You are only interested in the low spatial frequencies of your matrix, but you want a high resolution transform. Using fft2 with lots of zero-padding you get a decent resolution but it takes too long. What can you do? Try sft2_low!
This function uses the basic Fourier transform rather than the FFT, but because it is only evaluating it at a small number of points the total execution time is considerably less (it can be 20x faster or more).
The code is fully vectorised, so it's pretty quick even though it's doing a lot of arithmetic. At some point I may generalise the code to allow you to specify your exact frequencies of interest.
The output should be correct, but I recommend testing it and comparing the values with fft2. Please report problems and suggestions in the comments.
Code is now optimized to work specifically on real-valued inputs. It also now optionally gives the frequencies as outputs.