## DW Home Page
| Discovering Wavelets, the book
## maintained by Edward Aboufadel and Steven Schlicker |

You can order the book from an online bookstore.

Here is a review of the book from
*Zentralblatt Math*. There are a few reviews at Amazon.com. Here's a not-so-kind review from
Ian Kaplan.

**Here is the preface from the book, along with the Table of Contents.**

For the past several years, reflecting the excitement and creativity surrounding the subject of wavelets, articles about wavelets have appeared in professional publications, on the World Wide Web, and in mainstream magazines and newspapers. Much of this enthusiasm for wavelets comes from known and from potential applications. For example, wavelets have found use in image processing, in the restoration of recordings, and in seismology.

Many books are available on wavelets, but most are written at such a level that only research mathematicians can read them. The purpose of this book is to make wavelets accessible to anyone with a background in basic linear algebra (for example, graduate and undergraduate students), and to serve as an introduction for the nonspecialist. The level of the applications and the format of this book make it an excellent textbook for an introductory course on wavelets or as a supplement to a first or second course in linear algebra or numerical analysis. Potential readers would be intrigued by the discussion of the Wavelet/Scalar Quantization Standard, currently used by the Federal Bureau of Investigation to compress, transmit, and store images of fingerprints. In Britain, Scotland Yard also uses wavelets for the same purpose.

The projects that are contained within this book allow real applications to be incorporated into the mathematics curriculum. This fits well with the current trend of infusing mathematics courses with applications; an approach which is summarized well by Avner Friedman and John Lavery in their statement about an industrial mathematics program:

"[this approach provides] students an immense opportunity for greater and deeper contributions in all areas of the natural and social sciences, engineering and technology."The practice of either basing a mathematics course on applications or infusing a course with applications has been seen primarily in calculus courses. However, this enthusiasm for real applications has not been as obvious in our upper-level mathematics courses, such as linear algebra, abstract algebra, or number theory, which are taken primarily by mathematics majors. Many of these majors intend to be teachers, and applications such as wavelets can provide breadth to the curriculum. As stated in a recent report of the Mathematical Association of America:

"it is vitally important that [prospective math teachers'] undergraduate experience provide a broad view of the discipline, [since] for the many students who may never make professional use of mathematics, depth through breadth offers a strong base for appreciating the true power and scope of the mathematical sciences."The major benefit of this book is that it presents basic and advanced concepts of wavelets in a way that is accessible to anyone with only a basic knowledge of linear algebra. The discussion of wavelets begins with an idea of Gil Strang to represent the Haar wavelets as vectors in R

Chapter 2 builds on the ideas presented in chapter 1, developing more of the theory of wavelets along with function spaces. Readers get a better sense of how one might deduce the ideas presented in chapter 1. To fully understand the material in this chapter, a reader needs more sophisticated mathematics, such as inner product spaces and projections. The necessary background material from linear algebra that a reader needs to know to fully understand the discussion of wavelets in this book is contained in appendix A.

Chapter 3 features more advanced topics such as filters,
multiresolution analysis, Daubechies wavelets, and further applications.
These topics are all introduced by comparison to the material developed
with the Haar wavelets in chapters 1 and 2. These topics would be of interest
to anyone who desires to read some of the more technical books or papers
on wavelets, or to anyone seeking a starting point for research projects.
There are some new concepts introduced in this chapter (e.g. density, fixed-point
algorithms, and L^{2} spaces). They are discussed in enough detail
to allow the reader to understand these concepts and how they relate to
wavelets, but not in so much detail that these ancillary topics distract
from the major topic of wavelets. For example, the Fourier transform
does not appear until the final section of chapter 3.

Chapter 4 contains projects that could be used in linear
algebra courses. Along with these projects, some of the problems
introduce advanced topics that could be used as starting points for research
by undergraduates. There are also appendices that review linear algebra
topics and present *Maple* commands that are useful for some of the
problems.

These notes originated in courses (Linear Algebra II) that the authors taught during summers 1996 and 1997 at Grand Valley State University in Allendale, Michigan. Students in each of these courses had previously completed Linear Algebra I, where they learned to solve systems of linear equations, were introduced to matrix and vector operations, and encountered for the first time the fundamental ideas of spanning sets, linear independence and dependence, bases, and dimension. The topics for the second semester course were grouped around two major themes: linear transformations and orthogonal projections. In the 1996 course, about one of every five class meetings was set aside to learn about wavelets, and the material in these notes was timed to coincide with the flow of topics in the rest of the course. (For example, using orthogonal projections to approximate functions with wavelets was done as we studied orthogonal projections in inner product spaces.) Along with the information presented in these notes, various articles were distributed to students as required readings.

