On Finite Difference Jacobian Computation in Deformable Image Registration

Yihao Liu, Junyu Chen, Shuwen Wei, Aaron Carass, Jerry L. Prince

Abstract: Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant \(\vert J\vert \) everywhere, the number of pixels (2D) or voxels (3D) with \(\vert J\vert <0\) has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, \(\vert J\vert \) is commonly approximated using a central difference, but this strategy can yield positive \(\vert J\vert \)’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 \(\vert J\vert \). We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of \(\vert J\vert \) is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of \(\vert J\vert \)’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 \(\vert J\vert \)’s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of \(\vert J\vert \) and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.