  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.
