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 pixels (2D) or voxels (3D) 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 pixel or 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 .