Learning Diffeomorphism for Image Registration with Time-Continuous Networks using Semigroup Regularization

Diffeomorphic image registration (DIR) is a fundamental task in 3D medical image analysis that seeks topology-preserving deformations between image pairs. To ensure diffeomorphism, a common approach is to model the deformation field as the flow map solution of a differential equation, which is solved using efficient schemes such as scaling and squaring along with multiple smoothness regularization terms. In this paper, we propose a novel learning-based approach for diffeomorphic 3D image registration that models diffeomorphisms in a continuous-time framework using only a single regularization term, without requiring additional integration. We exploit the semigroup property-a fundamental characteristic of flow maps-as the sole form of regularization, ensuring temporally continuous diffeomorphic flows between image pairs. Leveraging this property, we prove that our formulation directly learns the flow map solution of an ODE, ensuring continuous inverse and cycle consistencies without explicit enforcement, while eliminating additional integration schemes and regularization terms. To achieve time-continuous diffeomorphisms, we employ time-embedded UNets, an architecture commonly used in diffusion models. Our results demonstrate that modeling diffeomorphism continuously in time improves registration performance. Experimental results on four public datasets demonstrate the superiority of our model over state-of-the-art diffeomorphic methods. Additionally, comparison to several recent non-diffeomorphic deformable image registration methods shows that our method achieves competitive Dice scores while significantly improving topology preservation.