The wavelet map is a graphical representation of the function
(sometimes called the wavelet coefficients) of two variables
and for a given wavelet. Large values of the function (the
so-called wavelet coefficients) reflect the combined effects of
a large fluctuation of the signal f at this time and of a
good matching of shape between the signal and the wavelet. It
can be noted that the Mexican hat wavelet used so far isolates
local minima and maxima of the signal at the selected duration.
This feature must be kept in mind when interpreting the wavelet map.
For example when we use the antisymmetric wavelet (Fig. 7)
and repeat the construction of the wavelet map, we observe many similarities and a general `phase shift' associated with the new emphasis on slopes of the signal (Fig. 8).
In fact, 
specializes in
the extraction of gradients from the signal.
Analytically, we get
The relation between the Gaussian wavelet transforms (those
with and
) and successive derivatives of a smoothed
signal is easily established. Define the smoothing by a
Gaussian kernel of duration 
by the formula
is not a wavelet, since the area under the
curve is non-zero.
Differentiations of 
and
integrations by parts yield, successively:
is the second derivative of the smoothed signal.