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Wavelets: A Concise Guide, by Amir-Homayoon Najmi
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Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction properties. Amir-Homayoon Najmi’s introduction to wavelet theory explains this mathematical concept clearly and succinctly.
Wavelets are used in processing digital signals and imagery from myriad sources. They form the backbone of the JPEG2000 compression standard, and the Federal Bureau of Investigation uses biorthogonal wavelets to compress and store its vast database of fingerprints. Najmi provides the mathematics that demonstrate how wavelets work, describes how to construct them, and discusses their importance as a tool to investigate and process signals and imagery. He reviews key concepts such as frames, localizing transforms, orthogonal and biorthogonal bases, and multi-resolution. His examples include the Haar, the Shannon, and the Daubechies families of orthogonal and biorthogonal wavelets.
Our capacity and need for collecting and transmitting digital data is increasing at an astonishing rate. So too is the importance of wavelets to anyone working with and analyzing digital data. Najmi’s primer will be an indispensable resource for those in computer science, the physical sciences, applied mathematics, and engineering who wish to obtain an in-depth understanding and working knowledge of this fascinating and evolving field.
- Sales Rank: #1238714 in Books
- Published on: 2012-02-24
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .74" w x 6.13" l, .92 pounds
- Binding: Paperback
- 304 pages
Review
A complete, concise and clear exposition of the more traditional tools related to linear filtering.
(Davide Barbieri Mathematical Reviews)Since their emergence in the last eighties and early nineties of the twentieth century, wavelets and other multi-scale transforms have become powerful signal and image processing tools. Najmi's book provides physicists and engineers with a clear and concise introduction to this fascinating field.
(Ignace Loris American Journal of Physics) About the Author
Amir-Homayoon Najmi completed the Mathematical Tripos at Cambridge University and obtained his D.Phil. at Oxford University. He is with the Johns Hopkins University's Applied Physics Laboratory and is a faculty member of the Whiting School of Engineering CE programs in applied physics and electrical engineering.
Most helpful customer reviews
10 of 10 people found the following review helpful.
Full of embarrassing mathematical errors
By David Lurie
I'm only on page 22, and already I have a long laundry list of mathematical errors.
The author can't decide on a notation for vectors in R^n. Are they row vectors? Column vectors? Do we use brackets or parentheses? Just pick one, for crying out loud.
Another example: "Hilbert spaces generalize the familiar Euclidean three-dimensional space R^3.... For instance, an orthogonal projection from the tip of a standard three dimensional vector onto an arbitrary plane through the origin has its analog in Hilbert spaces in the form of an orthogonal projection onto a linear subspace...." First of all, what the heck is the "tip" of a vector and how to you project from it? And what is the definition of projection of a vector onto a plane in R^3 if not exactly the definition given as the "analog" in Hilbert spaces?
The book gives a definition for a bounded linear operator, then inexplicably gives a different (though equivalent) definition for a bounded linear functional, which is a type of linear operator. Why not use the same definition that was already given? Why confuse matters by giving extra definitions for special cases when the original definition works just as well? To top it off, on the very next page we are given yet a third definition, this time for what it means for the point evaluation functional to be bounded. At this point the reader now has three (!!) definitions for a bounded point evaluation functional. I just don't even...
The author states that the integral notation will always refer to the Lebesgue integral but then uses the integral notation for inner product of distributions at the first possible chance without any explanation. This notation is actually standard, but it's nonsense without explanation, particularly when the text explicitly states that the notation means something different.
While many of my complaints above are arguably aesthetic, there are several serious mathematical problems with the text. Theorems are stated without proof, which may be fine for such a text, but some of the supplied proofs are a complete mess. One proof of an equivalence omits one of the directions. The author seems to not understand L_2 equivalence classes of functions. The most disturbing mathematical defect is the author's use of the word basis. Apparently the author does not know what a basis is, which is inexcusable given the subject. For example, the author states (incorrectly) that the complex exponentials are a basis (!!) for L_2(R). In fact, the complex exponentials are clearly not even members of L_2(R). (The difference between Schauder and Hamel bases is not mentioned.) These errors are embarrassing.
I think the reason I am so frustrated by these deficiencies is that the text would otherwise be fantastic. It would have been SO EASY for the author to have gotten these things right, and if he had, he would have written a beautiful text.
6 of 6 people found the following review helpful.
A Fine Introduction
By Illuminatus
With ever increasing frequency, wavelets are being successfully applied to virtually every aspect of signal and image processing, with a commensurate expansion in both the research and pedagogic literature. As a result, the novice can often find it challenging to achieve a rapid introduction to this increasingly important field. Dr. Najmi's new book, developed over more than a decade of teaching related courses at the Johns Hopkins University, provides an elegant solution to that problem.
The opening chapter offers a condensed but comprehensive introduction to the relevant mathematics, covering such necessary topics as Lebesgue integration, normed vector spaces, Fourier Transforms and frame representations. After a brief survey of linear time invariant systems, wavelets are introduced as a generalisation of the windowed Fourier Transform. Subsequent chapters explore the properties of the most common types of wavelets and several important applications. Although the language is mathematically precise, frequent use of terms such as "time series" and "filter banks", as well as the inclusion of many descriptive diagrams, will comfort the engineer as well. There are problems at the end of each chapter making this book suitable for a text in a one semester course or (possibly tutored) self study.
In short, Wavelets: A Concise Guide is an approachable and self contained introduction that will carry the reader from the absolute beginning to the capability to achieve informed solutions to many problems in modern signal processing.
It is not surprising that a book of this type, especially in a first edition, contains a number of typographical errors. Most have been discovered and they can be remedied by reference to the errata, which can be found on the Johns Hopkins University Press web site. (Amazon's policy forbids me from providing the URL.)
1 of 1 people found the following review helpful.
Wavelets: A Concise Guide
By Nasser M Nasrabadi, US Army Research Laboratory
This is an excellent introduction to the wavelet transform that combines a thorough discussion of essential mathematical structures (Hilbert spaces, orthogonal bases, frames), with all the practical discrete signal analysis concepts and methods of linear system theory (the discrete Fourier transform, discrete convolution for finite length discrete time data, the Z transform, etc.).
The continuous wavelet transform is introduced in analogy to the windowed Fourier transform. The Haar and the Shannon wavelets are used to motivate the general theory of multi-resolution analysis spaces, orthogonal and bi-orthogonal wavelets, and wavelet packets. A short chapter at the end provides an excellent exposition of two-dimensional application of wavelets to images. This is a highly recommended text for engineers, physical scientists, mathematicians, and everyone with a desire to have a firm theoretical and practical understanding of the foundations of wavelets.
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