The World According to Wavelets: The Story of a Mathematical Technique in the Making

Rating:
Summary: Marginally enlightening; journalistic style; overall poor
Review: ....

The first thing one notes is this book is written in a highly journalistic style. It reads as if a very long article in the New York Times. The author rarely explains anything herself; instead, she moves from one direct quote to another by various players in the wavelet field(s), with a small amount of glue in between. Although this lets the reader know he's probably getting some modicum of "truth," it leaves the material poorly flowing and difficult to read, let alone get any depth.

Speaking of depth, I would have to say I got very little depth out of this, in either a layman's sense (reading for pleasure or to find out a bit about wavelets) nor in a mathematical sense (reading to write software, a WinAMP plugin, an image compressor, or anything practical). For this reason, this book is relatively worthless, in my opinion. In fact, I feel I learned more about Fourier analysis than about wavelets, both mathematically and on a layman's basis. (For a good solid mathematical introduction to Fourier Analysis, see A First Course in Fourier Analysis by David Kammler; I recommend it.)

The first five chapters are a historical discussion of wavelets. By the fourth, I didn't really feel any urge to read the fifth (although I did). The remainder of the book was a series of slightly mathematical discussions on certain themes from the first five chapters (as D. Anderson refers to in his review). None of these interludes were either deep enough nor broad enough to give any real understanding, although some were somewhat interesting, such as the proof of the Uncertainty Principle for what can be learned from a signal and its Fourier transform.

Finally, this book was both too long (as I said, it wore me out 30% of the way through) and too short - it had hardly the depth or breadth that would allow any meaningful investigation of wavelets on one's own.

Note that the last page reveals that the author was, until this book, a journalist, and that this is her first book - and it was originally written in French and translated into English by her (it does not read like a translated book).

In sum, this book is better than total garbage (for which I reserve my one star rating and only believe the book "Java Design" by Peter Coad deserves, so far) but doesn't really enlighten interest nor knowledge.

Rating:
Summary: Overrated
Review: Excerpted with permission by UMAP Journal and Steven Krantz

"If you would like to know what wavelet theory is about--be you student, mathematician, or avid nonscientific reader with broad interests--then this book will give you more than a nodding acquaintance. It will take you to a level of depth that you choose, and it will introduce you to a modicum of fascinating mathematical culture in the process. The book is artfully written, contains amusing anecdotes and inspiring quotations, and is a fine model of scientific exposition. I recommend it with enthusiasm."

UMAP Journal 20 (1) (1999) 85-90

Steven G. Krantz, Mathematics Dept., Washington University, Box 1146, One Brookings Dr., St. Louis, MO 63130-4899; [e-mail]

The past fifteen years have seen an explosion of activity in the field of wavelets. Rarely have we seen ideas from theoretical mathematics filter so rapidly into the applied arena. Conferences on wavelets are attended in droves, both by mathematicians and by engineers. The online journal Wavelet Digest [Sweldens 1999] is said to have 12,000 subscribers. The "nattering nabob of negativism" (This phrase is due to Vice-President Spiro Agnew (1918-1996), September 1970) will point out that wavelets have their roots in Littlewood-Paley theory, and that that branch of harmonic analysis is 70 years old. With 20/20 hindsight, one can now see that wavelets were already contained in the Calderón reproducing formula of the 1960s. Old papers in artificial intelligence, in physics, and in signal processing can be said to have contained the germ of the idea of wavelets. True enough, but almost any idea in mathematics has roots that can be traced back some years. Wavelets offer a number of novelties to the Fourier analyst:

That he can choose "basis elements" for his Fourier analysis that are suited to the specific application at hand,

that he need not be slavishly tied to the notion that the basis elements must be orthogonal,

that simultaneous localization in both the space and phase variables is possible, and

that convergence issues for wavelets often are simpler to understand than the corresponding convergence issues for (as an example) Fourier series.

