The continuous wavelet transform

The Morlet-Grossmann definition of the continuous wavelet transform [17] for a 1D signal is:

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:

3.

it is covariant under dilations:

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:

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<#7103,#holschn<#7104] that f(x) can be restored using the formula:

where:

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 .