Collocation for Diffeomorphic Deformations in Medical Image Registration

Diffeomorphic deformation is a popular choice in medical image registration. A fundamental property of diffeomorphisms is invertibility, implying that once the relation between two points A to B is found, then the relation B to A is given per definition. Consistency is a measure of a numerical algorithm's ability to mimic this invertibility, and achieving consistency has proven to be a challenge for many state-of-the-art algorithms. We present CDD (Collocation for Diffeomorphic Deformations), a numerical solution to diffeomorphic image registration, which solves for the Stationary Velocity Field (SVF) using an implicit A-stable collocation method. CDD guarantees the preservation of the diffeomorphic properties at all discrete points and is thereby consistent to machine precision. We compared CDD's collocation method with the following standard methods: Scaling and Squaring, Forward Euler, and Runge-Kutta 4, and found that CDD is up to 9 orders of magnitude more consistent. Finally, we evaluated CDD on a number of standard bench-mark data sets and compared the results with current state-of-the-art methods: SPM-DARTEL, Diffeomorphic Demons and SyN. We found that CDD outperforms state-of-the-art methods in consistency and delivers comparable or superior registration precision.