Mexican hat transform
Rather than using the dilation coefficient (“a” in much of the literature), I started off using a `waveletnumber’ (k, a pseudo-frequency) until a tortuous path (most folks won’t want to read these papers…) led me to a simpler presentation. That is the one used here. Among its advantages are:
- Relation to the Gaussian filter: the wavelet transform is a difference of two filtered fields at nearby scales, divided by the linear or logarithmic difference in scales.
- The use of the “scale” of dimensions t2 simplifies substitutions in PDEs
- More importantly, the use of “scale” leads directly to a “multiscale” representation of the field (thus providing a simple implementation of Eyink’s 2005 idea)
Pdf document; the analytical relations (Parseval, inverse) are established in the Maple worksheet.
Corresponding software.
- Wavelet transform: compare the result with the conventional version: the information is the same, but here the wavelet coefficients are more simply related to their energy content, so we see the significant events.
- Inverse transform: modify the cutoff frequency in the code, and note how the spectral spread of the Mexican hat wavelet (good temporal resolution, poor spectral resolution) requires the inclusion of scales small than Nyquist’s; see also how 2 voices per octave give good accuracy (unlike Morlet, for which its good spectral resolution makes it miss information at this resolution).
- Spectra: the poor spectral resolution of the wavelet gives a very smooth version of the mean power spectrum, which must be interpreted carefully. See the discussion of Fourier spectra for the justification of the f.E(f) vs log(f) plots: all energies normalized per unit length of signal.
