Wavelet transform - Definition and Overview

The wavelet transform is a transformation to basis functions that are localized in frequency (similar in that sense to Fourier-related transforms). As basis functions one uses wavelets. The big advantage over the Fourier transform is the temporal (or spatial) locality of the base functions (see also short-time Fourier transform) and the smaller complexity (O(N) instead of O(N log N) for the fast Fourier transform (where N is the data size)).

In the likeness of the uncertainty principle the restriction for wavelet transform resolution can be written down:

<math> \Delta x\Delta\omega \ge \frac{1}{4\pi}<math>

and this result better in <math> 8\pi<math> times as compared to the Fourier transform Important applications are:

• image compression and video compression: wavelet compression
• Solving differential equations
• signal processing

Types of wavelet transforms:

• continuous wavelet transform (CWT)
• discrete wavelet transform (DWT)
• fast wavelet transform (FWT)
• wavelet packets
• complex wavelet transform

History

• First wavelet (Haar wavelet) by Alfred Haar (1909)
• Since the 1950s: Jean Morlet and Alex Grossman
• Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser

External links

• Wavelets for Kids (PDF file) (http://www.isye.gatech.edu/~brani/wp/kidsA.pdf) (introductory)
• link collection about wavelets (http://www.cosy.sbg.ac.at/~uhl/wav.html)
• wavelet digest homepage (http://www.wavelet.org/)
• Rob Polikar's Introduction to, and philosophy of, wavelet analysis (http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html)
• a really friendly guide to wavelets (http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html)
• Wavelet forums (French) (http://www.ondelette.com/index.html)
• Wavelet forum (English) (http://www.ondelette.com/indexen.html)