All wavelet transforms consider a function (taken to be a function of time) in terms of oscillations which are localized in both time and frequency. Wavelet transforms are most broadly classified into the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT).

As far as applications are concerned, the DWT is used for signal coding whereas the CWT is used for signal analysis. Consequently, the DWT is commonly used in engineering and computer science and the CWT is most often used in scientific research. Wavelet transforms are now being adopted for a vast number of different applications, often replacing the conventional Fourier transform in many applications. Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations, astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics, as well as many other fields including image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis.

In historical terms, the development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Goupillaud, Grossman and Morlet's formulation of what is now known as the CWT (1982), Jan Olov-Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Delprat 's time-frequency interpretation of the CWT (1991), Newland 's Harmonic wavelet transform and many others since.

Wavelet theory is related to several other subjects. All wavelet transforms may be considered to be forms of time-frequency representation and are, therefore, related to the subject of harmonic analysis. Discrete wavelet transforms are a form of finite impulse response filter. The wavelets forming a CWT are subject to Heisenberg's uncertainty principle and, equivalently, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.

One of the most important uses of wavelets are in data compression. Like several other transforms, the wavelet transform can be used to transform raw data (like images), then encode the transformed data, resulting in effective compression. JPEG 2000 is an image standard that uses wavelets.

See also

• Filter banks

List of Wavelets

Discrete wavelets

• Beylkin (18)
• Coiflet (6, 12, 18, 24, 30)
• Daubechies wavelet (2, 4, 6, 8, 10, 12, 14, 16, 18, 20)
• Haar wavelet
• Vaidyanathan filter (24)
• Symmlet

Continuous wavelets

• Mexican hat wavelet
• Hermitian wavelet
• Hermitian hat wavelet
• Complex Mexican hat wavelet
• Morlet wavelet
• Modified Morlet wavelet
• Addison wavelet
• Hilbert-Hermitian wavelet

References

• Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0750306920
• Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0898712742
• Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1568810415
• P. P. Viadyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0136057187

External links

• Wavelet Digest
• Wavelet Posting Board
• The Wavelet Tutorial by Polikar
• OpenSource Wavelet C Code
• An Introduction to Wavelets
• Filter Coefficients of Popular Wavelets
• Wavelet-based time-frequency analysis in Mathematica