Interpretation:



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Interpretation:

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)




Figure 7: The antisymmetric wavelet g1.


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).



Figure 8: g1-wavelet map of the cosine picks out gradients of the signal.

In fact, specializes in the extraction of gradients from the signal.

Analytically, we get



and the `phase shift' between the signal and transform is clear from the presence of the sine function.

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



where



For our cosine signal, we have



Note that is not a wavelet, since the area under the curve is non-zero.

Differentiations of and integrations by parts yield, successively:




i.e. the slope of the smoothed signal at scale k, and



is the second derivative of the smoothed signal.



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Jacques Lewalle
Mon Nov 13 10:51:25 EST 1995