Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity

Abstract

In this paper a variational method for registering or mapping like points in medical images is proposed and analyzed. The proposed variational principle penalizes a departure from rigidity and thereby provides a natural generalization of strictly rigid registration techniques used widely in medical contexts. Difficulties with finite displacements are elucidated, and alternative infinitesimal displacements are developed for an optical flow formulation which also permits image interpolation. The variational penalty against non-rigid flows provides sufficient regularization for a well-posed minimization and yet does not rule out irregular registrations corresponding to an object excision. Image similarity is measured by penalizing the variation of intensity along optical flow trajectories. The approach proposed here is also independent of the order in which images are taken. For computations, a lumped finite element Eulerian discretization is used to solve for the optical flow. Also, a Lagrangian integration of the intensity along optical flow trajectories has the advantage of prohibiting diffusion among trajectories which would otherwise blur interpolated images. The subtle aspects of the methods developed are illustrated in terms of simple examples, and the approach is finally applied to the registration of magnetic resonance images.