## Summary

I used **Hubbard, Frazier, and Nievergelt** as introductions. When I last touched wavelets, I was searching through all three of **Burrus et al., Strang & Ngygen, and Daubechies** to learn more about wavelets. I can’t say I was working through any one of them, but I was certainly trying to find things in all three of them.

## discussion

I can categorize my wavelet books as:

- for math people, for electrical engineering, and/or for general technical;
- textbook or monograph;
- elementary, upper division, or advanced.

I can also note whether they have code in some form, sample calculations, and how much start-up time they seem to take.

You will see what I mean as I introduce the books. Most of these classifications are not hard and fast; the clearest distinction is between **textbook or monograph — does it or does it not have exercises?** And the phrase **upper division should be interpreted as “upper division or early graduate student”.**

First, we have an elementary, general technical, monograph: “The World According to Wavelets” by Barbara **Hubbard**. It is an excellent, accurate introduction to wavelets: what they are, and what we can do with them. The book is divided into two parts: the first half requires no more than some high school math; the second half is a collection of short descriptions of the mathematics she left out of the first half. The only reason I call it a monograph is that it has no exercises. It is the only book which I call elementary.

If you are wondering, “What are wavelets?”, **Hubbard** is an excellent starting point. Let me quote one of the reviewers whom I recognize, T.W, Körner, the author of the wonderful books ‘Fourier Analysis’ and ‘A Companion to Analysis’: “(A) splendid little book… I can confidently recommend Barbara Hubbard’s book as mathematically reliable, historically accurate and free of hype.”

It would help if you have seen something of Fourier series — but it’s not necessary. The first half of the book is an English language introduction to the subject; the second half of the book supplies mathematical detail.

Some time ago, I encountered the phrase “matching pursuit” in a context which suggested that it had something to do with signal analysis. I, of course, did a Google search. And yet, if I had had sense enough to reach for Hubbard, I would have found a more appropriate answer. All I wanted to know was where in the universe “matching pursuit” lived; I wasn’t interested in the architectural details of its house. She had the answer I sought.

The book is an excellent overview. It may leave the reader hungry for detail, but detail is not what she is trying to provide. In fact, I think she’s trying to make the reader hungry.

On the downside, the book does not appear to be available in paperback, and at $60 list price (a little less on Amazon), it is a pricey overview.

That book did not have enough detail to get me started with wavelets. For that, I used “Wavelets Made Easy” by Yves **Nievergelt**. It is an upper division textbook, and I would call it general technical. Yes, it has a fair bit of mathematics in it, and it even has proofs; but its audience is more general than just math majors.

Nievergelt offers detailed calculations of wavelet coefficients for Haar and Daubechies D4. On the other hand, his code for Daubechies wavelets is sketchy. It’s a nice, clear, introductory book — but we’ll have to write our own code for carrying things out.

This book begins by showing, in almost gory detail, how to compute the wavelet coefficients for the Haar wavelets. It seems to be limited (!) to the Haar and Daubechies wavelets, but it goes into higher dimensions than 1D; it includes three chapters on basic Fourier analysis.

That book gave me a good feeling for the Haar wavelets, but left me wondering how it all applied to Daubechies wavelets — even though he did cover the Daubechies wavelets. I moved on to “An Introduction to Wavelets Through Linear Algebra” by Michael **Frazier**. It is an upper division textbook, more mathematics than general technical — but its mathematics is pretty well limited to linear algebra, which I would hope would give it a wide audience. It added more wavelets to my pantry, specifically Shannon. It showed me how to compute wavelet coefficients, but the efficient methods of computation were not emphasized.

It is a marvelous linear algebra book. Here’s how little I understood of the discrete Fourier transform (DFT): it is a change of basis. I didn’t know that until I started working in **Frazier** a few years ago. Since then, of course I’ve seen it everywhere. He does not emphasize the quick, slick way to compute wavelet coefficients, so I am looking forward to picking him up again after I have created working Mathematica® code to do that.

