Wavelet series

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

Formal definition

A function $psiin\; L^2(mathbb\{R\})$ is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space $L^2(mathbb\{R\})$ of square integrable functions. The Hilbert basis is constructed as the family of functions by means of dyadic translations and dilations of $psi,$,

:$psi\_\{jk\}(x)\; =\; 2^\{j/2\}\; psi(2^jx-k),$

for integers $j,kin\; mathbb\{Z\}$. This family is an orthonormal system if it is orthonormal under the inner product

:$langlepsi\_\{jk\},psi\_\{lm\}\; angle\; =\; delta\_\{jl\}delta\_\{km\}$

where $delta\_\{jl\},$ is the Kronecker delta and $langle\; f,g\; angle$ is the standard inner product $langle\; f,g\; angle\; =\; int\_\{-infty\}^infty\; overline\{f(x)\}g(x)dx$ on $L^2(mathbb\{R\}).$ The requirement of completeness is that every function $fin\; L^2(mathbb\{R\})$ may be expanded in the basis as

:$f(x)=sum\_\{j,k=-infty\}^infty\; c\_\{jk\}\; psi\_\{jk\}(x)$

with convergence of the series understood to be convergence in the norm. Such a representation of a function "f" is known as a wavelet series. This implies that an orthonormal wavelet is self-dual.

Wavelet transform

The integral wavelet transform is the integral transform defined as

:$left\; [W\_psi\; f\; ight]\; (a,b)\; =\; frac\{1\}\{sqrt\{|a|int\_\{-infty\}^infty\; overline\{psileft(frac\{x-b\}\{a\}\; ight)\}f(x)dx,$

The wavelet coefficients $c\_\{jk\}$ are then given by

:$c\_\{jk\}=\; left\; [W\_psi\; f\; ight]\; (2^\{-j\},\; k2^\{-j\})$

Here, $a=2^\{-j\}$ is called the binary dilation or dyadic dilation, and $b=k2^\{-j\}$ is the binary or dyadic position.

General remarks

Unlike the Fourier transform, which is an integral transform in both directions, the wavelet series is an integral transform in one direction, and a series in the other, much like the Fourier series.

The canonical example of an orthonormal wavelet, that is, a wavelet that provides a complete set of basis elements for $L^2(mathbb\{R\})$, is the Haar wavelet.

ee also

* Continuous wavelet transform
* Discrete wavelet transform
* Complex wavelet transform
* Dual wavelet
* Multiresolution analysis
* JPEG 2000, a wavelet-based image compression standard
* Some people generate spectrograms using wavelets, called scalograms. Other people generate spectrograms using a short-time Fourier transform
* Chirplet transform
* Time-frequency representation

References

* Charles K. Chui, "An Introduction to Wavelets", (1992), Academic Press, San Diego, ISBN 0121745848

External links

* cite web
author = Robi Polikar
date= 2001-01-12
title = The Wavelet Tutorial
url = http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html