More Fourier transform stuff.

Consider this image:

This is just a sine wave tipped at 45 degrees.  Now, look at the power spectrum:

Ok, the two spots make sense, as those correspond to the point in the (U,V) plane that has that frequency, along with it's reflection (the negative frequency).  But what's up with all that other crap?

FFTs make the assumption that the signal is periodic, and since it only has the image to work with, the default thing to do is to calculate a circular FFT, where the left and right edges wrap around, and the top and bottom do as well (making a video game torus).  However, since the sine signal doesn't nicely match up at the edges (most obvious at the top and bottom), this introduces a lot of extra Fourier components that correspond to those + shapes.

The solution to this problem is to taper the image down so that it is effectively periodic.  The easiest way to do this is to just multiply by a Gaussian so that the edges drop down to zero at the edges, like this:

Ok, what's this look like in (U,V)-space?

The two dots we expect.  There's some lower level noise, but this has the same colormap as the previous spectrum.  The dots aren't perfect points, which I think tells us something about the size of the Gaussian we used.

This is important, because the science stuff I've been playing with has odd edge effects that impose a strong sinc^2 signal onto the power spectrum.  This makes it difficult to isolate bad frequencies.  Remembering this trick solves that problem, and should make cleaning those images much easier.