Wavelet Overview

The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers in the wavelet field feel that, by using wavelets, one is adopting a whole new mindset or perspective in processing data.

Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analysis, the scale that one uses in looking at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with alarge "window," we would notice gross features. Similarly, if we look at a signal with a small "window," we would notice small discontinuities. The result in wavelet analysis is to "see the forest and the trees."

Wavelet Time-Freq Mapping

Can you see why these features make wavelets interesting and useful? For many decades, scientists have wanted more appropriate functions than the the sines and cosines which comprise the bases of Fourier analysis, to approximate choppy signals. By their definition, these functions are non-local (and stretch out to infinity), and therefore do a very poor job in approximating sharp spikes. But with wavelet analysis, we can use approximating functions that are contained neatly in finite domains. Wavelets are well-suited for approximating data with sharp discontinuities.

The wavelet analysis procedure is to adopt a wavelet prototype function, called an "analyzing wavelet" or "mother wavelet." Temporal analysis is performed with a contracted, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency version of the prototype wavelet. Because the original signal or function can be represented in terms of a wavelet expansion (using coefficients in a linear combination of the wavelet functions), data operations can be performed using just the corresponding wavelet coefficients. And if you further choose the best wavelets adapted to your data, or truncate the coefficients below a threshold, your data is sparsely represented. This "sparse coding" makes wavelets an excellent tool in the field of data compression.

Other applied fields that are making use of wavelets are: astronomy, acoustics, nuclear engineering, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and pure mathematics applications such as solving partial differential equations.

An Introduction to Wavelets Paper

The above is the first section from my paper: "An Introduction to Wavelets" which was published in the IEEE Computational Sciences and Engineering, Volume 2, Number 2, Summer 1995, pp 50-61.

For more:

Red Button Online version of this paper.
Blue Button PS, gzipped copy (compressed to 212 Kbytes, expands to 1.5Mbytes).
Blue Button PDF version (350 Kbytes).
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Wavelet Digest

If you are interested in wavelets, you should subscribe to the Wavelet Digest. The Wavelet Digest is a free newsletter which is sent about once every few months and contains many kinds of information concerning wavelets. You'll hear the latest announcements of available software, find out about errors in some of the wavelet texts, learn about wavelet conferences, learn answers to questions that you might have thought about, as well as ask questions of the experts that read it. The Wavelet Digest comes in three editions: Classical, HTML, and Executive. The Classic version is the full digest in simple ASCII text mail. The HTML version is the full digest in nicely formatted HTML including navigation links and images. The Executive version is a concise edition in HTML mail containing summaries and links to the site for full articles.

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Wavelet Blogsphere

A new blog is created every second, on average. This expanding blog universe could give you a hard-to-find wavelet idea, no dark energy necessary. The link below with keywords returns a Google blog search page with the results.

Blue Button Wavelet Blog search (courtesy of Google)
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Wavelet Patents

The wavelet patent market has been lucrative for some time. Curious? Click on the next link to learn something about the world of U.S wavelet patents!

Blue Button Wavelet Patents
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Fourier Trivia

From "The Hartley Transform"

by Ronald N. Bracewell. (Oxford Press, 1st ed, 1986, page 6)

"When the FFT was brought into the limelight by Cooley and Tukey in 1965 it had an enthusiastic reception in the populous world of electrical signal analysis as the news spread via tutorial articles and special issues of journals. This ferment occasioned mild surprise in the world of numerical analysis, where related techniques were already known. Admirable sleuthing by M.T. Heideman, C.S. Burrus, and D.H. Johnson (to appear in 'Archive for History of the Exact Sciences') has now traced the origins of the method back to a paper of C.F. Gauss (1777-1855) written in 1805, where he says, 'Experience will teach the user that this method will greatly lessen the tedium of mechanical calculation.' "

"A fascinating sidelight of the historical investigation is that Gauss' fast method for evaluating the sum of a Fourier series antedates the work on which Fourier's fame is based. We should hasten to add that Gauss' paper was not published until much later [Collected Works, Vol. 3 (Gottingen: Royal Society of Sciences, 1876)], and we should remember that, when Fourier introduced the idea of representing an arbitrary periodic function as a trigonometric series, eminent mathematicians such as Lagrange resisted it."

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One day in a land far away
Some mathematicians at play
Found a transform of convenient form
The basis of physics today.

Convolving would wreck people's brains
Still the advent of Fourier domains
for convolving in one
means multiplication
in the corresponding domain.

Got trouble with an ODE?
Fourier transforms will set you free
When once you would cry,
You now multiply,
by a constant times the frequency.

Fourier transforms backwards and forth
I hope that you now see their worth
For in every domain
Advantages reign
Fourier was the salt of the earth.

by Luke Krieg
Posted to sci.physics, 11 Sep 2000

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Blue Button Some Joseph Fourier history.
Blue Button An Introduction to Fourier Theory.
Blue Button Fast Fourier Transform and related code links.
Blue Button Fast Fourier Transform tutorial, code, and related code links.
Blue Button Fast Fourier Transform C library: "FFTW".

