Dear Paramonte, with respect to naming this is always a hard decision. The acronym CWT is well established so users look for it. Users also expressed some confusion at why we had CWT and CWTFT, and which one should they use. The new CWT algorithm also lends itself to an inverse algorithm which the old one did not, so now we have CWT and ICWT for the inverse. The ICWT accepts a frequency or period band input so you can do a frequency-localized reconstruction.
As I suggested, we specifically implemented the new CWT in a way that you can simply call the old one if you wish.
I would welcome a detailed discussion with you as I offered. But we have seen many users struggle with scale and specifying scales. For example, users will specify scales without taking into account the support of the analyzing wavelet. As a result they would have many scales where the wavelet extended well beyond the extent of their data. So in fact, they had no wavelet at all (there was not even one oscillation in the extent of the data). Or they would similarly use scales which were too small and therefore did not provide any wavelet analysis.
Also, as you are no doubt aware, the scale-to-frequency conversion is not equally good for all wavelets. For the Morse wavelets and the default we use, it is quite good because that wavelet is symmetric and well-localized in the Fourier domain. That is also true of the 'bump' wavelet and 'amor' (Morlet) wavelets supported by the new CWT.
For example, here is the default Morse wavelet in the Fourier domain with a vertical line at its predicted center frequency
omega = 0:0.001:(2*pi);
psihat = omega.^20.*0.0051.*exp(-omega.^3);
plot(omega,psihat); grid on; hold on;
title('Default Morse Wavelet -- Fourier Domain');
plot([(20/3)^(1/3) (20/3)^(1/3)],[0 2.5],'k')
You see that this is perfectly symmetric in the Fourier domain. That is certainly not the case with most of the orthogonal and biorthogonal wavelets, like the 'db' family. In those cases, scale to frequency is not so well defined because the wavelet is not symmetric in the Fourier domain and the bandwidth is much larger.
Also, in the new CWT we use L1 normalization which is much more natural in time-frequency analysis than the L2 normalization used in the older CWT. With L1 normalization, a sine wave of amplitude A, results in wavelet coefficients at the appropriate scale with magnitude A. In the older CWT, the magnitude of an oscillatory component in the wavelet coefficients depended on the scale.
We believe there are many other advantages of the new implementation in addition to the few I have mentioned. I would be very happy to discuss those and compare and contrast those at length with you.
Thank you again for your interest and for engaging in this way. As you said it is helpful for all.