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We present a postprocessing layer for deformable image registration to make aregistration field more diffeomorphic by encouraging Jacobians of thetransformation to be positive. Diffeomorphic image registration is importantfor medical imaging studies because of the properties like invertibility,smoothness of the transformation, and topology preservation/non-folding of thegrid. Violation of these properties can lead to destruction of theneighbourhood and the connectivity of anatomical structures during imageregistration. Most of the recent deep learning methods do not explicitlyaddress this folding problem and try to solve it with a smoothnessregularization on the registration field. In this paper, we propose adifferentiable layer, which takes any registration field as its input, computesexponential of the Jacobian matrices of the input and reconstructs a newregistration field from the exponentiated Jacobian matrices using Poissonreconstruction. Our proposed Poisson reconstruction loss enforces positiveJacobians for the final registration field. Thus, our method acts as apost-processing layer without any learnable parameters of its own and can beplaced at the end of any deep learning pipeline to form an end-to-end learnableframework. We show the effectiveness of our proposed method for a popular deeplearning registration method Voxelmorph and evaluate it with a datasetcontaining 3D brain MRI scans. Our results show that our post-processing caneffectively decrease the number of non-positive Jacobians by a significantamount without any noticeable deterioration of the registration accuracy, thusmaking the registration field more diffeomorphic. Our code is available onlineathttps://github.com/Soumyadeep-Pal/Diffeomorphic-Image-Registration-Postprocess.

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