Adaptive time-stepping in diffeomorphic image registration with bounded inverse consistency error

In a continuous setting, diffeomorphisms generated by stationary velocity fields (SVF) are invertible transformations with differentiable inverses. However, due to the numerical integration of the velocity field, inverse consistency is not achieved in practice. In SVF based image registration, inverse consistency is therefore often enforced through a penalty term. Existing penalty terms penalize the inverse consistency error generated by the composition of the forward and backward transformations. However, in such terms, a higher consistency requirement pushes the transformation towards linearity due to the discretization involved and fixed number of integration time-steps. In this paper, we propose a method to both penalize inverse consistency error and to adaptively set the number of integration time-steps required, so that the predicted maximum inverse consistency error is bounded, taking into account discretization errors. This formulation allows more flexibility in the transformation model to realize complex deformations while still achieving the desired level of inverse consistency. Using synthetic examples, we show that the measured inverse consistency and the predicted inverse consistency match. Also, the proposed method is able to achieve more accurate image registration. On the MGH10 dataset, the Jaccard index of the proposed method on inter-subject registration reaches the same level as the registration scheme using a fixed-time step and the conventional penalty term while using a lower number of integration time-steps, thus saving on the computational time.