In the 1997 course, groups of students submitted written reports
based on a subset of the problems in chapters 1 and 2. In addition, each
group created a gray-scale image in a 16-by-16 grid of pixels using a program,
*Pixel Images*, written by Schlicker. Each
group processed, compressed, and decompressed their image using Haar wavelets
and entropy coding. A modified version of this activity is included in
chapter 4.

While we initially introduced wavelets into the second semester linear algebra course, many of the tools and concepts can fit equally well in a first semester linear algebra course. In the last two years we have used various approaches in both semesters of linear algebra to expose our students to this valuable and exciting area of mathematics. In all of these courses, the students have expressed that they appreciated seeing a connection between what they were learning in college and what they saw happening in the world.

* A note on how to use this book.* To learn mathematics
it is important to become conversant with the terminology and to actually
work some problems. Throughout this book, key terms and phrases are highlighted
in italics. For pedagogical purposes, we have included some terminology
that is not standard in the literature: the phrase *daughter wavelets*
in chapter 1 is used to describe the functions that are obtained from dilations
and translations of the mother wavelet, *son wavelets* in chapter
2 describes the functions that are obtained from dilations and translations
of the father wavelet or scaling function, and *image box* in chapter
2 refers to any figure that contains projections and residuals of an original
image after processing with wavelets. We feel these terms provide appropriately
descriptive labels and use them without hesitation.

The problems posed in this text are distributed throughout the
reading rather than at the end of each chapter. This is important
because completing the problems is vital to learning about wavelets.
In fact, it is necessary to solve those problems marked with **bold**
numbers to completely understand the text. Also, a computer algebra
system is necessary to complete some problems, and hints and an appendix
are provided for those readers who have access to *Maple*. Answers
to selected problems are provided in an appendix. *Pixel
Images* and *Maple* worksheets are available at our web site:

Edward Aboufadel

Steven Schlicker

*Allendale, Michigan*
*July, 1999*

*Discovering Wavelets*
*Table of Contents***
**

Chapter 1 Wavelets, Fingerprints, and Image Processing

section 1.1 Problems of the Digital Age

section 1.2 Digitizing Fingerprint Images

section 1.3 Signals

section 1.4 The Haar Wavelet Family

section 1.5 Processing Signals

section 1.6 Thresholding and Compression of Data

section 1.7 The FBI Wavelet/Scalar Quantization Standard

Chapter 2 Wavelets and Orthogonal Decompositions

section 2.1 A Lego World

section 2.2 The Wavelet Sons

section 2.3 Sibling Rivalry: Two Bases for *V*_{n}

section 2.4 Averaging and Differencing

section 2.5 Projecting Functions Onto Wavelet Spaces

section 2.6 Function Processing and Image Boxes

section 2.7 A Summary of Two Approaches to Wavelets

Chapter 3 Multiresolutions, Cascades, and Filters

section 3.1 Extending the Haar Wavelets to the Real Line

section 3.2 Other Elementary Wavelet Families

section 3.3 Multiresolution Analysis

section 3.4 The Haar Scaling Function Rediscovered

section 3.5 Relationships Between the Mother and Father
Wavelets

section 3.6 Daubechies Wavelets

section 3.7 High and Low Pass Filters

section 3.8 More Problems of the Digital Age: Compact
Discs

Chapter 4 Sample Projects

section 4.1 Introduction: Overview of Projects

section 4.2 Linear Algebra Project: Image Processing and
Compression

section 4.3 A Wavelet-Based Search Engine

section 4.4 B-Splines

section 4.5 Processing with the *D*_{4} Wavelets

section 4.6 Daubechies Wavelets with Six Refinement Coefficients

Appendix A Vector Spaces and Inner Product Spaces

section A.1 Vector Spaces

section A.2 Subspaces

section A.3 Inner Product Spaces

section A.4 The Orthogonal Decomposition Theorem

Appendix B Maple Routines

section B.1 Matrix Generator

section B.2 Processing Sampled Data

section B.3 Projections onto Wavelet Spaces

section B.4 The Cascade Algorithm

section B.5 Processing an Image from *Pixel Images*

Appendix C Answers to Selected Problems

Appendix D Glossary of Symbols

References

Index