The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition [Hardcover]

Barbara Burke Hubbard (Author)

Book Description

ISBN-10: 1568810725 | ISBN-13: 978-1568810720 | Publication Date: May 30, 1998 | Edition: 2

This best-selling book introduces a broad audience including scientists and engineers working in a variety of fields as well as mathematicians from other subspecialties to one of the most active new areas of applied mathematics and the story of its discovery and development. Organized in "hypertext fashion," the book tells a story of scientific discovery with separate brief entries for technical terms and explicit appendices in a section called "Beyond Plain English."

Product Details

• Hardcover: 286 pages
• Publisher: A K Peters/CRC Press; 2 edition (May 30, 1998)
• Language: English
• ISBN-10: 1568810725
• ISBN-13: 978-1568810720
• Product Dimensions: 9 x 6 x 0.9 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (16 customer reviews)
• Amazon Best Sellers Rank: #728,217 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

66 of 67 people found the following review helpful:

5.0 out of 5 stars An introduction to wavelets at the college-freshman level, August 16, 2000

By 

Duwayne Anderson (Saint Helens, Oregon) - See all my reviews
(REAL NAME)   

This review is from: The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition (Hardcover)

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...

17 of 17 people found the following review helpful:

5.0 out of 5 stars Good for start, November 12, 2002

By 

Stana Zivanovic (Sheffield, UK) - See all my reviews

This review is from: The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition (Hardcover)

I was very happy reading this book. If you are familiar with the Fourier transform and don't know anything about wavelets, this is a book for you.

Actually, the book has got two parts. In the first part you can learn basic things about Fourier transform (about its usage but also about its limits), what we need wavelets for and what the wavelets are. It is explained in very simple language without any formulas. The second part contains basic formulas related to the topics in the first part. I find that the link between these two parts is very good. Also, the author gives physical explanation whenever it's possible.

If you are a specialist in the wavelets area, you probably know all these things but if you are new (like me!) you will find that this book is quite useful.

22 of 25 people found the following review helpful:

5.0 out of 5 stars It can be done!, August 17, 2002

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition (Hardcover)

I am a math professor,-- and I often wondered if it wouldn't be possible to get some essential math ideas accross to almost anyone, --and with fewer equations. Ideas can be burried in symbolism;-- not always! But it does happen. Many of my colleagues tell me that if it were possible, then it would be done. The author of this lovely little book didn't take math courses (she says!). Professional mathematicians would most likely agree with me that she (the author) did in fact communicate the essential ideas behind wavelets (and did it well!);- and so she must have understood them!! Perhaps, anyone who really wants to, can penetrate a specialized math discipline;-- I would guess. Perhaps it is not even hard!? At least this book proves that it is not impossible to communicate
the beauty of math;--and its uses. Take a look at the book, and judge for yourself!
It is fun too!