  Deterministic approaches using iterative optimisation have been historically
successful in diffeomorphic image registration (DiffIR). Although these
approaches are highly accurate, they typically carry a significant
computational burden. Recent developments in stochastic approaches based on
deep learning have achieved sub-second runtimes for DiffIR with competitive
registration accuracy, offering a fast alternative to conventional iterative
methods. In this paper, we attempt to reduce this difference in speed whilst
retaining the performance advantage of iterative approaches in DiffIR. We first
propose a simple iterative scheme that functionally composes intermediate
non-stationary velocity fields to handle large deformations in images whilst
guaranteeing diffeomorphisms in the resultant deformation. We then propose a
convex optimisation model that uses a regularisation term of arbitrary order to
impose smoothness on these velocity fields and solve this model with a fast
algorithm that combines Nesterov gradient descent and the alternating direction
method of multipliers (ADMM). Finally, we leverage the computational power of
GPU to implement this accelerated ADMM solver on a 3D cardiac MRI dataset,
further reducing runtime to less than 2 seconds. In addition to producing
strictly diffeomorphic deformations, our methods outperform both
state-of-the-art deep learning-based and iterative DiffIR approaches in terms
of dice and Hausdorff scores, with speed approaching the inference time of deep
learning-based methods.
