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.