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Title: The World According to Wavelets: The Story of a Mathematical Technique in the Making by Barbara Burke Hubbard ISBN: 1-56881-072-5 Publisher: AK Peters Ltd Pub. Date: May, 1998 Format: Hardcover Volumes: 1 List Price(USD): $40.00 |
Average Customer Rating: 4.4 (10 reviews)
Rating: 5
Summary: An introduction to wavelets at the college-freshman level
Comment: 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...
Rating: 5
Summary: A concise, non-technical introduction to wavelets.
Comment: This ambitious book attempts to explain to non-mathematicians the development and application of wavelet analysis, a recent branch of harmonic analysis. The book succeeds remarkably well. The author, an award-winning popular science writer, tells the story of the development of wavelet analysis in a compelling narrative. Once begun, the book is difficult to put down. The book's organization is unusual, but highly effective. The first part of the book is pure narrative, carefully avoiding all mathematical formalism, and presenting the ideas in Ms Hubbard's lucid prose. The second part of the book presents the mathematical details of wavelet analysis to those interested in a deeper understanding. What is unusual is that the author has seemlessly linked these two parts by including in the first boxed references to the second. This technique allows the reader to read the two parts individually or simultaneously, and is a very effective reading aid. I highly recommend this book for anyone interested in wavelet theory
Rating: 5
Summary: Good for start
Comment: 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.
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Title: A Primer on Wavelets and Their Scientific Applications by James S. Walker, Steve G. Krantz ISBN: 0849382769 Publisher: CRC Press Pub. Date: 26 March, 1999 List Price(USD): $41.95 |
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Title: Wavelets through a Looking Glass by Ola Bratteli, Palle Jorgensen ISBN: 0817642803 Publisher: Birkhauser Boston Pub. Date: 12 July, 2002 List Price(USD): $59.95 |
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Title: Introduction to Wavelets and Wavelets Transforms by C. Sidney Burrus, C. Sidney Burrus, Ramesh A. Gopinath ISBN: 0134896009 Publisher: Prentice Hall Pub. Date: 14 August, 1997 List Price(USD): $55.00 |
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Title: Wavelets Made Easy by Yves Nievergelt ISBN: 0817640614 Publisher: Birkhauser Boston Pub. Date: 28 April, 1999 List Price(USD): $49.95 |
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Title: An Introduction to Wavelets and Other Filtering Methods in Finance and Economics by Ramazan Gençay, Faruk Selçuk, Brandon Whitcher ISBN: 0122796705 Publisher: Academic Press Pub. Date: 12 September, 2001 List Price(USD): $84.95 |
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