Inferring motion from the rank constraint of the phase matrix

We investigate the rank constraint of the discrete phase difference, and derive its exact parametric model. We show that the discrete phase difference of two shifted images, or their subregions, is a 2-dimensional sawtooth signal. This allows us to determine the motion parameters to subpixel accuracy by simply counting the number of cycles of the phase difference along each frequency axis. The subpixel portion is given by the non-integer fraction of the last cycle along each axis. The problem is formulated as a homogeneous cost function under rank constraint for the phase matrix, and the shape constraint for the filter that computes the group delay, and is solved using a robust technique.