Convert x and y axis units after using continuous wavelet transform (cwt)

I'm analysing some images by running a cwt ('morse' for example) on them. The original image has a scale set to 21.45 pixels/μm, with resolution 1944x2292. After performing the cwt column by column and averaging them, the outputted image looks as so:

I'm having trouble understanding what the units of the x and y axis are representing, as far as I can tell they are representing some scaled unit of time and frequency respectively. If so, what scaling is being done? I would like to end up with both the x and y axis both being measured in μm. Apologies if I have something wrong, I'm new to both MATLAB and this type of analysis.

Answer 1

[Shubh Dhyani]

Hi Harry,

I understand that you are facing a problem interpreting the graph after continous wavelet transform.

When you apply the CWT to an image column by column, you're essentially treating each column as a 1D signal and analyzing its local frequency content. The CWT decomposes a signal into its constituent wavelets at different scales (often interpreted as frequencies) and positions.

In the context of your image:

1. The x-axis typically represents the spatial position in your image, which, given your description, is essentially the column index.
2. The y-axis generally represents the scale (or pseudo-frequency) of the wavelets.

If the x-axis represents column indices, then converting it to micrometers (μm) would involve using the scale you provided (21.45 pixels/μm).

The y-axis is a bit more involved. The exact conversion from wavelet scale to frequency (or wavelength) depends on the specific wavelet being used. For the Morse wavelet in MATLAB, the scales are dimensionless numbers. To translate these scales into real-world units like μm, you'd typically need to understand the relationship between the wavelet scales and the frequencies they represent in the context of your data. If you have a specific reference frequency or wavelength in your data, you can potentially use that to anchor your scale-to-frequency conversion. You can refer to this link for more information:

https://www.mathworks.com/help/wavelet/continuous-wavelet-transforms.html