  Classical deformable registration techniques achieve impressive results and
offer a rigorous theoretical treatment, but are computationally intensive since
they solve an optimization problem for each image pair. Recently,
learning-based methods have facilitated fast registration by learning spatial
deformation functions. However, these approaches use restricted deformation
models, require supervised labels, or do not guarantee a diffeomorphic
(topology-preserving) registration. Furthermore, learning-based registration
tools have not been derived from a probabilistic framework that can offer
uncertainty estimates.
  In this paper, we build a connection between classical and learning-based
methods. We present a probabilistic generative model and derive an unsupervised
learning-based inference algorithm that uses insights from classical
registration methods and makes use of recent developments in convolutional
neural networks (CNNs). We demonstrate our method on a 3D brain registration
task for both images and anatomical surfaces, and provide extensive empirical
analyses. Our principled approach results in state of the art accuracy and very
fast runtimes, while providing diffeomorphic guarantees. Our implementation is
available at http://voxelmorph.csail.mit.edu.
