PREFACE
PREFACE TO THE SECOND EDITION
NOTATION
I INTRODUCTION TO A TRANSIENT WORLD
1.1 Fourier Kingdom
1.2 Time-Frequency Wedding
1.2.1 Windowed Fourier Transform
1.2.2 Wavelet Transform
1.3 Bases of Time-Frequency Atoms
1.3.1 Wavelet Bases and Filter Banks
1.3.2 Tilings of Wavelet Packet and Local Cosine Bases
1.4 Bases for What?
1.4.1 Approximation
1.4.2 Estimation
1.4.3 Compression
1.5 Travel Guide
1.5.1 Reproducible Computational Science
1.5.2 Road Map
II FOURIER KINGDOM
2.1 Linear Time-Invariant Filtering
2.1.1 Impulse Response
2.1.2 Transfer Functions
2.2 Fourier Integrals
2.2.1 Fourier Transform in L1(R)
2.2.2 Fourier Transform in L2(R)
2.2.3 Examples
2.3 Properties
2.3.1 Regularity and Decay
2.3.2 Uncertainty Principle
2.3.3 Total Variation
2.4 Two-Dimensional Fourier Transform
2.5 Problems
III DISCRETE REVOLUTION
3.1 Sampling Analog Signals
3.1.1 Whittaker Sampling Theorem
3.1.2 Aliasing
3.1.3 General Sampling Theorems
3.2 Discrete Time-Invariant Filters
3.2.1 Impulse Response and Transfer Function
3.2.2 Fourier Series
3.3 Finite Signals
3.3.1 Circular Convolutions
3.3.2 Discrete Fourier Transform
3.3.3 Fast Fourier Transform
3.3.4 Fast Convolutions
3.4 Discrete Image Processing
3.5 Problems
IV TIME MEETS FREQUENCY
4.1 Time-Frequency Atoms
4.2 Windowed Fourier Transform
4.2.1 Completeness and Stability
4.2.2 Choice of Window 2
4.2.3 Discrete Windowed Fourier Transform
4.3 Wavelet Transforms
4.3.1 Real Wavelets
4.3.2 Analytic Wavelets
4.3.3 Discrete Wavelets
4.4 Instantaneous Frequency
4.5 Quadratic Tune-Frequency Energy
4.6 Problems
V FRAMES
5.1 Frame Theory
5.2 Windowed Fourier Frames
5.3 Wavelet Frames
5.4 Translation Invariance
5.5 Dyadic Wavelet Transform
5.6 Problems
VI WAVELET ZOOM
6.1 Lipschitz Regularity
6.2 Wavelet Transform Modulus Maxima
6.3 Multiscale Edge Detection
6.4 Multifractals
6.5 Problems
VII WAVELET BASES
7.1 Orthogonal Wavelet Bases
7.2 Classes of Wavelet Bases
7.3 Wavelets and Filter Banks
7.4 Biorthogonal Wavelet Bases
7.5 Wavelet Bases on an Interval
IX AN APPROXIMATION TOUR
9.1 Linear Approximations
9.2 Non-Linear Approximations
9.3 Image Approximations with Wavelets
9.4 Adaptive Basis Selection
ESTIMATIONS ARE APPROXIMATIONS
XI TRANSFORM CODING
Appendix A MATHEMATICAL COMPLEMENTS
Appendix B SOFTWARE TOOLBOXES
BIBLIOGRAPHY
INDEX