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ABOUT THE BOOK

This treatment is vastly different from traditional math-oriented wavelet books in that we use examples, figures, and computer demonstrations to show how to understand and work with wavelets. This is a comprehensive, in-depth. up-to-date treatment of the subject, but from an intuitive, conceptual point of view.

We do look at a few key equations from the traditional literature but only AFTER the concepts are demonstrated and understood. If desired, further study from scholarly texts and papers is then made much easier and more palatable when you already understand fundamentals of wavelets and how they relate to the real world.

Preface
Understanding & Harnessing Wavelet “Elephants”
How this Book Differs from Other Wavelet Texts
How this Book is Laid Out—Study Suggestions

Acknowledgments

CHAPTER 1 - Preview of Wavelets, Wavelet Filters, and Wavelet Transforms
1.1 What is a Wavelet?
1.2 What is a Wavelet Filter and how is it different from a Wavelet?
1.3 The value of Transforms and Examples of Everyday Use
1.4 Short-Time Transforms, Sheet Music, and a first look at Wavelet Transforms
1.5 Example of the Fast Fourier Transform (FFT) with an Embedded Pulse Signal
1.6 Examples using the Continuous Wavelet Transform
1.7 A First Glance at the Undecimated Discrete Wavelet Transform (UDWT)
1.8 A First Glance at the conventional Discrete Wavelet Transform (DWT)
1.9 Examples of use of the conventional DWT
1.10 Summary
CHAPTER 2 - The Continuous Wavelet Transform (CWT) Step-by-Step
2.1 Simple Scenario: Comparing Exam Scores using the Haar Wavelet
2.2 Above Comparison Process seen as simple Correlation or Convolution
2.3 CWT Display of the Exam Scores using the Haar Wavelet Filter
2.4 Summary
CHAPTER 3 - The Undecimated Discrete Wavelet Transform (UDWT) Step-by-Step
3.1 Single-Level Undecimated Discrete Wavelet Transform (UDWT) of Exam Data
3.2 Frequency Allocation of a Single-Level UDWT
3.3 Multi-Level Undecimated Discrete Wavelet Transform (UDWT)
3.4 Frequency Allocation of a Multiple-Level UDWT
3.5 The Haar UDWT as a Moving Averager
3.6 Summary
CHAPTER 4 - The Conventional (Decimated) DWT Step-by-Step
4.1 Single-Level (Decimated) Discrete Wavelet Transform (DWT) of Exam Data
4.2 Additional Example of Perfect Reconstruction in a Single-Level DWT
4.3 Compression and Denoising Example using the Single-Level DWT
4.4 Multi-Level Conventional (Decimated) DWT of Exam Data using Haar Filters
4.5 Frequency Allocation in a (Conventional, Decimated) DWT
4.6 Final Approximations and Details and how to read the DWT Display
4.7 Denoising using a Multi-Level DWT
4.8 Summary
CHAPTER 5 - Obtaining Discrete Wavelet Filters from “Crude” Wavelet Equations
5.1 Review of Familiar DSP Truncated Sinc Function
5.2 Adding More Points at the Ends for Better Filter Performance
5.3 Adding More Points by Interpolation for Lower Cutoff Frequency
5.4 Multi-Point Stretched Filters (“Crude Wavelets”) from Explicit Equations
5.5 Mexican Hat Wavelet Filter as an Example of a Stretched Crude Filter
5.6 Morlet Wavelet as another example of Stretched Crude Filters
5.7 Bandpass Characteristics of the Mexican Hat and Morlet Wavelet Filters
5.8 Summary
CHAPTER 6 - Obtaining Variable Length Filters from Basic Fixed Length Filters
6.1 Review of Conventional Interpolation Techniques from DSP
6.2 Interpolating the Basic “Mother” Wavelet by Upsampling and Lowpass Filtering
6.3 Frequency Characteristics of the Basic and Stretched Haar Filters
6.4 Perfect Overlay of Filter Points on the “Continuous” Wavelet Estimation
6.5 Frequency Characteristics of some of the Basic Filters
6.6 Summary
CHAPTER 7 - Comparison of the Major Types of Wavelet Transforms
7.1 Advantages and Disadvantages of the Continuous Wavelet Transform
7.2 Stretching the Wavelet—The Undecimated Discrete Wavelet Transform
