Stepwise Inverse Consistent Euler's Scheme for Diffeomorphic Image Registration

Theoretically, inverse consistency in an image registration problem can be achieved by employing a diffeomorphic scheme that uses transformations parametrized by stationary velocity fields (SVF). The displacement from a given SVF, formulated as a series of self compositions of a transformation function, can be obtained by Euler integration in the time domain. However in practice, the discrete time integration produces results that are inverse inconsistent, and inverse consistency in the final solution needs to be explicitly ensured. One way of achieving this is to penalize the endpoint displacement offset obtained by evaluating a composition of the transformation with its inverse at an arbitrary point. In this paper, we propose a variation in which the displacement penalization is required only in the first composition step of the transformation thereby bringing down the computational complexity. We compare these two ways of enforcing inverse consistency by applying the registration framework on four pairs of brain magnetic resonance images. We observe that the proposed stepwise scheme maintains both precision and level of inverse consistency similar to the endpoint scheme.