Another reason why Fourier transforms are magical.

Let's pretend we have an image, and that image is just a bunch of dots:

Here are three, with different numbers and positions for the dots (randomly distributed), except for the centered single dot case.

I've chosen Gaussians for the shape of the dots, because those are simple to work with.  Now, take the 2D Fourier transform of these three images.  What does the power spectrum look like?

Those curly-cue shapes are ~20 orders of magnitude smaller than the peak.  That's what I get for plotting logarithms.

To within the floating point accuracy of my computer, these power spectra are all identical.  No matter where the dots are on the image, the power spectrum is identical.  How does that work?  Basically, the power spectrum just says "the image is this big, and the things that show up on it look like this."  All of the position information is stored in the phase information between the real and imaginary components:

The single dot has a constant phase image.  The other two have phase images that look like random noise, but are actually encoding the position information for each dot.  Next interesting thing:  what happens when we increase the size of the dots:

Bigger dots have a smaller sized dot in the power spectrum, as wider things are dominated by smaller frequency terms (also the fact you can directly calculate that F(Gaussian(sigma)) = Gaussian(1/sigma)).