7.3 Shrinking the Signal—The Conventional Discrete Wavelet Transform
7.4 Relating the Conventional DWT to the Continuous Wavelet Transform
7.5 Decomposing All the Frequencies—The Wavelet Packet Transform
7.6 Summary
CHAPTER 8 - PRQMF and Halfband Filters and How they are Related
8.1 Perfect Reconstruction Quadrature Mirror Filters and their Inter-Relationships
8.2 Perfect Reconstruction Begins with the Halfband Filters
8.3 Properties of the Halfband Filters
8.4 “Reverse Engineering” Perfect Reconstruction to Produce the Basic Filters
8.5 Orthogonal Vectors, Sinusoids, and Wavelets
8.6 Biorthogonal Filters—Another Way to Factor the Halfband Filters
8.7 Summary
CHAPTER 9 - Highlighting Additional Properties by using “Fake” Wavelets
9.1 Matching the Wavelet to the Signal and the Concept of Regularity
9.2 Customized Wavelets, Best Basis, and the “Sport of Basis Hunting”
9.3 Vanishing Moments and another Fake Wavelet
9.4 Examples of Use of Vanishing Moments
9.5 Finding the “Magic Numbers” of Basic Db4 Filters using Wavelet Properties
9.6 Summary
CHAPTER 10 - Specific Properties and Applications of Wavelet Families
10.1 (Real) Crude Wavelets
- MEXICAN HAT WAVELET
- MORLET WAVELET
- GAUSSIAN WAVELETS
- MEYER WAVELETS
10.2 Complex Crude Wavelets
- SHANNON (“SINC”) WAVELET
- COMPLEX FREQUENCY B-SPLINE WAVELETS
- COMPLEX MORLET WAVELET
- COMPLEX GAUSSIAN WAVELETS
10.3 Orthogonal Wavelets
- HAAR WAVELETS
- DAUBECHIES WAVELETS
- SYMLETS
- COIFLETS
- DISCRETE MEYER WAVELETS
10.4 Biorthogonal and Reverse Biorthogonal Wavelets
- BIORTHOGONAL WAVELETS
- REVERSE BIORTHOGONAL WAVELETS
10.5 Summary and Table of Wavelets and their Properties
- TABLE 10.5–1 - ATTRIBUTES OF THE VARIOUS WAVELETS (FILTERS)
CHAPTER 11 - Case Studies of Wavelet Applications
11.1 White Noise in a Chirp Signal
11.2 Binary Signal Buried in Chirp Noise
11.3 Binary Signal with White Noise
11.4 Image Compression/De-noising
11.5 Improved Performance using the UDWT
11.6 Summary
CHAPTER 12 - Alias Cancellation in the Conventional (Decimated) DWT
12.1 DWT Alias Cancellation Demonstrated in the Time Domain.
12.2 DWT Alias Cancellation Demonstrated in the Frequency Domain.
12.3 Relating the Above Concepts to Equations Found in the Traditional Literature
12.4 Summary
CHAPTER 13 - Relating Key Equations to Conceptual Understanding
13.1 Building the Scaling Function from The “Dilation Equation”
13.2 Building the Scaling Function Using Upsampling and Simple Convolution
13.3 Building the Wavelet Function from the Dilation Equation
13.4 Building the Wavelet Function Using Upsampling and Simple Convolution
13.5 “Forward DWT”, “Inverse DWT” and Other Terms from Wavelet Literature
13.6 Summary
Postscript
Appendix A: Relating Wavelet Transforms to Fourier Transforms
A.1 Example of a Pathological Case Using the Fast Fourier Transform
A.2 FFT and STFT Results Shown In Continuous Wavelet Transform Format
A.3 The Wavelet Terms “Approximation” and “Details” Shown in FFT Format.
A.4 The FFT Presented as a Sinusoid Correlation (Similar to Wavelet Correlation)
A.5 The Ordinary Acoustic Piano: An Audio Fourier Transform
Appendix B: Heisenberg Boxes and the Heisenberg Uncertainty Principle
B.1 Natural Order of Time and Frequency
B.2 Heisenberg Boxes (Cells) and the Uncertainty Principle
B.3 Short Time Fourier Transforms are Constrained to Fixed Heisenberg Boxes
Appendix C: Reprint of Article “Wavelets: Beyond Comparison”
The Discrete Fourier Transform/Fast Fourier Transform (DFT/FFT)
The Continuous Wavelet Transform (CWT)
Discrete Wavelet Transforms Overview
Undecimated or “Redundant” Discrete Wavelet Transforms (UDWT/RDWT)
Conventional (Decimated) Discrete Conventional Transforms (DWT)
Appendix D: Further Resources for the Study of Wavelets
D.1 Wavelet Books
D.2 Wavelet Articles
D.3 Wavelet Websites

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