Sub-pixel registration and estimation of local shifts directly in the Fourier domain

In this paper, we propose a new approach to sub-pixel registration, and the estimation of local shifts between a pair of images, directly in the Fourier domain. For this purpose, we establish the exact relationship between the continuous and the discrete Fourier phase of two shifted images or their subregions. In particular, we show that the discrete phase difference of two shifted images is a 2-dimensional sawtooth signal, with the exact shifts determined to sub-pixel accuracy by the number of periods of the phase difference along each frequency axis. The sub-pixel portions of the shifts are determined by the non-integer fraction of a period of the phase difference.