The Morlet-Grossmann definition of the continuous wavelet transform [17] for a 1D signal is:
where 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:
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 which is the wavelet transform of a given function . It has been shown [,] that 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 0.