On Finite Difference Jacobian Computation in Deformable Image Registration

Producing spatial transformations that are diffeomorphic has been a central problem in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant $|J|$ everywhere, the number of voxels with $|J|<0$ has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, $|J|$ is commonly approximated using central difference, but this strategy can yield positive $|J|$'s for transformations that are clearly not diffeomorphic -- even at the voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of $|J|$. We show that to determine diffeomorphism for digital images, use of any individual finite difference approximations of $|J|$ is insufficient. We show that for a 2D transformation, four unique finite difference approximations of $|J|$'s must be positive to ensure the entire domain is invertible and free of folding at the pixel level. We also show that in 3D, ten unique finite differences approximations of $|J|$'s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of $|J|$ and accurately detects non-diffeomorphic digital transformations.

Unpaired Brain MR-to-CT Synthesis using a Structure-Constrained CycleGAN

The cycleGAN is becoming an influential method in medical image synthesis. However, due to a lack of direct constraints between input and synthetic images, the cycleGAN cannot guarantee structural consistency between these two images, and such consistency is of extreme importance in medical imaging. To overcome this, we propose a structure-constrained cycleGAN for brain MR-to-CT synthesis using unpaired data that defines an extra structure-consistency loss based on the modality independent neighborhood descriptor to constrain structural consistency. Additionally, we use a position-based selection strategy for selecting training images instead of a completely random selection scheme. Experimental results on synthesizing CT images from brain MR images demonstrate that our method is better than the conventional cycleGAN and approximates the cycleGAN trained with paired data.