  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.
