(1)
Review of Fourier based energetics	268	(2)
Basic concepts from the theory of wavelet analysis	270	(3)
Energetics in the domain of wavelet indices (or any orthogonal basis)	273	(1)
Kinetic energy localized flux functions	274	(2)
Results and interpretation	276	(19)
Time averaged statistics	276	(3)
Time dependent multiresolution analysis at fixed (φ, p)	279	(4)
Kinetic energy transfer functions	283	(12)
Concluding remarks	295	(4)
References	296	(3)
Wavelets in atomic physics and in solid state physics	299	(40)
J.-P. Antoine
Ph. Antoine
B. Piraux
Introduction	299	(2)
Harmonic generation in atom-laser interaction	301	(13)
The physical process	301	(1)
Calculation of the atomic dipole for a one-electron atom	302	(2)
Time-frequency analysis of the dipole acceleration: H(1s)	304	(9)
Extension to multi-electron atoms	313	(1)
Calculation of multi-electronic wave functions	314	(4)
The self-consistent Hartree-Fock method (HF)	315	(2)
Beyond Hartree-Fock: inclusion of electron correlations	317	(1)
CWT realization of a 1-D HF equation	317	(1)
Other applications in atomic physics	318	(2)
Combination of wavelets with moment methods	318	(1)
Wavelets in plasma physics	319	(1)
Electronic structure calculations	320	(7)
Principle	320	(1)
A non-orthogonal wavelet basis	321	(3)
Orthogonal wavelet bases	324	(2)
Second generation wavelets	326	(1)
Wavelet-like orthonormal bases for the lowest Landau level	327	(7)
The Fractional Quantum Hall Effect setup	328	(1)
The LLL basis problem	329	(1)
Wavelet-like bases	330	(3)
Further variations on the same theme	333	(1)
Outcome: what have wavelet brought to us?	334	(5)
References	335	(4)
The thermodynamics of fractals revisited with wavelets	339	(52)
A. Arneodo
E. Bacry
J.F. Muzy
Introduction	340	(3)
The multifractal formalism	343	(5)
Microcanonical description	343	(3)
Canonical description	346	(2)
Wavelets and multifractal formalism for fractal functions	348	(12)
The wavelet transform	348	(1)
Singularity detection and processing with wavelets	349	(1)
The wavelet transform modulus maxima method	350	(7)
Phase transition in the multifractal spectra	357	(3)
Multifractal analysis of fully developed turbulence data	360	(6)
Wavelet analysis of local scaling properties of a turbulent velocity signal	361	(2)
Determination of the singularity spectrum of a turbulent velocity signal with the WTMM method	363	(3)
Beyond multifractal analysis using wavelets	366	(11)
Solving the inverse fractal problem from wavelet analysis	367	(6)
Wavelet transform and renormalization of the transition to chaos	373	(4)
Uncovering a Fibonacci multiplicative process in the arborescent fractal geometry of diffusion-limited aggregates	377	(7)
Conclusion	384	(7)
References	385	(6)
Wavelets in medicine and physiology	391	(30)
P.Ch. Ivanov
A.L. Goldberger
S. Havlin
C.-K. Peng
M.G. Rosenblum
H.E. Stanley
Introduction	391	(3)
Nonstationary physiological signals	394	(2)
Wavelet transform	396	(1)
Hilbert transform	397	(3)
Universal distribution of variations	400	(5)
Wavelets and scale invariance	405	(2)
A diagnostic for health vs. disease	407	(1)
Information in the Fourier phases	408	(4)
Concluding remarks	412	(9)
References	413	(8)
Wavelet dimension and time evolution	421	(28)
Ch.-A. Guerin
M. Holschneider
Introduction	421	(4)
The lacunarity dimension	425	(4)
Quantum chaos	429	(1)
The generalized wavelet dimensions	430	(3)
Time evolution and wavelet dimensions	433	(2)
Appendix	435	(14)
References	446	(3)
Index	449