Wavelets


Wavelets are a relatively recent arrival on the scene, starting to make their mark in both the mathematical and applied communities in the mid 1980s. They provide an alternative to classical Fourier methods for one- and multi-dimensional data analysis and synthesis, and have numerous applications both within mathematics (e.g., to partial differential operators) and in areas as diverse as physics, seismology, medical imaging, digital image processing, signal processing and computer graphics and video.

The most popular and accessible application of wavelets is probably to image compression.



Emmy Noether (original, 25 to 1 and 100 to 1 normalized Haar wavelet compressed images).


See Matlab M-files if you want to generate similar compression pictures for yourself.

Unlike their Fourier cousins, wavelet methods make no assumptions concerning periodicity of the data at hand. As a result, wavelets are particularly suitable for studying data exhibiting sharp changes or even discontinuities. Wavelets allow information to be encoded according to "levels of detail" - in one sense this parallels the way in which we often process information in our everyday lives.

Contrary to popular belief, wavelet basics can be explored keeping the mathematical prerequisites to a minimum - namely, familiarity with the elements of linear algebra. In particular, no knowledge of Fourier analysis is necessary to grasp the main concepts. We first became aware of this, in the case of simple Haar wavelets, via the wonderful paper Wavelets for Computer Graphics: A Primer, by Eric Stollnitz, Tony DeRose and David Salesin at the University of Washington. The accessibilty of advanced wavelet families without Fourier techniques is made abundantly clear by Wim Sweldens' Lifting Scheme.


Publications:

An elementary, student-friendly approach is taken in both of the papers discussed below:

The first version of this was written in two weeks in August 1995, upon returning from SIGGRAPH '95 in Los Angeles.

Note: The above ps file contains some minor corrections to the published MAA version; see the Errata page for details.

"Plotting and Scheming with Wavelets" was awarded a Carl B. Allendoerfer Award (for articles of expository excellence) by the MAA at MathFest '97 in Atlanta, August 1997.

The image from the cover of that issue of Mathematics Magazine is available in both postcript and PDF formats. A related image which did not appear on the cover of Mathematics Magazine - or anywhere else for that matter - can be found here (ps) or here (pdf).

A variation of the above wavelets paper, from an even more elementary point of view, is:

Wavelets themselves are not mentioned at all! More worked details are given for Haar wavelet transformations of matrices. This was written in a week or two in July 1996. It has been claimed that any smart 15 year old could follow this introduction to wavelets.


Getting students to explore wavelets:

Wavelets have been explored by undergraduate mathematics students at Spelman, specifically with a view to image compression, starting with Jimmitria Ford ('97) and Tamara Pearson ('96) in the summer of 1995. (Tamara is now in the computer science doctoral program at the University of Florida at Gainesville.) This work was done with the support of Spelman College's Center For Scientific Applications of Mathematics (CSAM). In the 1996-97 academic year, Camille Daniel ('98) and (Yaletha) Latoria Thomas ('98) took up the baton, funded by MIE and CSAM.


The definitive World Wide Web resource for wavelets is The Wavelet Digest.


Workshops and Short Courses:

A two-hour tutorial on ``Image Compression Using Wavelets'' was given at the International Conference CIMAF 2001 (6th Symposium on Mathematics and 4th Italo-Latinoamerican Conference on Industrial and Applied Mathematics) in Havana, Cuba, 19th-23rd March, 2001. A longer tutorial was given there two years later, at CIMAF in March 2003.

The most recent (four-hour) minicourse introducing wavelets given under the auspices of the MAA was at the Joint Winter Meeting in San Antonio, in January 1999.

An invited talk on Wavelets - a new tool for imaging, graphics and Hollywood was given on Friday 19th March 1999, during the 16th Annual Conference on Undergraduate Mathematics at Rose-Hulman Institute of Technology (in Terre Haute, IN).

Two full-day workshops on wavelets, the first listed aimed at mathematicians, and the second at electrical engineers, were given in 1998 in Ireland:




The first (half-day) wavelets short course given was:


Symposium:

On Tuesday, 17th February 1998, a half-day (three hour) wavelet symposium entitled Wavelets: The Latest Big Splash in Science, Engineering, Imaging and Graphics (co-chaired with Farid Dowla of Livermore Lab) was held, in Philadelphia, as part of the AAAS Annual Meeting and Science Innovation Exposition.

For detailed information, click here.


(This is http://www.spelman.edu/~colm/wav.html, click here to return to main page.)

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