    Next: Examples of Wavelets Up: The Wavelet Transform Previous: Introduction

The continuous wavelet transform

The Morlet-Grossmann definition of the continuous wavelet transform  for a 1D signal is: (14.1)

where z* denotes the complex conjugate of z, is the analyzing wavelet, a (>0) is the scale parameter and b is the position parameter. The transform is characterized by the following three properties:
1.
it is a linear transformation,
2.
it is covariant under translations: (14.2)

3.
it is covariant under dilations: (14.3)

The last property makes the wavelet transform very suitable for analyzing hierarchical structures. It is like a mathematical microscope with properties that do not depend on the magnification.

In Fourier space, we have: (14.4)

When the scale a varies, the filter is only reduced or dilated while keeping the same pattern.

Now consider a function W(a,b) which is the wavelet transform of a given function f(x). It has been shown [#grossmann<#14252,#holschn<#14253] that f(x) can be restored using the formula: (14.5)

where: (14.6)

Generally , but other choices can enhance certain features for some applications.

The reconstruction is only available if is defined (admissibility condition). In the case of , this condition implies , i.e. the mean of the wavelet function is 0.    Next: Examples of Wavelets Up: The Wavelet Transform Previous: Introduction
Petra Nass
1999-06-15