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| List of Figures | |
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| List of Acronyms | |
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| Preface | |
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| Acknowledgments | |
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| Analysis in Vector and Function Spaces | |
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| Introduction | |
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| The Lebesgue Integral | |
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| Discrete Time Signals | |
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| Vector Spaces | |
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| Linear Independence | |
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| Bases and Basis Vectors | |
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| Normed Vector Spaces | |
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| Inner Product | |
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| Banach and Hilbert Spaces | |
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| Linear Operators, Operator Norm, the Adjoint Operator | |
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| Reproducing Kernel Hilbert Space | |
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| The Dirac Delta Distribution | |
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| Orthonormal Vectors | |
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| Orthogonal Projections | |
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| Multi-Resolution Analysis Subspaces | |
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| Complete and Orthonormal Bases in L<sub>2</sub> (R) | |
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| The Dirac Notation | |
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| The Fourier Transform | |
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| The Fourier Series Expansion | |
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| The Discrete Time Fourier Transform | |
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| The Discrete Fourier Transform | |
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| Band-Limited Functions and the Sampling Theorem | |
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| The Basis Operator in L<sub>2</sub>(R) | |
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| Biorthogonal Bases and Representations in L<sub>2</sub> (R) | |
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| Frames in a Finite Dimensional Vector Space | |
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| Frames in L<sub>2</sub> (R) | |
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| Dual Frame Construction Algorithm | |
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| Exercises | |
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| Linear Time-Invariant Systems | |
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| Introduction | |
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| Convolution in Continuous Time | |
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| Convolution in Discrete Time | |
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| Convolution of Finite Length Sequences | |
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| Linear Time-Invariant Systems and the Z Transform | |
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| Spectral Factorization for Finite Length Sequences | |
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| Perfect Reconstruction Quadrature Mirror Filters | |
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| Exercises | |
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| Time, Frequency, and Scale Localizing Transforms | |
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| Introduction | |
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| The Windowed Fourier Transform | |
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| The Windowed Fourier Transform Inverse | |
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| The Range Space of the Windowed Fourier Transform | |
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| The Discretized Windowed Fourier Transform | |
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| Time-Frequency Resolution of the Windowed Fourier Transform | |
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| The Continuous Wavelet Transform | |
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| The Continuous Wavelet Transform Inverse | |
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| The Range Space of the Continuous Wavelet Transform | |
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| The Morlet, the Mexican Hat, and the Haar Wavelets | |
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| Discretizing the Continuous Wavelet Transform | |
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| Algorithm A' Trous | |
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| The Morlet Scalogram | |
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| Exercises | |
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| The Haar and Shannon Wavelets | |
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| Introduction | |
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| Haar Multi-Resolution Analysis Subspaces | |
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| Summary and Generalization of Results | |
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| The Spectra of the Haar Filter Coefficients | |
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| Half-Band Finite Impulse Response Filters | |
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| The Shannon Scaling Function | |
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| The Spectrum of the Shannon Filter Coefficients | |
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| Meyer's Wavelet | |
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| Exercises | |
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| General Properties of Scaling and Wavelet Functions | |
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| Introduction | |
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| Multi-Resolution Analysis Spaces | |
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| The Inverse Relations | |
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| The Shift-Invariant Discrete Wavelet Transform | |
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| Time Domain Properties | |
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| Examples of Finite Length Filter Coefficients | |
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| Frequency Domain Relations | |
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| Orthogonalization of a Basis Set: b1 Spline Wavelet | |
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| The Cascade Algorithm | |
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| Biorthogonal Wavelets | |
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| Multi-Resolution Analysis Using Biorthogonal Wavelets | |
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| Exercises | |
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| Discrete Wavelet Transform of Discrete Time Signals | |
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| Introduction | |
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| Discrete Time Data and Scaling Function Expansions | |
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| Implementing the DWT for Even Length h0 Filters | |
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| Denoising and Thresholding | |
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| Biorthogonal Wavelets of Compact Support | |
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| The Lazy Filters | |
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| Exercises | |
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| Wavelet Regularity and Daubechies Solutions | |
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| Introduction | |
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| Zero Moments of the Mother Wavelet | |
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| The Form of H<sub>0</sub>(z) and the Decay Rate of �(�) | |
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| Daubechies Orthogonal Wavelets of Compact Support | |
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| Wavelet and Scaling Function Vanishing Moments | |
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| Biorthogonal Wavelets of Compact Support | |
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| Biorthogonal Spline Wavelets | |
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| The Lifting Scheme | |
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| Exercises | |
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| Orthogonal Wavelet Packets | |
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| Introduction | |
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| Review of the Orthogonal Wavelet Transform | |
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| Packet Functions for Orthonormal Wavelets | |
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| Discrete Orthogonal Packet Transform of Finite Length Se-quences | |
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| The Best Basis Algorithm | |
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| Exercises | |
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| Wavelet Transform in Two Dimensions | |
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| Introduction | |
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| The Forward Transform | |
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| The Inverse Transform | |
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| Implementing the Two-Dimensional Wavelet Transform | |
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| Application to Image Compression | |
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| Image Fusion | |
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| Wavelet Descendants | |
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| Exercises | |
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| Bibliography | |
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| Index | |