Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 61) 1st Edition by Ingrid Daubechies (PDF)

Description

This monograph contains 10 lectures presented by Dr. Daubechies as the principal speaker at the 1990 CBMS-NSF Conference on Wavelets and Applications. Wavelets are a mathematical development that many experts think may revolutionize the world of information storage and retrieval. They are a fairly simple mathematical tool now being applied to the compression of data, such as fingerprints, weather satellite photographs, and medical x-rays – that were previously thought to be impossible to condense without losing crucial details. The opening chapter provides an overview of the main problems presented in the book. Following chapters discuss the theoretical and practical aspects of wavelet theory, including wavelet transforms, orthonormal bases of wavelets, and characterization of functional spaces by means of wavelets. The last chapter presents several topics under active research, as multidimensional wavelets, wavelet packet bases, and a construction of wavelets tailored to decompose functions defined in a finite interval.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As stated by the author in the first sentence of her redoubtable treatise (p.1), “The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale.” She goes on to trace the evolution of this mathematical tool, that is seen to rely (in the case of Meyer wavelets, p.119) on “quasi-miraculous cancellations.” Sigrid Daubechies’ contribution was that of developing an algorithm for defining a wavelet function that did not suffer from the drawbacks of previously-defined analytical or digital functions. Analytical functions, while usually sufficiently smooth, are not truncated in temporal or configuration space–nor are the associated temporal or spacial frequencies commonly accessed by Fourier transforms (p.4). On the other hand, digital functions such as square waves or step functions (or the Haar wavelet, p.15) are often not sufficiently smooth. As the author observes on the first line of Chapter 6 (p.167) “Except for the Haar basis, all examples of orthonormal wavelet bases in the previous chapter consisted of infinitely supported functions.” Daubechies, however, was able to unravel the Gordian knot by defining discrete/digital wavelets (pp.53-105) that are both sufficiently smooth and naturally truncated or “compact” (i.e. “compactly supported,” pp.194-199).The author lucidly derives and proves the incredible properties of what have come to be called Daubechies wavelets–thereby convincing the reader that these amazing mathematical entities actually exist! It is difficult, however, to digest every proof laid out by the author. I would therefore suggest that the reader begin with the most interesting proofs, which are found in the first section (5.1, pp.127-137) of Chapter 5: “Orthonormal bases of wavelets and multiresolution analysis.” The reader should pay particular attention to p.132, which introduces the trigonometric polynomial m0(u), the properties of which are further detailed on pp.155, 168, and 216.For those of us who are not heavily involved in signal and image processing, however, Daubechies’ book is a difficult place to begin one’s study of wavelets. A better starting place for the novice would be an introductory textbook such as that of Burrus et al.:

⭐. In reading Burrus along with Daubechies, however, the reader needs to keep in mind that the two books use different notations for expressing wavelets and scaling functions (Daubechies p.130; Burrus p.15), such that scaling parameter as used in one book’s notation is the reciprocal of that defined by the other.A good overview of the field is also provided by Hubbard’s

⭐. It is also necessary that the reader have an understanding of Fourier transforms at the level of Bracewell’s

⭐.I should also note that Daubechies’ subject index spans only two pages–which is not nearly long enough! The shortcomings of the index and the detailed nature of this monograph limit its usefulness as a reference book. Therefore, the reader needs to buy his own copy of Daubechies and mark it up to his heart’s content, including notes on the title page (and the pages immediately following it) that may supplement the subject index.

⭐This is an excellent book, very lucid and rigorous. I took Daubechies’ course on wavelet analysis while I was at Princeton some years ago, and her course essentially follows this book. So this book makes a lot of sense to me, after taking her class. But beware, this is a book on mathematical analysis. It is not a book to learn about wavelets from a practical standpoint. If you are engineer, most likely you do not have the required mathematical background to understand anything from this book. This is really for math people, or engineers/physicists who are mathematically inclined. The pre-requisite are real analysis, complex analysis and analysis in several variable, and maybe a slight amount of functional analysis, although the latter is not really needed because most of the theorems are derived in this book. So to re-iterate, this is an excellent book, but it is not for learning about wavelets. Read Stephane Mallat’s textbook instead, which was written to teach the topic from a more practical standpoint.

⭐I think the book is very good. I know some people say it’s difficult too read, old and has good reviews just because it’s a “classical” text. What you need to understand is that the book emphasis the theory of frames and wavelet and does not focus on image processing, for example. I can honestly say that I learned a lot from the book, back then when I was a student and even today when I need to refresh my memory with the theory I always go back to it.

⭐Thank you

⭐Original papers … a good starting point to learn about wavelets.

⭐What can I say, Wavelets explained by Ingrid Daubechies! You can’t go wrong.

⭐This book has become a classic,– and a hit;– for more than ten reasons. It is multilayered, and yet presents a unity of ideas: The material, and the writing is engaging for the beginner, and for the research mathematician alike. When I used it in my teaching, it was equally popular with the math students, and those from engineering. I don’t know if I can say this about any other book I have taught from. The students could follow all the carefully presented proofs, and the engineer could generate algorithms from the applied chapters.

⭐There are so many well written books on Wavelts out nowadays. Don’t waste your money on this one. It’s famous because it was first (or one of the first). On the positive side, it does present a strong mathematical foundation. I recommend you buy a readable book (just do a search on Amazon.com and you’ll find half a dozen great books on Wavelets), then when you approach the “expert” level, use this one a s a reference (if at all).

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Free Download Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 61) 1st Edition in PDF format
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