Unsupervised Deep-Learning Based Deformable Image Registration: A Bayesian Framework

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.