The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance Paul S Addison Napier University, Edinburgh, UK IoP Institute of Physics Publishing Bristol and Philadelphia
Preface xi 1 Getting started T 1.1 Introduction 1 1.2 The wavelet transform 2 1.3 Reading the book 3 2 The continuous wavelet transform 6 2.1 Introduction 6 2.2 The wavelet 6 2.3 Requirements for the wavelet 9 2.4 The energy spectrum of the wavelet 9 2.5 The wavelet transform 11 2.6 Identification of coherent structures 14 2.7 Edge detection 21 2.8 The inverse wavelet transform 25 2.9 The signal energy: wavelet-based energy and power spectra 28 2.10 The wavelet transform in terms of the Fourier transform 33 2.11 Complex wavelets: the Morlet wavelet 35 2.12 The wavelet transform, short time Fourier transform and Heisenberg boxes 45 2.13 Adaptive transforms: matching pursuits 51 2.14 Wavelets in two or more dimensions 55 2.15 The CWT: computation, boundary effects and viewing 56 2.16 Endnotes 63 2.16.1 Chapter keywords and phrases 63 2.16.2 Further resources 63 3 The discrete wavelet transform 65 3.1 Introduction 65 3.2 Frames and orthogonal wavelet bases 65 3.2.1 Frames 65 vi l
Vlll 3.2.2 Dyadic grid scaling and orthonormal wavelet transforms 67 3.2.3 The scaling function and the multiresolution representation 69 3.2.4 The scaling equation, scaling coefficients and associated wavelet equation 72 3.2.5 The Haar wavelet 73 3.2.6 Coefficients from coefficients: the fast wavelet transform 75 3.3 Discrete input signals of finite length 77 3.3.1 Approximations and details 77 3.3.2 The multiresolution algorithm an example 81 3.3.3 Wavelet energy 83 3.3.4 Alternative indexing of dyadic grid coefficients 85 3.3.5 A simple worked example: the Haar wavelet transform 87 3.4 Everything discrete 91 3.4.1 Discrete experimental input signals 91 3.4.2 Smoothing, thresholding and denoising 96 3.5 Daubechies wavelets 104 3.5.1 Filtering 112 3.5.2 Symmlets and coiflets 115 3.6 Translation invariance 117 3.7 Biorthogonal wavelets 119 3.8 Two-dimensional wavelet transforms 121 3.9 Adaptive transforms: wavelet packets 133 3.10 Endnotes 141 3.10.1 Chapter keywords and phrases 141 3.10.2 Further resources 141 4 Fluids 144 4.1 Introduction 144 4.2 Statistical measures 145 4.2.1 Moments, energy and power spectra 145 4.2.2 Intermittency and correlation 152 4.2.3 Wavelet thresholding 153 4.2.4 Wavelet selection using entropy measures 159 4.3 Engineering flows 160 4.3.1 Jets, wakes, turbulence and coherent structures 160 4.3.2 Fluid-structure interaction 171 4.3.3 Two-dimensional flow fields 174 4.4 Geophysical flows 178 4.4.1 Atmospheric processes 178 4.4.2 Ocean processes 186 4.5 Other applications in fluids and further resources 187 5 Engineering testing, monitoring and characterization 189 5.1 Introduction 189 5.2 Machining processes: control, chatter, wear and breakage 189 5.3 Rotating machinery 195
5.4 5.5 5.6 5.7 5.8 5.3.1 Gears 5.3.2 Shafts, bearings and blades Dynamics Chaos Non-destructive testing Surface characterization Other applications in engineering and further resources 5.8.1 Impacting 5.8.2 Data compression 5.8.3 Engines 5.8.4 Miscellaneous ix 195 199 202 208 211 221 224 224 225 228 229 Me( 6.1 6.2 6.3 6.4 6.5 6.6 6.7 licine Introduction The electrocardiogram 6.2.1 ECG timing, distortions and noise 6.2.2 Detection of abnormalities 6.2.3 Heart rate variability 6.2.4 Cardiac arrhythmias 6.2.5 ECG data compression Neuroelectric waveforms 6.3.1 Evoked potentials and event-related potentials 6.3.2 Epileptic seizures and epileptogenic foci 6.3.3 Classification of the EEG using artificial neural networks Pathological sounds, ultrasounds and vibrations 6.4.1 Blood flow sounds 6.4.2 Heart sounds and heart rates 6.4.3 Lung sounds 6.4.4 Acoustic response Blood flow and blood pressure Medical imaging 6.6.1 Ultrasonic images 6.6.2 Magnetic resonance imaging, computed tomography and other radiographic images 6.6.3 Optical imaging Other applications in medicine 6.7.1 Electromyographic signals 6.7.2 Sleep apnoea 6.7.3 DNA 6.7.4 Miscellaneous 6.7.5 Further resources Fractals, finance, geophysics and other areas 7.1 Introduction 7.2 Fractals 7.2.1 Exactly self-similar fractals 231 234 236 239 248 248 249 252 255 258 259 260 263 264 267 273 276 276 277 279
X 7.2.2 Stochastic fractals 282 7.2.3 Multifractals 292 7.3 Finance 294 7.4 Geophysics 298 7.4.1 Properties of subsurface media 299 7.4.2 Surface feature analysis 305 7.4.3 Climate, clouds, rainfall and river levels 307 7.5 Other areas 309 7.5.1 Astronomy 309 7.5.2 Chemistry and chemical engineering 310 7.5.3 Plasmas 311 7.5.4 Electrical systems 311 7.5.5 Sound and speech 312 7.5.6 Miscellaneous 313 Appendix Useful books, papers and websites 314 1 Useful books and papers 314 2 Useful websites 315 References 317 Index 351