Why are the Continuous Wavelet Transforms of the same signal drastically different?

Question

I'm currently studying wavelets and am running into confusion with regards to CWT coefficients. Ideally, I want a CWT algorithm that produces outputs similar to that of a STFT - i.e. produces coefficients that correspond to frequency magnitudes of the input signal. However, I'm finding conflicting results.

From the CWT Wikipedia page, I'm given the illustration below. This illustration shows that the wavelet coefficients are essentially just sampling different subbands of the signal. This is not what I'm looking for.

However, when I look at a MATLAB web page discussing the CWT and DWT, I find a different illustration of the CWT when operating on the same input signal. This diagram looks more similar to what I was hoping for where each frequency has a constant coefficient.

What is the difference between CWTs?

Answer 1

There are differences in axis-scales. However, I suspect that the main difference is that the first picture relates to either a real CWT (or time-frequency transform) or the real-part of a complex CWT (or time-frequency transform, again), while the second seems to be a magnitude.

In the first image, the apparently wiggling part could "planified" by combining it to some imaginary part. Then changing the $y$-scale from linear to logarithmic could make the plots look closer.

Apart from representation issues (scales, magnitudes), with admissible continuous wavelets, there exists a wide variety thereof. Factors that affect scalograms in practice are also the global shape of wavelets (symmetry and anti-symmetry for instance), their regularity, and the ways it is discretized (number and sampling of voices).