On Finite Difference Jacobian Computation in Deformable Image  Registration

Abstract

Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> everywhere, the number of voxels with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> <mml:mo><</mml:mo> <mml:mn>0</mml:mn></mml:math> has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> is commonly approximated using a central difference, but this strategy can yield positive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> '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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> . We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> '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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> 's are required to be positive. Our proposed <i>digital diffeomorphism</i> criteria solves several errors inherent in the central difference approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo></mml:math> and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.