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Wavelets: A Concise Guide


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Table of Contents

List of TablesList of FiguresList of AcronymsPrefaceAcknowledgments1. Analysis in Vector and Function Spaces1.1. Introduction1.2. The Lebesgue Integral1.3. Discrete Time Signals1.4. Vector Spaces1.5. Linear Independence1.6. Bases and Basis Vectors1.7. Normed Vector Spaces1.8. Inner Product1.9. Banach and Hilbert Spaces1.10. Linear Operators, Operator Norm, the Adjoint Operator1.11. Reproducing Kernel Hilbert Space1.12. The Dirac Delta Distribution1.13. Orthonormal Vectors1.14. Orthogonal Projections1.15. Multi-Resolution Analysis Subspaces1.16. Complete and Orthonormal Bases in L2 (R)1.17. The Dirac Notation1.18. The Fourier Transform1.19. The Fourier Series Expansion1.20. The Discrete Time Fourier Transform1.21. The Discrete Fourier Transform1.22. Band-Limited Functions and the Sampling Theorem1.23. The Basis Operator in L2(R)1.24. Biorthogonal Bases and Representations in L2 (R)1.25. Frames in a Finite Dimensional Vector Space1.26. Frames in L2 (R)1.27. Dual Frame Construction Algorithm1.28. Exercises2. Linear Time-Invariant Systems2.1. Introduction2.2. Convolution in Continuous Time2.3. Convolution in Discrete Time2.4. Convolution of Finite Length Sequences2.5. Linear Time-Invariant Systems and the Z Transform2.6. Spectral Factorization for Finite Length Sequences2.7. Perfect Reconstruction Quadrature Mirror Filters2.8. Exercises3. Time, Frequency, and Scale Localizing Transforms3.1. Introduction3.2. The Windowed Fourier Transform3.3. The Windowed Fourier Transform Inverse3.4. The Range Space of the Windowed Fourier Transform3.5. The Discretized Windowed Fourier Transform3.6. Time-Frequency Resolution of theWindowed Fourier Transform3.7. The Continuous Wavelet Transform3.8. The Continuous Wavelet Transform Inverse3.9. The Range Space of the Continuous Wavelet Transform3.10. The Morlet, the Mexican Hat, and the Haar Wavelets3.11. Discretizing the Continuous Wavelet Transform3.12. Algorithm A' Trous3.13. The Morlet Scalogram3.14. Exercises4. The Haar and Shannon Wavelets4.1. Introduction4.2. Haar Multi-Resolution Analysis Subspaces4.3. Summary and Generalization of Results4.4. The Spectra of the Haar Filter Coefficients4.5. Half-Band Finite Impulse Response Filters4.6. The Shannon Scaling Function4.7. The Spectrum of the Shannon Filter Coefficients4.8. Meyer's Wavelet4.9. Exercises5. General Properties of Scaling and Wavelet Functions5.1. Introduction5.2. Multi-Resolution Analysis Spaces5.3. The Inverse Relations5.4. The Shift-Invariant Discrete Wavelet Transform5.5. Time Domain Properties5.6. Examples of Finite Length Filter Coefficients5.7. Frequency Domain Relations5.8. Orthogonalization of a Basis Set: b1 Spline Wavelet5.9. The Cascade Algorithm5.10. Biorthogonal Wavelets5.11. Multi-Resolution Analysis Using Biorthogonal Wavelets5.12. Exercises6. Discrete Wavelet Transform of Discrete Time Signals6.1. Introduction6.2. Discrete Time Data and Scaling Function Expansions6.3. Implementing the DWT for Even Length h0 Filters6.4. Denoising and Thresholding6.5. Biorthogonal Wavelets of Compact Support6.6. The Lazy Filters6.7. Exercises7. Wavelet Regularity and Daubechies Solutions7.1. Introduction7.2. Zero Moments of the Mother Wavelet7.3. The Form of H0(z) and the Decay Rate of F(?)7.4. Daubechies Orthogonal Wavelets of Compact Support7.5. Wavelet and Scaling Function Vanishing Moments7.6. Biorthogonal Wavelets of Compact Support7.7. Biorthogonal Spline Wavelets7.8. The Lifting Scheme7.9. Exercises8. Orthogonal Wavelet Packets8.1. Introduction8.2. Review of the Orthogonal Wavelet Transform8.3. Packet Functions for Orthonormal Wavelets8.4. Discrete Orthogonal Packet Transform of Finite Length Sequences8.5. The Best Basis Algorithm8.6. Exercises9. Wavelet Transform in Two Dimensions9.1. Introduction9.2. The Forward Transform9.3. The Inverse Transform9.4. Implementing the Two-Dimensional Wavelet Transform9.5. Application to Image Compression9.6. Image Fusion9.8. ExercisesBibliographyIndex

About the Author

Amir-Homayoon Najmi completed the Mathematical Tripos at Cambridge University and obtained his D.Phil. at Oxford University. He is with the Johns Hopkins University's Applied Physics Laboratory and is a faculty member of the Whiting School of Engineering CE programs in applied physics and electrical engineering.


A complete, concise and clear exposition of the more traditional tools related to linear filtering. -- Davide Barbieri * Mathematical Reviews *
Since their emergence in the last eighties and early nineties of the twentieth century, wavelets and other multi-scale transforms have become powerful signal and image processing tools. Najmi's book provides physicists and engineers with a clear and concise introduction to this fascinating field. -- Ignace Loris * American Journal of Physics *

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