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

where

- 1.
- it is a linear transformation,
- 2.
- it is covariant under translations:

(14.2)

- 3.
- it is covariant under dilations:

(14.3)

In Fourier space, we have:

(14.4) |

When the scale

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.