Subpixel estimation of shifts directly in the Fourier domain

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 two-dimensional sawtooth signal. Subpixel registration can, thus, be performed directly in the Fourier domain by counting the number of cycles of the phase difference matrix along each frequency axis. The subpixel portion is given by the noninteger fraction of the last cycle along each axis. The problem is formulated as an overdetermined homogeneous quadratic cost function under rank constraint for the phase difference, and the shape constraint for the filter that computes the group delay. The optimal tradeoff for imposing the constraints is determined using the method of generalized cross validation. Also, in order to robustify the solution, we assume a mixture model of inlying and outlying estimated shifts and truncate our quadratic cost function using expectation maximization.