Estimating sub-pixel shifts directly from the phase difference

In this paper, we establish the exact relationship between the continuous and the discrete phase-difference of two shifted images, and show that their discrete phase difference is a 2-dimensional sawtooth signal. As a result, the exact shifts between two images can be determined to sub-pixel accuracy by counting the number of cycles of the phase difference matrix along each frequency axis. The sub-pixel portion is represented by a fraction of a cycle corresponding to the non-integer part of the shift. The problem is formulated as an over-determined system of equations, and is solved by imposing a regularity constraint, using the method of generalized cross validation (GCV).