# A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints

arXiv: https://arxiv.org/abs/2308.02393 · HTML: https://mlsanity.com/paper/2308.02393/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 67 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 14 of 14)

- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks](https://mlsanity.com/paper/2003.09514/) (2020)
- [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017)
- [CNN-based Lung CT Registration with Multiple Anatomical Constraints](https://mlsanity.com/paper/2011.14372/) (2020)
- [The varifold representation of non-oriented shapes for diffeomorphic registration](https://mlsanity.com/paper/1304.6108/) (2013)
- [Estimation of Large Motion in Lung CT by Integrating Regularized Keypoint Correspondences into Dense Deformable Registration](https://mlsanity.com/paper/1807.00467/) (2018)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [A Lagrangian Gauss-Newton-Krylov Solver for Mass- and Intensity-Preserving Diffeomorphic Image Registration](https://mlsanity.com/paper/1703.04446/) (2017)
- [A New Variational Model for Joint Image Reconstruction and Motion Estimation in Spatiotemporal Imaging](https://mlsanity.com/paper/1812.03446/) (2018)
- [Diffeomorphic density matching by optimal information transport](https://mlsanity.com/paper/1501.07635/) (2015)
- [A unifying framework for n-dimensional quasi-conformal mappings](https://mlsanity.com/paper/2110.10437/) (2021)
- [Automatic Landmark Detection and Registration of Brain Cortical Surfaces via Quasi-Conformal Geometry and Convolutional Neural Networks](https://mlsanity.com/paper/2208.07010/) (2022)
- [Indirect Image Registration with Large Diffeomorphic Deformations](https://mlsanity.com/paper/1706.04048/) (2017)
- [Spatiotemporal Imaging with Diffeomorphic Optimal Transportation](https://mlsanity.com/paper/2011.11906/) (2020)

## Built on by (most cited first, 1 of 1)

- [Two-chart Beltrami Optimization for Distortion-Controlled Spherical Bijection with Application to Brain Surface Registration](https://mlsanity.com/paper/2602.01589/) (2026)

## Abstract (arXiv)

Diffeomorphic registration is a widely used technique for finding a smooth and invertible transformation between two coordinate systems, which are measured using template and reference images. The point-wise volume-preserving constraint det(nablabmvarphi(bmx)) =1 is effective in some cases, but may be too restrictive in others, especially when local deformations are relatively large. This can result in poor matching when enforcing large local deformations. In this paper, we propose a new bi-variant diffeomorphic image registration model that introduces a soft constraint on the Jacobian equation det(nablabmvarphi(bmx)) = f(bmx) > 0. This allows local deformations to shrink and grow within a flexible range 0<kappam<det(nablabmvarphi(bmx))<kappaM. The Jacobian determinant of transformation is explicitly controlled by optimizing the relaxation function f(bmx). To prevent deformation folding and improve the smoothness of the transformation, a positive constraint is imposed on the optimization of the relaxation function f(bmx), and a regularizer is used to ensure the smoothness of f(bmx). Furthermore, the positivity constraint ensures that f(bmx) is as close to one as possible, which helps to achieve a volume-preserving transformation on average. We also analyze the existence of the minimizer for the variational model and propose a penalty-splitting algorithm with a multilevel strategy to solve this model. Numerical experiments demonstrate the convergence of the proposed algorithm and show that the positivity constraint can effectively control the range of relative volume without compromising the accuracy of the registration. Moreover, the proposed model generates diffeomorphic maps for large local deformations and outperforms several existing registration models in terms of performance.