Blue Button Lennart Carleson wins 2006 Abel Prize (Harmonic Analysis)!
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Sound Fun

Click HERE to see some sound fun with wavelets.

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The Wavelet IDR Center

The Wavelet IDR Center is a consortium of institutions that is aimed at developing new wavelet/multiscale theories and technologies based on the principle of redundant representations. Its secondary goal is to serve as a national center for the entire development in the area of multiscale/wavelet methods. The Center brings its members together for meetings and workshops, offers postdoc positions, and you can download publications from the Center. The headquarters of the Wavelet IDR Center is in Madison, Wisconsin.

http://www.waveletidr.org/
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NuHAG Gabor Server

The NuHAG Gabor Server from the Numerical Harmonic Analysis Group (NUHAG) is a private initiative inspired by the success of Wim Sweldens' Wavelet Digest (see above), but not competing with it. It is hoped to be an informal platform for the exchange of ideas related to Gabor analysis, Weyl-Heisenberg frames and its applications. In addition it should be a central node on the Web for those seeking information in Gabor analysis.

http://www.mat.univie.ac.at/~nuhag/Gabor/
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Discovering Wavelets

Edward Aboufadel from Grand Valley State University, Michigan has established a Web site with the goal of being a center of activity for incorporating wavelets into the undergraduate curriculum.

http://faculty.gvsu.edu/aboufade/web/dw.htm

Some of the offerings on the site are:

You can contact Edward Aboufadel if you have contributions for the site.

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Wavelet Software

Click HERE to see a list of wavelet software.

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Wavelet Introductions on the WWW

There has been a surge in growth of wavelet Web-based tutorials. And in these last years several good tutorials have dropped away. I have collected what I know is on the Web for beginners tutorials. If you are a beginner, I suggest looking at more than one of them because it is useful to learn about a new subject from different perspectives. The ordering below has no particular meaning.