The applications of wavelet theory to image processing, signal processing, fingerprint analysis and compression, filtering of musical signals, electrocardiography, magnetic resonance imaging, biological neural systems, and many other branches of hard science have been profound and influential. These are just a few of the dozens of examples. Wavelets have actually changed the way that people think about harmonic analysis problems. ... there is a paucity of books introducing wavelets to the layman--perhaps (very roughly speaking) as James Gleick [1988] introduced the public to the putative theory of chaos. The book under review is actually much better than Gleick [1998], because it is written by someone who has worked hard to understand the subject matter, who has checked and verified her work with numerous authorities, who has studiously avoided flash and glitz, and who can present the subject with some depth and texture. As a signal of the quality of Hubbard's book, it was recently awarded the Prix d'Alembert. (Note that d'Alembert is famous in part for his studies of the heat equation, one of the first scientific problems to be treated with Fourier analysis.) Hubbard begins in a gentle manner, acquainting the reader with the anomalous life of Fourier and his astonishing contributions to scientific thought. She describes Fourier series and how they are used. She segues into a history of nineteenth-century analysis, telling of the development of the Fourier transform. Along the way, she skillfully introduces the problem of simultaneously localizing both the space and the phase variables; as she puts it, the Fourier transform tells us what (musical) note is being played, but not when. This setup gives a beautiful entrée to the basic idea of wavelets. Most of us are aware of wavelets today because star Fourier analyst Yves Meyer was comfortable swapping ideas with engineers and physicists. A signal-processing problem that had captured their attention struck a chord with Meyer, and the rest is history. Hubbard gives a charming rendition of this critical part of the wavelet legend. ... There are many delightful figures, computer printouts, and illustrations to amplify the principles being discussed. Hubbard wisely divides her book into three parts. The first is truly heuristic: It is written for the science "groupie" who has no technical knowledge whatever. Beginning on p. 111 is the portion entitled "Beyond Plain English," which actually begins to show some mathematics. Here we encounter the language of functions, function spaces, bases, filters, and transform theory. Section 17 of this part contains a very pretty collection of tables summarizing the key features of various transforms. The last part of the book--actually a collection of Appendices--becomes quite technical, and even includes a proof of the Heisenberg uncertainty principle. The jaded theoretical mathematician might object that this book concentrates too much on applications and too little on the beautiful wavelet theory. But we must keep in mind the audience: A person who hardly knows what a function is will not appreciate that wavelets provide an unconditional basis for the Hardy space H1, nor appreciate the finer properties of a multiresolution analysis. Instead, such a person will appreciate the mathematics because it can be used for something more familiar. That is the point of view of this book. I hate advertisements that cheerily profess their product to offer "something for everyone." Such hyperbole oft bespeaks a tired and obvious disingenuousness: that the artifact being promoted offers nothing for anyone. But I might dare apply the phrase to this book. If you would like to know what wavelet theory is about--be you student, mathematician, or avid nonscientific reader with broad interests--then this book will give you more than a nodding acquaintance. It will take you to a level of depth that you choose, and it will introduce you to a modicum of fascinating mathematical culture in the process. The book is artfully written, contains amusing anecdotes and inspiring quotations, and is a fine model of scientific exposition. I recommend it with enthusiasm.

Rating:
Summary: Attractive format & nicely laid out but just didn't hook me
Review: I agree with most of what the favorable reviewers had to say but my experience led me to the opinion that the book didn�t meet my attention while sincerely trying to get into the subject. I�m not knocking the book, its format is nicely laid out; the author's style is friendly, and the book follows a seemingly logical path - I just wasn't hooked by it. My imagination was not stimulated by the examples used by the author to illustrate her subject. I resorted to occasionally skipping around the paragraphs and chapters until I finally set it down for a long rest. So, as you can see, this is just the opinion of a guy who's probably being finicky. It wasn�t what I was looking for. Maybe I�ll re-visit it later in my pursuit of intellectual expansion and get more out of it. �

May I suggest that you look into the book titled �Who Is Fourier?: A Mathematical Adventure� by the Transnational College of LEX. It is a Japanese workbook wonderfully translated into English which will 'hold your hand' through the mathematical learning curve with regards to waves. I think that 'Fourier' is better suited for learning about the mathematical treatment of Waves in it's extremely friendly walk-through of Trigonometry and light elements of Calculus - 'Fourier' is even easy for those, like myself, who're borderline incompetent in technical mathematics!

