Ripples in Mathematics

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 Definition 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 Properties of Orthogonal Filters.- 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 Definition 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 Packet Bases.- 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 Coiflet 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 Web Sites.- References.

From the reviews:

"This book is a very well-written introduction to discrete wavelet transforms, very convenient for students in electrical engineering, computer science, and applied mathematics. It is based on the lifting approach to discrete wavelet transforms ... . MATLAB is used as the computational environment for examples and implementations of discrete wavelet transforms." (Manfred Tasche, Zentralblatt MATH, Vol. 989 (14), 2002)

"This is an algorithm based, completely elementary introduction to the discrete wavelet transform (DWT) and wavelet packet transform, easy to read and easy to understand, well suited for an introductory course on wavelets for undergraduate students of applied sciences or mathematics. ... Implementations and examples using basic Matlab (TM) as well as the public domain ubi-wave wavelet toolbox help to further a deeper understanding of the algorithms." (C. Cenker, Internationale Mathematische Nachrichten, Vol. 56 (189), 2002)

"This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. ... MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry." (ETDE Energy Database, December, 2001)