Inspired by Henrik's idea, I came up with the following scheme (magnitude on the left, magnitude + phase ramps on the right):

I take the phase mod Pi/2 to get four sets of phase ramps and after mapping into the range [0,1] I also raise to the 10th power so the ramps, when added to the magnitudes, don't distract from the magnitudes.

Thanks! I should have emphasized that the plot also shows the magnitude (which diverges in your example). Here's how my method renders your function:

My functions don't diverge as they are defined by the Fourier transform. Here's how I've been generating them:

supp = 10; size = 400; prob = .2;

image = RotateLeft[ PadRight[ RandomInteger[BernoulliDistribution[prob], {supp, supp}], {size, size}], Floor[{supp, supp}/2]];

fourier = RotateRight[Fourier[image], Floor[{size, size}/2]];