前进 (2022-12-31 11:39):
#paper Liu Y, Chen J, Wei S, et al. On Finite Difference Jacobian Computation in Deformable Image Registration[J]. arXiv preprint arXiv:2212.06060, 2022. 产生微分同胚的空间变换一直是变形图像配准的中心问题。作为一个微分同胚变换,应在任何位置都具有正的雅可比行列式|J|。|J|<0的体素数已被用于测试微分同胚性,也用于测量变换的不规则性。 对于数字变换,|J|通常使用中心差来近似,但是对于即使在体素分辨率级别上也明显不具有差分同胚性的变换,这种策略可以产生正的|J|。为了证明这一点,论文首先研究了|J|的不同有限差分近似的几何意义。为了确定数字图像的微分同胚性,使用任何单独的有限差分逼近|J|是不够的。论文证明对于2D变换,|J|的四个唯一的有限差分近似必须是正的,以确保整个域是可逆的,并且在像素级没有折叠。在3D中,|J|的十个唯一的有限差分近似值需要是正的。论文提出的数字微分同胚准则解决了|J|的中心差分近似中固有的几个误差,并准确地检测非微分同胚数字变换。
On Finite Difference Jacobian Computation in Deformable Image Registration
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Abstract:
Producing spatial transformations that are diffeomorphic has been a central problem in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant |J| everywhere, the number of voxels 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 central difference, but this strategy can yield positive |J|'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 |J|. We show that to determine diffeomorphism for digital images, use of any individual finite difference approximations of |J| is insufficient. We show that for a 2D transformation, four unique finite difference approximations of |J|'s must be positive to ensure the entire domain is invertible and free of folding at the pixel level. We also show that in 3D, 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.
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