  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.
