  Image registration is an inherently ill-posed problem that lacks the
constraints needed for a unique mapping between voxels of the two images being
registered. As such, one must regularize the registration to achieve physically
meaningful transforms. The regularization penalty is usually a function of
derivatives of the displacement-vector field, and can be calculated either
analytically or numerically. The numerical approach, however, is
computationally expensive depending on the image size, and therefore a
computationally efficient analytical framework has been developed. Using cubic
B-splines as the registration transform, we develop a generalized mathematical
framework that supports five distinct regularizers: diffusion, curvature,
linear elastic, third-order, and total displacement. We validate our approach
by comparing each with its numerical counterpart in terms of accuracy. We also
provide benchmarking results showing that the analytic solutions run
significantly faster -- up to two orders of magnitude -- than finite
differencing based numerical implementations.
