If you want only the spectrum from within the green circle regions, you can just zero out everything outside of those. You can either use drawcircle to manually do it (demo attached), or find some automatic way to find the spikes. See attached demos.
The "crosses" are caused by the finite size of your image and the fact that the intensity at the top of the image is substantially different than the bottom. To reduce that you can use a Hamming window to taper down the intensity near the edges. However if you're zeroing out everything outside of the green circles, there won't be a mismatch at the top and bottom since the intensities will now be zero.
