Figure 4 shows an example of a cross-correlation function for a sum of squared differences. The reference image is a bright galaxy, and the compared image is a shifted version of the reference with a translation vector of (+5,-5). This plot clearly shows that the cross-correlation criterion is minimal at the expected point. A search for a local minimum will yield the right translation vector.
At this point, a subpixel offset detection is applied by fitting a parabola to the cross-correlation signal in x and y, and looking for the subpixel minimum. This method is theoretically precise up to a 1/100 pixel precision in the ideal case of noiseless images. The precision obtained with SOFI data is about 1/10 pixel.