(1) "Wavelets: Seeing the Forest and the Trees" by Dana Mackenzie
This article from the National Academy of Sciences in the U.S. is a professionally-produced popular science article about wavelets by Dana Mackenzie written with the assistance of Ingrid Daubechies, Daniel Kleppner, Stephane Mallat, Yves Meyer, Mary Beth Ruskai, and Guido Weiss. Inside you'll find the several sections describing wavelets (no equations) using descriptive words and colorful graphics. This introduction might be good for anyone not wishing to work with wavelets, but who wants to know why they are a valuable tool; for example, this article would be good introduction for journalists. It has a web format and can be downloaded as a PDF, as well.
Blue Button Wavelets: Seeing the Forest and the Trees by Mackenzie
Blue Button and PDF.
(2) "The Little Wave with the Big Future" by P.S. Addison,
This article from Physics World, March 2004, Vol.17(3), 35-39 has the same audience as the above article. It is a professionally-produced popular science article about wavelets with some recent examples of their use, including the author's own work detecting and monitoring the breathing of newborn babies. The Institute of Physics in the UK gave permission for the article to be placed at the authors site in the UK; a PDF is availble to download.
Blue Button The Little Wave with the Big Future by Addison (PDF)
(3) "A Really Friendly Guide to Wavelets" by Valens
C. Valens has written a digital signal engineer's overview of wavelets, covering topics such as the continuous wavelet transform, band-pass filters, scaling function, subband coding. He has a quirky sense of humor too.
Blue Button A Really Friendly Guide ot Wavelets by Valens.
Blue Button Zipped, and Blue Button PDF.
(4) "The Engineer's Ultimate Guide to Wavelet Analysis" by Polikar
Robi Polikar from Iowa State University of Science and Technology has written an introduction to wavelets from an engineering point of view. I truthfully don't know the difference between the engineer's perspective and the non-engineer's perspective of wavelets. Be prepared as this is a graphics-intensive site.
Blue Button Engineer's Guide to Wavelet Analysis by Polikar.
(5) "Surfing the Wavelets."
From the Center of Machine Condition Monitoring (CMCM) in Austrailia. This is my favorite of the WWW tutorial sites listed in this section for introductory wavelet material on the Web. I like it because it has just the right amount of wavelet information, it has the best graphics, and I thought that the examples were interesting. The CMCM closed in 1998, however J. Altmann's tutorial was kindly collected and made available to me and the wavelets community by J. Stecki and his colleagues at Monash University. Wim Sweldens generously placed the tutorial at his wavelets site at Lucent.
Blue Button "Surfing the Wavelets."
(6) "Wavelet Wading Pool" by Alex Nicolaou
Alex Nicolaou has written a Web-based tutorial for the beginner demonstrating wavelets for image compression.
Blue Button Nicolaou's Wavelet Wading Pool.
(7) "Tutorial on Wavelet Analysis of Experimental Data."
Jacques Lewalle has written the most thorough introduction to wavelets site on the WWW. Jacques Lewalle's pages cover twenty-four categories from continuous wavelet transforms, to mean power spectrum to the Morlet transform of modulated oscillations to feature enhancement with filtering and denoising. This site appears to be a LaTeX2HTML version of a written document.
Blue Button Lewalle's Wavelet Analysis of Experimental Data Tutorial.
(8) "A Practical Guide to Wavelet Analysis" by Torrence and Compo
Christopher Torrence and Gilbert P. Compo have written (with a reprint available to download) a practical guide to using wavelets. I believe this information to be quite useful to the scientist beginning to use wavelets as one's new analysis tool. I learned good things from this page. In addition to a tutorial and paper to download, the authors also provide Fortran, Matlab, and IDL continuous wavelet transform code. And don't miss their beautiful continuous wavelet transform demonstration using IDL's ION.
Blue Button A Practical Guide to Wavelet Analysis.
(9) "2D and 3D Progressive Transmission Using Wavelets" by Benj Lipchak
This talk by Benj Lipchak for a computer graphics class makes some preliminary introductions to wavelets, but then it goes quickly into the realm of computer graphics and may be too detailed for the beginning waveleteer. Much of this presentation is derived from the Wavelets Course given at SIGGRAPH 96. The 3D surface material comes both from the above course notes and from the paper by Certain et. al. entitled, "Interactive Multiresolution Surface Viewing," presented at SIGGRAPH 96.
Blue Button Lipchak's 2D and 3D Progressive Transmission Using Wavelets.
(10) "Wavelets and Signal Processing" by Ian Kaplan
These descriptions by Ian Kaplan appear to have started from the author's interest in applying wavelets to financial data, and after he caught the wavelet bug, the web pages grew organically, exponentially ever since. The topics covered include: 1) Applying the Haar Wavelet Transform to Time Series Information 2) wavelets Java source, 3) A Linear Algebra View of the Wavelet Transform 4) Daubechies Wavelets. 5) The Wavelet Lifting Scheme. 6) Spectral Analysis and Filtering with the Wavelet Transform 7) Calculating of the Hurst Exponent using the Wavelet Transform 8) Histogram Smoothing via the Wavelet Transform 9) The Wavelet Packet Transform 10) Wavelet compression, determinism and time series forcasting. The author has an engaging and personal style of writing and his descriptions are clear and interesting.
Blue Button Kaplan's Wavelets and Signal Processing.
(11) "A Gentle Introduction to Wavelets" by Rehmi Post
This paper appears to be a Latex2HTML version of a document written several years ago by Rehmi Post. One might consider this a mathematician's introduction to wavelets.
Blue Button Rehmi Post's A Gentle Introduction to Wavelets.
(12) "Fundamentals of Wavelet Transform Theory" by Andrey Kiselev
This introduction by the Andrey Kiselev of BaseGroup Lab in Russia offers a concise introduction to wavelets, giving a list of their practical applications.
Blue Button Kiselev's Fundamentals of Wavelet Transform Theory
(13) "Wavelets versus Fourier Course" by Martin Vetterli
The purpose of this course is to establish the theory necessary to understand wavelets and related constructions. A particular emphasis will be put on constructions that are amenable to efficient algorithms.
Blue Button Vetterli's Wavelets versus Fourier Course
(14) "A short presentation" by F. Chaplais
This presentation is based/inspired by one of my favorite books: A Wavelet Tour of Signal Processing by S. Mallat. In this presentation, Chaplais covers: Fourier analysis, time-frequency analysis, frames, singularity analysis and reconstruction, and wavelet bases and filter banks.
Blue Button Chaplais' Short Presentation of Wavelet Transform Theory.
(15) "Wavelets Transformationskodierung" by group at FH-Jena
This presentation in German shows the background behind the JPEG2000 standard, that is, wavelet compression (displaying different wavelet compression ratios), discrete wavelet transforms what some of the different wavelets look like. The ideas are conveyed concisely via images and some equations.
Blue Button Wavelets Transformationskodierung.
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Wavelet Beginners Bibliography, Some with Code

Click HERE to see a wavelet beginner's bibliography.

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Wavelet Bibliographies on WWW

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Wavelet WWW Sites

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My Introductory Seminars

If your hunger for beginner's wavelet information is not satiated, I sometimes give introductory wavelet seminars, now to hundreds of people over the years (see my resume for details). The duration of my seminars have ranged from thirty minutes to five hours. My audiences up to now have been astronomers, atmospheric scientists, seismologists, engineers, and computer scientists. My fees are negotiable and flexible; I've been known to give a seminar for the cost of a weekend attendance at Munich hackbrett festival, for example. Locations are similarly flexible; I've given them in computer scientist's homes (Ted Kaehler: to Alan Kay and his Squeak team) to distant Italian islands (Pantelleria). Please contact me if you are interested. (contact me at amara.graps at ifs-roma.inaf.it)

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Personal Correspondence

I am glad that you found this page useful! Please be aware that the Internet can be a very demanding place for individuals like myself who provide pages like this one. I have limited or non-existent time, money, typing hands, Internet access to maintain these Web pages and to answer email. Please don't be offended (i.e. don't take it personally) if I don't respond (or I respond sporadically) to your request for assistance or for making changes to my site or for friendly correspondance. Sometimes I have weeks with no Internet access and most of my life takes precedence over this page. I do file away email messages regarding updates, but I might not make the change that you suggest for a year or two. And if you find these pages are really useful, then please consider a Paypal donation. Grazie!


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