Subpixel Alignment of MRI Data Under Cartesian and Log-Polar Sampling

Magnetic resonance imaging (MRI) allows numerous Fourier domain sampling schemes such as Cartesian and non-Cartesian trajectories (e.g. Polar, circular, and spherical). On the other hand, it provides directly the Fourier spectrum of the field of view (FOV) in the corresponding sampling scheme. Motivated by these characteristic features of MRI, we have developed a new scheme for direct Fourier domain registration of MRI data based on the phase difference matrix. We derive the exact relationship between the continuous and the discrete Fourier phase-difference for Cartesian or polar sampling schemes, and demonstrate that in each case the discrete phase difference is a 2D sawtooth signal. Subpixel alignment under rotation, translation and scale variation is then established simply by counting the number of cycles of the sawtooth signal. 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)