Nature will quite readily calculate the 2D Fourier transform and its inverse for us without the use of a digital computer.
This is because lenses effectively do a Fourier transform at the focal point. With the right setup, you can apply filters at the focal point and get pretty much exactly what you would expect. An example of such a setup is the 4F Correlator. [0]
Fourier optics is a whole subfield within optics, and it really is rather fascinating.
Exposure to Fourier optics really helped develop my intuition around the Fourier transform.
This is how Electron Crystallograpy works. You can choose to get half the Fourier transform (aka the diffraction pattern) with phase information lost, or use a secondary lens to get the full picture back after correction. It's quite magical.
Then you can do FT on that final image on a computer and then modify that pattern in reciprocal space to fix flaws with the image like astigmatism and noise.
I was introduced to this idea in a really cool video about using Fourier Optics for optical pattern recognition.[0] The video happens to have one of the best explanations of Fourier transforms I've yet encountered.
This is because lenses effectively do a Fourier transform at the focal point. With the right setup, you can apply filters at the focal point and get pretty much exactly what you would expect. An example of such a setup is the 4F Correlator. [0]
Fourier optics is a whole subfield within optics, and it really is rather fascinating.
Exposure to Fourier optics really helped develop my intuition around the Fourier transform.
[0] https://en.wikipedia.org/wiki/Fourier_optics#4F_Correlator