Wavelet transform

The wavelet transform is a form of a frequency transform. 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).

Types of wavelet transforms:

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

Important applications are:

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

History:

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

Links:

• Wavelets for Kids (http://www.isye.gatech.edu/~brani/wp/kidsA.ps) (introductory)
• link collection about wavelets (http://www.cosy.sbg.ac.at/~uhl/wav)
• wavelet digest homepage (http://www.wavelet.org/)
• Rob Polikar's Introduction to, and philosophy of, wavelet analysis (http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial)