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 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 a 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 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.
Code (1)
Tasks
Image RegistrationMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Achieving ${O}(ε^{-1.5})$ Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization
In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has …
Bilevel OptimizationDARE: A Deformable Adaptive Regularization Estimator for Learning-Based Medical Image Registration
Deformable medical image registration is a fundamental task in medical image analysis. While deep learning-based methods have demonstrated superior accuracy and computational efficiency compared to traditional techniques…
Medical Image RegistrationComputational EfficiencyAchieving $\mathcal{O}(\epsilon^{-1.5})$ Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization
In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has …
Deformable Registration Using Average Geometric Transformations for Brain MR Images
Accurate registration of medical images is vital for doctor's diagnosis and quantitative analysis. In this paper, we propose a new deformable medical image registration method based on average geometric transformations a…
Deformable Medical Image RegistrationImage RegistrationMedical Image RegistrationPrescribed Performance Control of Deformable Object Manipulation in Spatial Latent Space
Manipulating three-dimensional (3D) deformable objects presents significant challenges for robotic systems due to their infinite-dimensional state space and complex deformable dynamics. This paper proposes a novel model-…
Dimensionality Reduction