At about the same time, I picked up “Wavelets and Filter Banks” by Gilbert **Strang** and Truong **Nguyen**. This is an upper division textbook, both mathematics and electrical engineering. It uses the languages of both subjects. Although I had never heard of a filter bank when I saw the book, that Strang was one of the authors meant that I had to have the book. (All of you electrical engineers are allowed to snicker at my struggling through filter banks, quadrature mirror filters, the polyphase matrix, etc.)

It is a remarkable book. In its language and style, it is bedtime reading. In content, however, it is a textbook for MIT. It reminds me of sitting in the mathematics department lounge listening to a professor talking casually about a subject — except that this casual talk is accompanied by detailed notes, which I get to look at later, at my leisure.

I come to it knowing the vocabulary rather than the experience of discrete signal processing (DSP); this appears to be enough, for me. I bring to it more mathematics than they are assuming, and, almost certainly, less electrical engineering. One of their purposes was to describe filter banks using the languages of both mathematics and signal processing.

Formally, I suspect that this book presupposes an introductory course in signals and systems. I think it is being used as an upper division book at MIT.

I found it difficult, however, to see how to do computations. In particular, infinite dimensional matrices bother me, computationally speaking. The book that has finally let me sink my teeth into wavelets is “Introduction to Wavelets and Wavelet Transforms: a Primer” by **Burrus, Gopinath, and Guo**, which I also refer to as **Burrus et al**. This is an upper division monograph — there are no exercises — for a general technical audience. Now, I have already had to use **Strang & Nguyen** to figure out exactly how to do what **Burrus et al.** called “a similar calculation”; and it will take some work to extract their examples, because they usually only provide pictures; but by and large this is a wonderful book, and I recommend it highly for hands-on wavelets. As I have said in this post, they provide both printed and electronic Matlab code. They also describe a wide collection of wavelets.

**Burrus et al**. is a step deeper into the details. This is the book that shows me four ways to compute the D4 scaling function (two of which I’ve shown you), and provides the code. I like this book because it appears to be a very nice companion to **Strang and Nguyen**, and a very nice introduction to **Daubechies**. I think I can say that it is aimed at technical readers, but not specifically at mathematicians or electrical engineers.

On the downside, although the book provides code for computing the discrete wavelet transform (DWT) and its inverse, it has no examples. That is, it has no data against which you can check the code. On the upside, what code I have tried, works — which is more than I can say for some books.

Maybe now is a good time to point out that once we understand the wavelet decomposition, we can verify our code on any examples we choose, because the wavelet coefficients we compute are the components of our data wrt a basis. That is, we can confirm that given function or data is equal to the computed expansion in wavelets. The lack of examples isn’t all that crucial.

Oh, I am in their Chapter 5 at present, and Chapter 6 looks pretty reasonable, too. Chapter 7 looks like rather more complicated — but not particularly advanced — mathematics, and Chapter 8 is about filter banks. If the book holds true to form, chapter 8 will be a reasonable first introduction to filter banks.

But every once in a while **Burrus** is sadly incomplete. As I said earlier, I have had to use **Strang & Nguyen** to find the details of “a similar calculation” – they were not obvious! – and how to compute the mother wavelet was a bit obscure.

Okay, so much for the auxiliary material, for right now. “Read the masters,” they say. That brings us to “Ten lectures on Wavelets” by Ingrid **Daubechies**. This is, strictly speaking, an advanced mathematical monograph. Nevertheless, the book is aimed at a wide technical audience, and she did a remarkable job of explaining what she was doing. Remarkable, I say. She is worth quoting on this:

“This book is a mathematics book: it states and proves many theorems.

“… most of the book can be followed with just the basic notions of Fourier analysis. Moreover, I have tried to keep a very pedestrian pace in almost all proofs, at the risk of boring some mathematically sophisticated readers. I hope therefore that these lecture notes will interest people other than mathematicians…. I hope to succeed in sharing with my readers some of the excitement that this interdisciplinary subject has brought into my scientific life.”

