Wavelet?
Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, and earthquake prediction. This paper introduces wavelets to the interested technical person outside of the digital signal processing field. I describe the history of wavelets beginning with Fourier, compare wavelet transforms with Fourier transforms, state properties and other special aspects of wavelets, and finish with some interesting applications such as image compression, musical tones, and de-noising noisy data.
More Wavelet?
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 we use to look at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with a large "window," we would notice gross features. Similarly, if we look at a signal with a small "window," we would notice small features. The result in wavelet analysis is to see both the forest and the trees, so to speak. This makes wavelets interesting and useful. For many decades, scientists have wanted more appropriate functions than the sines and cosines which comprise the bases of Fourier analysis, to approximate choppy signals (1). By their definition, these functions are non-local (and stretch out to infinity). They 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 same 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 include 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.
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Fingerprints Go Digital by Christopher M. Brislawn, in the Notices of the A.M.S., November 1995. (PDF file)
Fingerprint references from the Southern California Association of Fingerprint Officers.
GFF Format Summary: FBI Fingerprint Format -- very technical.
Wavelet Image File Format -- who needs GIF?
JPEG 2000 (wavelet-based) -- who needs the old JPG?
Wavelet-based Image Compression System by David Wanqian Liu.
Wavelet-based Image Search Engine -- by James Z. Wang and Gio Wiederhold.
Listening for Defects: Wavelet-Based Acoustical Signal Processing in Japan by Mei Kobayashi, in SIAM News, Vol. 29, No. 2, March 1996.
Wavelet Applications Come to the Fore by Barry Cipra, in SIAM News, Volume 26, Number 7, November 1993.
Wavelet Image Compression: Beating the Bandwidth Bottleneck By Peter Schr�der, in Wired Vol. 3, No. 5, May 1995.
Wavelet in Computer Graphics: A Primer.
An Introduction to Wavelets, by Amara Graps.
Plotting & Scheming with Wavelets, by Colm Mulchay, a Postscript File of his Mathematics Magazine article, along with an Errata Page.
Wavelet Resources Web Page, managed by Mathsoft.
Washington University's Wavelet NetCare with lots of links!
Discussion on Haar Wavelets by Don Morgan of Embedded.com.
The World According to Wavelets : The Story of a Mathematical Technique in the Making a book by Barbara Burke Hubbard. (2nd edition)
Wavelet Explorer for Mathematica.
A Wavelet Tutorial (based on S. Mallat's book)
Gilbert Strang's Home Page. Strang�s idea to connect Haar wavelets with real-valued vectors motivated our work.