On Finite Difference Jacobian Computation in Deformable Image Registration

Producing spatial transformations that are diffeomorphic is a key goal 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 a 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 if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of |J| is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of |J|'s must be positive to ensure that the entire domain is invertible and free of folding at the pixel level. For a 3D transformation, 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. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.