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.