Pattern recognition using 2D fft in matlab
DFT is a great tool to find periodic patterns in signals.  However, you should understand what kind of period you want to find. If you want to find a series of points that lie on a regular grid (with even spacing), or more mathematically speaking, a subset of points $(x_i,y_i)$, such that there exists $(dx,dy)$ for which the following equations hold: 

$x_i = x_1 + dx*i $
   $ y_i =y_1 + dy*j $ 

(The patterns can also be rotated, and DFT will still find them). 

You should compute a 2D histogram of your data, with large number of bins, and run DFT on it. Afterwards, you should look for a peak in any location expect the DC. An ideal noise should have no patters, thus the contrast of the frequencies responses should be very low.