Phase Correlation Algorithm

Idea: To exploit the link between (spatial) shift and (Fourier) phase of two blocks

How does it works?: Let b₀(x,y) and b₁(x,y) be two aligned blocks that satisfy b₁(x,y)= b₀(x-d_x,y-d_y), where b₀(x,y) is the reference block, M x M is the block size while d=(d_x,d_y) is the motion vector. Then, their normalized cross power spectrum is:

where ^, Arg and * indicate the Fourier transform, the phase the complex conjugate. The inverse FFT of the equality above gives the phase correlation surface that has a peak in correspondence to the motion d.

Drawback: There may be more peaks when there is more than one motion in the block (with a difficulty in the automatic estimation)

Improvements:

Motion estimation in the DCT domain: oriented to avoid complex numbers
Motion estimation in the Fourier domain: oriented to avoid the inverse FFT (The proposed model)

Reference:

C. D. Kuglin and D. C. Hines, The phase correlation image alignment method, Proc. of IEEE Int. Conf. Systems, Man and Cybernetics, pp. 163-165, September 1975.