Towards Positive Jacobian: Learn to Postprocess Diffeomorphic Image Registration with Matrix Exponential

We present a postprocessing layer for deformable image registration to make a registration field more diffeomorphic by encouraging Jacobians of the transformation to be positive. Diffeomorphic image registration is important for medical imaging studies because of the properties like invertibility, smoothness of the transformation, and topology preservation/non-folding of the grid. Violation of these properties can lead to destruction of the neighbourhood and the connectivity of anatomical structures during image registration. Most of the recent deep learning methods do not explicitly address this folding problem and try to solve it with a smoothness regularization on the registration field. In this paper, we propose a differentiable layer, which takes any registration field as its input, computes exponential of the Jacobian matrices of the input and reconstructs a new registration field from the exponentiated Jacobian matrices using Poisson reconstruction. Our proposed Poisson reconstruction loss enforces positive Jacobians for the final registration field. Thus, our method acts as a post-processing layer without any learnable parameters of its own and can be placed at the end of any deep learning pipeline to form an end-to-end learnable framework. We show the effectiveness of our proposed method for a popular deep learning registration method Voxelmorph and evaluate it with a dataset containing 3D brain MRI scans. Our results show that our post-processing can effectively decrease the number of non-positive Jacobians by a significant amount without any noticeable deterioration of the registration accuracy, thus making the registration field more diffeomorphic. Our code is available online at https://github.com/Soumyadeep-Pal/Diffeomorphic-Image-Registration-Postprocess.