I have now added symmetry options to my Fourier Transform painting application. As PC pointed out in a comment to my previous post, examples with higher frequency information are more interesting. Here are some images of the background tile of this website. The magnitude is on the left and the phase is on the right. You can click them for the full-sized images.
This is the original:
This is the Fourier transform of the original:
This is my edited version of that transform. I wiped out lots of the off-axis frequencies. I tinted the center red. And, I added some green blotches along the diagonals.
And, here is the inverse transform of my edited data.