The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition 2nd Edition

This book represents one of the wonderfull introducing courses for wavelets and wavelet transforms. Ideas that come from different parts of our material world are connected and visualized in mathematical models. These models show unexpected new relationships among phenomenons and deliver to us world in a new way.
Author deservs all compliments for his concise, charming and inspiring style of writing with clear and understandable explanations of complex and difficult mathemathics terminology.
It is recomended for those who are new in digital signal processing and for those experienced who look for new ideas and ideas burried in history of mathematics in all fields of human activities.
Gratulations to the author.
Dragan Matkovic, lecturer of the Polytechnic of Varazdin, Croatia

This volume bridges the gap between the necessarily mathematically-dense expositions characteristic of books on wavelets appropriate for graduate students and advanced undergraduates in mathematics, science, and engineering--and the need for a more intuitive description of the important concepts. In so doing the author serves a purpose surprisingly close to that of a professor teaching a course on wavelets. The success of this interpolation, however, depends very strongly on the background and taste of the reader in question. I don't think that I would have found this treatment very satisfying if I had read it as a Freshman in college who was just beginning calculus--there are far too many concepts addressed in this book for a mathematical novice to feel sufficiently comfortable. I suspect that most readers would need to have the equivalent of a course on Fourier transforms under their belt in order to for this book to achieve its purpose.

Complementing the intuitive and somewhat informal discussion of wavelet-related issues provided by this volume are a number of interesting and relevant historical vignettes. Among these pieces are pocket biographies of the physicists, engineers, and mathematicians who have contributed to the wavelet concept and its mathematical underpinnings, from Fourier's work in the days of the French Revolution, Napoleon, and the Bourbon restoration, to Ingrid Daubechies' contribution to this field in the late 1980's. To her credit, the author conducted many interviews of those who had pioneered the field of wavelets in the last quarter of the twentieth century, and includes many illuminating quotes from these important researchers.

I have four books in my personal library (in addition to Hubbard's)
that deal with wavelets: "Wavelet Analysis With Applications to
Image Processing," by L. Prasad and S. S. Iyengar, "Joint
Time-Frequency Analysis," by Shie Qian and Dapang Chen,"A
Friendly Guide to Wavelets," by Gerald Kaiser, and "Wavelets:
an Analysis Tool," by M. Holschneider. While these are good
"introductory" books for people already deeply familiar with
orthogonal bases and mathematics in general, I think they are
inadequate for someone wanting a truly fresh introduction to the
subject.
Hubbard's book, though, was just what I'd been looking for.
My wife bought it for me after dinner and a movie as we were browsing
the local bookstore in celebration of my 45th birthday. Hubbard wrote
her book with the idea in mind that it is possible to describe
accurately and in principle many mathematical concepts that are often
made incomprehensible, or nearly so, through technical jargon. The
technical jargon is necessary, of course, among professional
mathematicians, but it need not, and should not, get in the way of
conveying the basic ideas and concepts in an introductory text. As a
science writer, Hubbard has done a masterful job of doing just that.
This book gives me the intuitive, spatial understanding of wavelets
that I just could not find in the other books I listed above. It
helps form the basis for understanding the more detailed books, and it
also provides some interesting historical information.
The book is
divided into two parts. Part 1, called "The World According to
Wavelets," is essentially devoid of any mathematical formulas.
Instead of using mathematical symbols it uses imagery and verbal
explanation. This is likely to be somewhat frustrating for those who
have a mathematical background. Indeed, there were times when I found
myself trying to figure out which of several possibilities Hubbard was
talking about. Mostly, part one introduces the reader to the idea of
separating a signal into its Fourier components, and then it extends
this basic idea - that signals can be expressed in different
"languages" to the notion of the wavelet.
Sprinkled
throughout part 1 are references to part 2, which is titled
"Beyond Plain English." Unlike Part 1, Part 2 is full of
mathematical equations and terminology (though not at the same level
as the other books I mentioned above). The level of mathematics is
mostly limited to what you'd expect to find in an undergraduate class
in physics or mathematics.
Even with the mathematical detail,
Hubbard presents Part 2 with the same sensitivity toward the
explanation of new ideas as she uses in Part 1. The first chapter in
part 2 reviews the Fourier series and the Fourier transform. This
chapter is less than ten pages long, but it's one of the best short
summaries I've seen. It does not skimp on the mathematical details
but it's clear and understandable to a fault.
Chapter 2 talks about
the convergence of the Fourier series and has some nice (you've seen
them before, I suspect) illustrations showing how the Fourier series
of a train of square pulses converges. There is some interesting
explanation of the Gibb's effect, as well as an interesting section on
stability of the solar system. Hubbard does a nice job of explaining
how Fourier methods can be applied to studies of the stability of the
solar system, and how uncertainty arises from small divisors.
I
have another book in my personal library by E. Oran Brigham called the
"Fast Fourier Transform." This is another great book, with
very good background material (succinct) on the Fourier series and
transform. However, I found Brigham's explanation of the FFT harder
to follow than the one Hubbard gives in chapter 3 of Part 2. Granted,
Brigham's explanation goes into more detail (part of what makes it
harder to follow) but Hubbard, as she does throughout the book, does a
better job of illustrating the problem from the 50,000-foot
level.
Chapter 5 introduces the continuous wavelet transform in
integral form. Chapter 6 returns to ideas developed qualitatively in
Part 1 about orthogonal bases. Hubbard does a nice job of explaining
orthogonality by extension of the dot product between two-dimensional
vectors. She also has a short description of non-orthogonal
bases.
Chapter 7 is pivotal, and describes multiresolution. Hubbard
shows how the Haar function (a simple, orthogonal wavelet) and its
scaling function can be derived by using Fourier analysis and low-and
high-pass filters. This was the chapter that I'd been looking for
when I bought the book - a simple (but not stupid) explanation of what
and how a wavelet is/works, written for an engineer who might want,
some day, to actually use them to do something useful.
Chapter 8 is
an explanation of the fast wavelet transform and is written in the
same understandable manner (and same high-level position) as the
chapter on the FFT. Following it are several small chapters on
wavelets in two dimensions, pyramid algorithms, and
multiwavelets.
Chapter 12 is short (like most of the chapters) but
has one of the nicest explanations of the Heisenberg uncertainty
principle I've ever seen. This is accompanied later in the book with
a nice proof in the appendix. Chapter 13 helps tie it all together
with discussions about probability, the Heisenberg uncertainty
principle, and quantum mechanics.
The appendixes in this book are
especially useful and there is a nice list of wavelet software and
electronic resources at the end...

I really like this book. It's probably not for mathematicians or anyone seeking a rigorous technical treatment. It's unusual in that it's written by a non-specialist... but in this case this is a feature, not a bug. For a quantitative person seeking a good overview — not just of wavelet transforms but of signal processing in general — with some technical leads and plenty of solid references, it's spot on. Caveat: the book may be a little dated at this point; I first read it in about 2005.