**Daubechies** is the place to look for the mathematics of wavelets. It is readable — perhaps as readable as any mathematics book can be. **Burrus** looks like a nice introduction to it.

Let me also say: **Daubechies** is not easy reading, but it’s accessible because she describes so clearly what she is doing. (I’m about to see if she tells me how to compute the mother wavelet from the scaling function…. Yes.)

The primer by **Burrus, Gopinath, and Guo** is an excellent introduction to **Daubechies**, but no substitute for it. That’s one of the reasons I hope that they will also make **Strang and Nguyen** more accessible to me.

## Summary repeated

I used **Hubbard, Frazier, and Nievergelt** as introductions. When I last touched wavelets, I was searching through all three of **Burrus et al., Strang & Ngygen, and Daubechies** to learn more about wavelets. I can’t say I was working through any one of them, but I was certainly trying to find things in all three of them.

## the books added

Burrus, C. Sidney; Gopinath, Ramesh A.; Guo, Haitao. **Introduction to Wavelets and Wavelet Transforms, A Primer.** Prentice Hall, 1998.

ISBN 0 13 489600 9.

[wavelets; 20 Dec 2009]

Finally! A book that shows me how to compute things properly. “We tried to present this in a way that is accessible to the engineer, scientist, and applied mathematician…” I think it is accessible to a technical undergraduate, but I could be underestimating the mathematics. Guide to further reading. No answers as such, but MATLAB code and pictures drawn using it.

Daubechies, Ingrid; **Ten Lectures on Wavelets.** Society for Industrial & Applied Mathematics, 1992;

ISBN 0 89871 274 2.

[wavelets; 20 Dec 2009]

This is one of “the books”, an essential reference for the mathematics of wavelets. She is now famous for early work on wavelets. She is a fine expositor, and although I often do not understand what she is doing, she explains why she is doing it – and that is a tremendous help for reading other books.

Frazier, Michael W.; **An Introduction to Wavelets Through Linear Algebra.** Springer, 1999.

ISBN 0 387 98639 1.

[wavelets, linear algebra; 20 Dec 2009]

“Undergraduate Texts in Mathematics”. Requires only linear algebra and some calculus; a “topics” course for math majors. I had no idea that the Discrete Fourier Transform was a change-of-basis! This is where I began to learn the theory of computing wavelets, although it didn’t take me far enough. Guide to further reading.

Hubbard, Barbara Burke; **The World According to Wavelets.** A.K. Peters, 1998.

ISBN 1 56881 072 5.

[wavelets; 20 Dec 2009]

This is an extraordinary semi-popular introduction to wavelets and the underlying mathematics. It is written on two levels. Part I is written in English, with some graphs; Part II provides the equations and more detail. Part I is at a “popular” level; Part II pretty much requires a technical background, but might serve to whet someone’s appetite for mathematics or applied mathematics.

Nievergelt, Yves. **Wavelets Made Easy**. Birkhäuser, 2001 (2nd printing with corrections).

ISBN 0 8176 4061 4.

[wavelets; 20 Dec 2009]

The first two chapters give us the fast wavelet transform for Haar and Daubechies wavelet coefficients. This is where I did my very first computations in wavelets. The rest of the book looks like the “advanced calculus” of wavelets.

Strang, Gilbert; Nguyen, Truong.**Wavelets and Filter Banks.**Wellesley-Cambridge Press, 1997 (revised edition).

ISBN 0 9614088 7 1.

[wavelets, filter banks; 20 Dec 2009]

This is challenging material presented with a light-hearted style, and I can enjoy the style while wondering what is going on. I have to think that this text presupposes some exposure to digital signal processing; some, but perhaps not a lot. “Our text explains filter banks and wavelets from the beginning — in several ways and at least two languages (mathematics and signal processing). A small collection of papers at the end, and the final chapter (Applications) are effectively an epilog for what’s next.

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