Ripples in Mathematics
The Discrete Wavelet Transform

1. Introduction

1.1 Prerequisites
1.2 Guide to the Book
1.3 Background Information

2. A First Example

2.1 The Example
2.2 Generalizations
Exercises

3. The Discrete Wavelet Transform via Lifting

3.1 The First Example Again
3.2 Denition of Lifting
3.3 A Second Example
3.4 Lifting in General
3.5 DWT in General
3.6 Further Examples
Exercises

4. Analysis of Synthetic Signals

4.1 The Haar Transform
4.2 The CDF(2,2) Transform
Exercises

5. Interpretation

5.1 The First Example
5.2 Further Results on the Haar Transform
5.3 Interpretation of General DWT
Exercises

6. Two Dimensional Transforms

6.1 One Scale DWT in Two Dimensions
6.2 Interpretation and Examples
6.3 A 2D Transform Based on Lifting
Exercises

7. Lifting and Filters I

7.1 Fourier Series and the z -Transform
7.2 Lifting in the z -Transform Representation
7.3 Two Channel Filter Banks
7.4 Orthonormal and Biorthogonal Bases
7.5 Two Channel Filter Banks in the Time Domain
7.6 Summary of Results on Lifting and Filters
7.7 PropertiesofOrthogonalFilters
7.8 Some Examples
Exercises

8. Wavelet Packets

8.1 From Wavelets to Wavelet Packets
8.2 Choice of Basis
8.3 Cost Functions
Exercises

9. The Time-Frequency Plane

9.1 Sampling and Frequency Contents
9.2 Denition of the Time-Frequency Plane
9.3 Wavelet Packets and Frequency Contents
9.4 More about Time-Frequency Planes
9.5 More Fourier Analysis. The Spectrogram
Exercises

10. Finite Signals

10.1 The Extent of the Boundary Problem
10.2 DWT in Matrix Form
10.3 Gram-Schmidt Boundary Filters
10.4 Periodization
10.5 Moment Preserving Boundary Filters
Exercises

11. Implementation

11.1 Introduction to Software
11.2 Implementing the Haar Transform Through Lifting
11.3 Implementing the DWT Through Lifting
11.4 The Real Time Method
11.5 Filter Bank Implementation
11.6 Construction of Boundary Filters
11.7 Wavelet Packet Decomposition
11.8 Wavelet PacketBases
11.9 Cost Functions
Exercises

12. Lifting and Filters II

12.1 The Three Basic Representations
12.2 From Matrix to Equation Form
12.3 From Equation to Filter Form
12.4 From Filters to Lifting Steps
12.5 Factoring Daubechies 4 into Lifting Steps
12.6 Factorizing Coi et 12 into Lifting Steps
Exercises

13. Wavelets in Matlab

13.1 Multiresolution Analysis
13.2 Frequency Properties of the Wavelet Transform
13.3 Wavelet Packets Used for Denoising
13.4 Best Basis Algorithm
13.5 Some Commands in Uvi Wave
Exercises

14. Applications and Outlook

14.1 Applications
14.2 Outlook
14.3 Some WebSites

References

Index

Ripples in Mathematics - The Discrete Wavelet Transform. Springer Verlag Berlin Heidelberg 2001
Web master: Anders la Cour-Harbo, Aalborg University, Denmark.
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