On Finite Difference Jacobian Computation in Deformable Image Registration
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Published:2024-04-18
Issue:9
Volume:132
Page:3678-3688
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ISSN:0920-5691
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Container-title:International Journal of Computer Vision
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language:en
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Short-container-title:Int J Comput Vis
Author:
Liu YihaoORCID, Chen Junyu, Wei Shuwen, Carass Aaron, Prince Jerry
Abstract
AbstractProducing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant $$\vert J\vert $$
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J
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everywhere, the number of pixels (2D) or voxels (3D) with $$\vert J\vert <0$$
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J
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<
0
has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, $$\vert J\vert $$
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J
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is commonly approximated using a central difference, but this strategy can yield positive $$\vert J\vert $$
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J
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’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 $$\vert J\vert $$
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J
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. We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of $$\vert J\vert $$
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J
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is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of $$\vert J\vert $$
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J
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’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 $$\vert J\vert $$
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J
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’s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of $$\vert J\vert $$
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J
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and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.
Funder
National Eye Institute
Publisher
Springer Science and Business Media LLC
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