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 $${psi_{jk}:j,kin} by means of dyadic translations and dilations of $$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