Discrete Wavelet Transformations: An Elementary Approach with Applications, Second Edition
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More About This Title Discrete Wavelet Transformations: An Elementary Approach with Applications, Second Edition

English

Updated and Expanded Textbook Offers Accessible and Applications-First Introduction to Wavelet Theory for Students and Professionals

The new edition of Discrete Wavelet Transformations continues to guide readers through the abstract concepts of wavelet theory by using Dr. Van Fleet’s highly practical, application-based approach, which reflects how mathematicians construct solutions to challenges outside the classroom. By introducing the Haar, orthogonal, and biorthogonal filters without the use of Fourier series, Van Fleet allows his audience to connect concepts directly to real-world applications at an earlier point than other publications in the field.

Leveraging extensive graphical displays, this self-contained volume integrates concepts from calculus and linear algebra into the constructions of wavelet transformations and their applications, including data compression, edge detection in images and denoising of signals. Conceptual understanding is reinforced with over 500 detailed exercises and 24 computer labs. 

The second edition discusses new applications including image segmentation, pansharpening, and the FBI fingerprint compression specification. Other notable features include:

  • Two new chapters covering wavelet packets and the lifting method
  • A reorganization of the presentation so that basic filters can be constructed without the use of Fourier techniques
  • A new comprehensive chapter that explains filter derivation using Fourier techniques
  • Over 120 examples of which 91 are “live examples,” which allow the reader to quickly reproduce these examples in Mathematica or MATLAB and deepen conceptual mastery
  • An overview of digital image basics, equipping readers with the tools they need to understand the image processing applications presented
  • A complete rewrite of the DiscreteWavelets package called WaveletWare for use with Mathematica and MATLAB
  • A website, www.stthomas.edu/wavelets, featuring material containing the WaveletWare package, live examples, and computer labs in addition to companion material for teaching a course using the book 

Comprehensive and grounded, this book and its online components provide an excellent foundation for developing undergraduate courses as well as a valuable resource for mathematicians, signal process engineers, and other professionals seeking to understand the practical applications of discrete wavelet transformations in solving real-world challenges.

English

PATRICK J. VAN FLEET is Professor and Chair of the Department of Mathematics at the University of St. Thomas in St. Paul, Minnesota. He has authored several journal articles on (multi)wavelets and conducted sponsored workshops for developing and teaching an applications-first course on wavelets. He received his PhD in Mathematics from Southern Illinois University-Carbondale in 1991.

English

Preface to the First Edition

Preface

Acknowledgements

Chapter 1: Introduction: Why Wavelets?

Chapter 2: Vectors and Matrices

2.1 Vectors, Inner Products, and Norms

Problems

2.2 Basic Matrix Theory

Problems

2.3 Block Matrix Arithmetic

Problems

2.4 Convolution and Filters

Problems

Chapter 3: An Introduction to Digital Images

3.1 The Basics of Grayscale Digital Images

Problems

Computer Lab

3.2 Color Images and Color Spaces

Problems

Computer Lab

3.3 Huffman Coding

Problems

Computer Lab

3.4 Qualitative and Quantitative Measures

Problems

Computer Labs

Chapter 4: The Haar Wavelet Transformation

4.1 Constructing the Haar Wavelet Transformation

Problems

Computer Lab

4.2 Iterating the Process

Problems

Computer Lab

4.3 The Two–Dimensional Haar Wavelet Transformation

Problems

Computer Lab

4.4 Applications: Image Compression and Edge Detection

Problems

Computer Labs

Chapter 5: Daubechies Wavelet Transformations

5.1 Daubechies Filter of Length Four

Problems

Computer Lab

5.2 Daubechies Filter of Length Six

Problems

Computer Lab

5.3 Daubechies Filters of Even Length

Problems

Computer Lab

Chapter 6: Wavelet Shrinkage: An Application to Denoising

6.1 An Overview of Wavelet Shrinkage

Problems

Computer Lab

6.2 VisuShrink

Problems

Computer Lab

6.3 SureShrink

Problems

Computer Labs

Chapter 7: Biorthogonal Wavelet Transformations

7.1 The (5; 3) Biorthogonal Spline Filter Pair

Problems

Computer Lab

7.2 The (8; 4) Biorthogonal Spline Filter Pair

Problems

Computer Lab

7.3 Symmetry and Boundary Effects

Problems

Computer Lab

7.4 Image Compression and Image Pansharpening

Computer Lab

Chapter 8: Complex Numbers and Fourier Series

8.1 The Complex Plane and Arithmetic

Problems

8.2 Fourier Series

Problems

8.3 Filters and Convolution in the Fourier Domain

Problems

Chapter 9: Filter Construction in the Fourier Domain

9.1 Filter Construction in the Fourier Domain

Problems

9.2 Daubechies Filters

Problems

9.3 Coiflet Filters

Problems

Computer Lab

9.4 Biorthogonal Spline Filter Pairs

Problems

Computer Lab

9.5 The Cohen–Daubechies–Feauveau 9/7 Filter

Problems

Computer Lab

Chapter 10: Wavelet Packets

10.1 The Wavelet Packet Transform

Problems

10.2 Cost Functions and the Best Basis Algorithm

Problems

10.3 The FBI Fingerprint Compression Specification

Computer Lab

Chapter 11: Lifting

11.1 The LeGall Wavelet Transform

Problems

Computer Lab

11.2 Z–Transforms and Laurent Polynomials

Problems

11.3 A General Construction of the Lifting Method

Problems

11.4 The Lifting Method – Examples

Problems

Computer Lab

Chapter 12: The JPEG2000 Image Compression Standard

12.1 An Overview of JPEG

Problems

12.2 The Basic JPEG2000 Algorithm

Problems

12.3 Examples

Appendix A: Basic Statistics

A.1Descriptive Statistics

Problems

A.2 Sample Spaces, Probability, and Random Variables

Problems

A.3 Continuous Distributions

Problems

A.4 Expectation

Problems

A.5 Two Special Distributions

Problems

References

Index

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