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About this book
About this book
This book provides an overview of the theory and practical application of wavelet transform methods. It is designed specifically for the 'applied' reader in science, engineering, medicine, finance or any other of the growing number of application areas. New comers to the subject will find an accessible and clear account of the theory of continuous and discrete wavelet transforms with a large number of examples of their use across a wide range of disciplines. Readers already acquainted with wavelets working in a particular application area can use the book to broaden their perspective. A strength of the book is the author's use of several hundred illustrations, some in colour, to convey mathematical concepts and the results of applications. The application chapters make the reader aware of the similarities that exist in the use of wavelet transform analysis across disciplines. A comprehensive list of more than 700 references provides a valuable resource for further study. Chapter 1 is a brief overview of the wavelet transform including a short history. The rest of the book is split into two parts: the first (chapters 2 and 3) deals with the mathematics of both the discrete and continuous wavelet transforms; and the second (chapters 4 - 8) deals with applications in a variety of subject areas: including geophysics, medicine, fluid turbulence, engineering testing, speech and sound analysis, image analysis, and data compression.
PREFACECHAPTER 1: GETTING STARTEDCHAPTER 2: THE CONTINUOUS WAVELET TRANSFORM2.1 Introduction2.2 The Wavelet2.3 Requirements for the Wavelet2.4 The Energy Spectrum of the Wavelet2.5 The Wavelet Transform2.6 Identification of Coherent Structures2.7 Edge Detection2.8 The Inverse Wavelet Transform2.9 The Signal Energy: Wavelet Based Energy and Power Spectra2.10 The Wavelet Transform in Terms of the Fourier Transform2.11 Complex Wavelets: The Morlet Wavelet2.12 The Wavelet Transform, Short Time Fourier Transform and Heisenberg Boxes2.13 Adaptive Transforms: Matching Pursuits2.14 Wavelets in Two or More Dimensions2.15 The CWT: Computation, Boundary Effects and Viewing2.16 Endnotes2.16.1 Chapter Key Words and Phrases2.16.2 Further ResourcesCHAPTER 3:THE DISCRETE WAVELET TRANSFORM3.1 Introduction3.2 Frames and Orthogonal Wavelet Bases3.2.1 Frames3.2.2 Dyadic Grid Scaling and Orthonormal Wavelet Transforms3.2.3 The Scaling Function and the Multiresolution Representation3.2.4 The Scaling Equation, Scaling Coefficients and Associated Wavelet Equation3.2.5 The Haar Wavelet3.2.6 Coefficients from Coefficients: The Fast Wavelet Transform3.3 Discrete Input Signals of Finite Length3.3.1 Approximations and Details3.3.2 The Multiresolution Algorithm - An Example3.3.3 Wavelet Energy3.3.4 Alternative Indexing of Dyadic Grid Coefficients3.3.5 A Simple Worked Example: The Haar Wavelet Transform3.4 Everything Discrete3.4.1 Discrete Experimental Input Signals3.4.2 Smoothing, Thresholding and Denoising3.5 Daubechies Wavelets3.5.1 Filtering3.5.2 Symmlets and Coiflets3.6 Translation Invariance3.7 Biorthogonal Wavelets3.8 Two-Dimensional Wavelet Transforms3.9 Adaptive Transforms: Wavelet Packets3.10 Endnotes3.10.1 Chapter Key Words and Phrases3.10.2 Further ResourcesCHAPTER 4: FLUIDS4.1 Introduction4.2 Statistical Measures4.2.1 Moments, Energy and Power Spectra4.2.2 Intermittency and Correlation4.2.3 Wavelet Thresholding4.2.4 Wavelet Selection using Entropy Measures4.3 Engineering Flows4.3.1 Jets, Wakes, Turbulence and Coherent Structures4.3.2 Fluid-Structure Interaction4.3.3 Two-Dimensional Flow Fields4.4 Geophysical Flows4.4.1 Atmospheric Processes4.4.2 Ocean Processes4.5 Other Applications in Fluids and Further ResourcesCHAPTER 5: ENGINEERING TESTING, MONITORING AND CHARACTERISATION5.1 Introduction5.2 Machining Processes: Control, Chatter, Wear and Breakage5.3 Rotating Machinery5.3.1 Gears5.3.2 Shafts, Bearings and Blades5.4 Dynamics5.5 Chaos5.6 Non-Destructive Testing5.7 Surface Characterisation5.8 Other Applications in Engineering and Further Resources5.8.1 Impacting5.8.2 Data Compression5.8.3 Engines5.8.4 MiscellaneousCHAPTER 6: MEDICINE6.1 Introduction6.2 The Electrocardiogram6.2.1 ECG Timing, Distortions and Noise6.2.2 Detection of Abnormalities6.2.3 Heart Rate Variability6.2.4 Cardiac Arrhythmias