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Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant everywhere, the number of pixels (2D) or voxels (3D) with has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, is commonly approximated using a central difference, but this strategy can yield positive’s for transformations that are clearly not diffeomorphic—even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of. We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of …