Short description This self-contained, elementary introduction to wavelet theory and its applications provides a comprehensive treatment of the topic using two real-life applications: the FBI Wavelet Scalar Quantization Standard and image segmentation. Over 400 exercises plus extensive details and examples help readers gain a full understanding of the material. Supplemented with a software package called DiscreteWavelets, which helps both students and professionals to visualize the applications addressed throughout the book.
From the contents Preface.
Acknowledgments.
1 The Complex Plane and the Space L²(R).
1.1 Complex Numbers and Basic Operations.
Problems.
1.2 The Space L²(R).
Problems.
1.3 Inner Products.
Problems.
1.4 Bases and Projections.
Problems.
2 Fourier Series and Fourier Transformations.
2.1 Euler's Formula and the Complex Exponential Function.
Problems.
2.2 Fourier Series.
Problems.
2.3 The Fourier Transform.
Problems.
2.4 Convolution and B-Splines.
Problems.
3 Haar Spaces.
3.1 The Haar Space V0.
Problems.
3.2 The General Haar Space Vj.
Problems.
3.3 The Haar Wavelet Space W0.
Problems.
3.4 The General Haar Wavelet Space Wj.
Problems.
3.5 Decomposition and Reconstruction.
Problems.
3.6 Summary.
4 The Discrete Haar Wavelet Transform and Applications.
4.1 The One-Dimensional Transformation.
Problems.
4.2 The Two-Dimensional Transformation.
Problems.
4.3 Edge Detection and Naive Image Compression.
5 Multiresolution Analysis.
5.1 Multiresolution Analysis.
Problems.
5.2 The View from the Transform Domain.
Problems.
5.3 Examples of Multiresolution Analyses.
Problems.
5.4 Summary.
6 Daubechies Scaling Functions and Wavelets.
6.1 Constructing the Daubechies Scaling Functions.
Problems.
6.2 The Cascade Algorithm.
Problems.
6.3 Orthogonal Translates, Coding and Projections.
Problems.
7 The Discrete Daubechies Transformation and Applications.
7.1 The Discrete Daubechies Wavelet Transform.
Problems.
7.2 Projections and Signal and Image Compression.
Problems.
7.3 Naive Image Segmentation.
Problems.
8 Biorthogonal Scaling Functions and Wavelets.
8.1 A Biorthogonal Example and Duality.
Problems.
8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces.
Problems.
8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair.
Problems.
8.4 Decomposition and Reconstruction.
Problems.
8.5 The Discrete Biorthogonal Wavelet Transformation.
Problems.
8.6 Riesz Basis Theory.
Problems.
9 Wavelet Packets.
9.1 Constructing Wavelet Packet Functions.
Problems.
9.2 Wavelet Packet Spaces.
Problems.
9.3 The Discrete Packet Transform and Best Basis Algorithm.
