  Unsupervised deep-learning (DL) models were recently proposed for deformable
image registration tasks. In such models, a neural-network is trained to
predict the best deformation field by minimizing some dissimilarity function
between the moving and the target images. After training on a dataset without
reference deformation fields available, such a model can be used to rapidly
predict the deformation field between newly seen moving and target images.
Currently, the training process effectively provides a point-estimate of the
network weights rather than characterizing their entire posterior distribution.
This may result in a potential over-fitting which may yield sub-optimal results
at inference phase, especially for small-size datasets, frequently present in
the medical imaging domain. We introduce a fully Bayesian framework for
unsupervised DL-based deformable image registration. Our method provides a
principled way to characterize the true posterior distribution, thus, avoiding
potential over-fitting. We used stochastic gradient Langevin dynamics (SGLD) to
conduct the posterior sampling, which is both theoretically well-founded and
computationally efficient. We demonstrated the added-value of our Basyesian
unsupervised DL-based registration framework on the MNIST and brain MRI (MGH10)
datasets in comparison to the VoxelMorph unsupervised DL-based image
registration framework. Our experiments show that our approach provided better
estimates of the deformation field by means of improved mean-squared-error
($0.0063$ vs. $0.0065$) and Dice coefficient ($0.73$ vs. $0.71$) for the MNIST
and the MGH10 datasets respectively. Further, our approach provides an estimate
of the uncertainty in the deformation-field by characterizing the true
posterior distribution.
