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

Diffeomorphic image registration (DIR) is a critical task in 3D medical image analysis, aimed at finding topology preserving deformations between pairs of images. Focusing on the solution of the flow map differential equation as the diffeomorphic deformation, recent methods use discrete timesteps along with various regularization terms to penalize the negative determinant of Jacobian and impose smoothness of the solution vector field. In this paper, we propose a novel learning-based approach for diffeomorphic 3D-image registration which finds the diffeomorphisms in the time continuum with fewer regularization terms and no additional integration. As one of the fundamental properties of flow maps, we exploit the semigroup property as the only form of regularization, ensuring temporally continuous diffeomorphic flows between pairs of images. Leveraging this property, our method alleviates the need for additional regularization terms and scaling and squaring integration during both training and evaluation. To achieve time-continuous diffeomorphisms, we employ time-embedded UNets, a technique commonly utilized in diffusion models. The proposed method reveals that ensuring diffeomorphism in a continuous time interval leads to better registration results. Experimental results on two public datasets (OASIS and CANDI) demonstrate the superiority of our model over both learning-based and optimization-based methods.