# On Finite Difference Jacobian Computation in Deformable Image Registration

arXiv: https://arxiv.org/abs/2212.06060 · HTML: https://mlsanity.com/paper/2212.06060/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2022.

## Built on by (most cited first, 23 of 23)

- [A survey on deep learning in medical image registration: new technologies, uncertainty, evaluation metrics, and beyond](https://mlsanity.com/paper/2307.15615/) (2023)
- [Beyond the LUMIR challenge: The pathway to foundational registration models](https://mlsanity.com/paper/2505.24160/) (2025)
- [Vector Field Attention for Deformable Image Registration](https://mlsanity.com/paper/2407.10209/) (2024)
- [Deformable Cross-Attention Transformer for Medical Image Registration](https://mlsanity.com/paper/2303.06179/) (2023)
- [Mamba? Catch The Hype Or Rethink What Really Helps for Image Registration](https://mlsanity.com/paper/2407.19274/) (2024)
- [From Registration Uncertainty to Segmentation Uncertainty](https://mlsanity.com/paper/2403.05111/) (2024)
- [Spatially-varying Regularization with Conditional Transformer for Unsupervised Image Registration](https://mlsanity.com/paper/2303.06168/) (2023)
- [TimeFlow: Temporal Conditioning for Longitudinal Brain MRI Registration and Aging Analysis](https://mlsanity.com/paper/2501.08667/) (2025)
- [Pretraining Deformable Image Registration Networks with Random Images](https://mlsanity.com/paper/2505.24167/) (2025)
- [Encoder-Only Image Registration](https://mlsanity.com/paper/2509.00451/) (2025)
- [Unsupervised learning of spatially varying regularization for diffeomorphic image registration](https://mlsanity.com/paper/2412.17982/) (2024)
- [Large Scale Unsupervised Brain MRI Image Registration Solution for Learn2Reg 2024](https://mlsanity.com/paper/2409.00917/) (2024)
- [NestedMorph: Enhancing Deformable Medical Image Registration with Nested Attention Mechanisms](https://mlsanity.com/paper/2410.02550/) (2024)
- [Improving Instance Optimization in Deformable Image Registration with Gradient Projection](https://mlsanity.com/paper/2410.15767/) (2024)
- [Strategies for Robust Deep Learning Based Deformable Registration](https://mlsanity.com/paper/2510.23079/) (2025)
- [Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization](https://mlsanity.com/paper/2511.11679/) (2025)
- [Deformable Image Registration with Multi-scale Feature Fusion from Shared Encoder, Auxiliary and Pyramid Decoders](https://mlsanity.com/paper/2408.05717/) (2024)
- [Fine-Tuning TransMorph with Gradient Correlation for Anatomical Alignment](https://mlsanity.com/paper/2412.20822/) (2024)
- [Surrogate Supervision for Robust and Generalizable Deformable Image Registration](https://mlsanity.com/paper/2509.09869/) (2025)
- [Zero-shot Multi-Contrast Brain MRI Registration by Intensity Randomizing T1-weighted MRI (LUMIR25)](https://mlsanity.com/paper/2602.06292/) (2026)
- [Diffeomorphism-Equivariant Neural Networks](https://mlsanity.com/paper/2602.06695/) (2026)
- [Adapting Frozen Mono-modal Backbones for Multi-modal Registration via Contrast-Agnostic Instance Optimization](https://mlsanity.com/paper/2603.26393/) (2026)
- [Registration is a Representation Problem: Solving Whole-Body PET/CT Registration](https://mlsanity.com/paper/2610.18804/) (2026)

## Abstract (arXiv)

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.