But what do I know? I�m just another tourist trudging along the road of happy destiny with what I�ve picked up along the way. The other reviewers are better suited to elucidate the pedagogical aspects of this book and, therefore, I recommend paying close attention to what they have to say.

�The World According to Wavelets : The Story of a Mathematical Technique in the Making� might be just what you�re looking for. I know that each of us look for something which appeals to our particular (dare I say peculiar?) tastes. Check it out. Do the math. Look with your own eyes & decide for yourself. Remember, 'There is a principle which is a bar against all information, which is proof against all arguments and which cannot fail to keep a man in everlasting ignorance. That principle is contempt prior to investigation.' - Herbert Spencer.

I truly hope that you get more out of it than I did. Enjoy all your reading experiences.

Adventure Master IndiAndy: Investigating opportunities for intellectual growth...

�The most important thing we can know about a man is what he takes for granted and the most elemental and important facts about a society are those that are seldom debated and generally regarded as settled.� Louis Wirth

Rating:
Summary: It can be done!
Review: 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!

Rating:
Summary: An introduction to wavelets at the college-freshman level
Review: 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:
Summary: Outstanding overview
Review: I thing this is a fabulous book. I would not have it as your only book on wavelets. I would read it first to get the big picture. Then get a more applicative or theoretical book as you detail reference. The author has a knack for explaining the basic ideas clearly and simply. Easy and entertaining to read but it isn't all fluff. You learn the sweeping and critical ideas and terminology. Has broad coverage. Get this book... and one other.

Rating:
Summary: Good for start
Review: 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.

Rating:
Summary: Overrated
Review: If you're looking for an mathematical introduction to wavelets to be on your way to using it as a tool, then find something else. This is not a math book. Sure, it has formulas and stuff, but you won't find any new or deep insights beyond the common ones found in most introductory chapters in most books on wavelets. But you shouldn't expect to anyway; the author is a journalist, not a mathematician. But if you want to read about the "story" of wavelets, then this is the book. I'd give it 5 stars for that. Which is what the author intends it to be. Just don't let the other reviewers lull you into thinking it's more than it really is. Check out Mallat's book if you want a good introductory math book on wavelets.

Rating:
Summary: A MUST-HAVE fundamental book on Wavelets
Review: Mathematics texts, as a rule, tend to be organized along the lines of presenting postulates, theorems, proofs, and examples in a sequential order. There are symbols to decipher, equations to be analyzed, proofs to verify, etc. On the other hand, mathematics books written in plain prose tend to be too general (good enough to provide an overview but usually not enough detail to really learn the subject matter well). Ms. Hubbard's book is a rare one indeed-- one that provides the reader with an intuitively solid overview of wavelets followed by a more traditional and substantial mathematical presentation. A unique (and very effective) feature of Ms. Hubbard's book is the way she links the mathematical details relevant to her more general discussions (towards the beginning of the text) with boxed references. Also, another clever feature that the author employs is the use of underscore braces to nonintrusively insert comments to equations. For those readers who desire more mathematical rigor and detail, Ms. Hubbard provides a very comprehensive reference to other sources. Whether you're a student, mathematician, engineer, scientist, or just a beginner who wants to learn wavelets, this book is a definite must have! In the vast woodland of wavelets Ms. Hubbard provides you with a map of the forest as well as a description of the trees-- a very rare combination for a math book!

Rating:
Summary: A concise, non-technical introduction to wavelets.
Review: 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