2.3.1 Interpretation of the CWT as a singularity scanner

In order to get a physical interpretation of the CWT, we notice that in signal analysis, as in classical electromagnetism, the L ² norm is interpreted as the total energy of the signal. Therefore, the relation (2.24) suggests we interpret S( , a, ?) ² as the energy density in the wavelet parameter space [284].

Assume now, as in 1-D, that the wavelet ? is fairly well localized both in position space ( ) and in spatial frequency space ( ). Then so is the transformed wavelet , with effective support suitably translated by , rotated by ? and dilated by a. Because (2.19) is essentially a convolution with a function ? of zero mean, the transform S( , a, ?) is appreciable only in those regions of parameter space ( , a, ?) where the signal is. Thus we get an appreciable value of S only where the wavelet matches the features of the signal s. In other words, the CWT acts on a signal as a local filter in all four variables , a, ?: S( , a, ?) sees only that portion of the signal that lives around , a, ? and filters out the rest. Therefore, if the wavelet is well localized, the energy density of the transform...