# Two-Parameter Flow Map Learning for Continuous-Time Diffeomorphic Image Registration

arXiv: https://arxiv.org/abs/2609.10789 · HTML: https://mlsanity.com/paper/2609.10789/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 52 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Builds on (most cited first, 24 of 24)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Deep Learning for Segmentation using an Open Large-Scale Dataset in 2D Echocardiography](https://mlsanity.com/paper/1908.06948/) (2019)
- [An Unsupervised Learning Model for Deformable Medical Image Registration](https://mlsanity.com/paper/1802.02604/) (2018)
- [Large Deformation Diffeomorphic Image Registration with Laplacian Pyramid Networks](https://mlsanity.com/paper/2006.16148/) (2020)
- [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)
- [CycleMorph: Cycle Consistent Unsupervised Deformable Image Registration](https://mlsanity.com/paper/2008.05772/) (2020)
- [Conditional Deformable Image Registration with Convolutional Neural Network](https://mlsanity.com/paper/2106.12673/) (2021)
- [DiffuseMorph: Unsupervised Deformable Image Registration Using Diffusion Model](https://mlsanity.com/paper/2112.05149/) (2021)
- [textttGradICON: Approximate Diffeomorphisms via Gradient Inverse Consistency](https://mlsanity.com/paper/2206.05897/) (2022)
- [Correlation-aware Coarse-to-fine MLPs for Deformable Medical Image Registration](https://mlsanity.com/paper/2406.00123/) (2024)
- [NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration](https://mlsanity.com/paper/2108.03443/) (2021)
- [Multi-scale Neural ODEs for 3D Medical Image Registration](https://mlsanity.com/paper/2106.08493/) (2021)
- [Adaptive Correspondence Scoring for Unsupervised Medical Image Registration](https://mlsanity.com/paper/2312.00837/) (2023)
- [textttNePhi: Neural Deformation Fields for Approximately Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2309.07322/) (2023)
- [DiffuseReg: Denoising Diffusion Model for Obtaining Deformation Fields in Unsupervised Deformable Image Registration](https://mlsanity.com/paper/2410.05234/) (2024)
- [Guiding Registration with Emergent Similarity from Pre-Trained Diffusion Models](https://mlsanity.com/paper/2506.02419/) (2025)
- [Towards Saner Deep Image Registration](https://mlsanity.com/paper/2307.09696/) (2023)
- [PULPo: Probabilistic Unsupervised Laplacian Pyramid Registration](https://mlsanity.com/paper/2407.10567/) (2024)
- [SACB-Net: Spatial-awareness Convolutions for Medical Image Registration](https://mlsanity.com/paper/2503.19592/) (2025)

## Abstract (arXiv)

Diffeomorphic image registration is central to medical image analysis, enabling anatomically consistent alignment across subjects. Most learning-based diffeomorphic methods model autonomous ODEs(ordinary differential equations) by parameterizing a stationary velocity field and recovering deformations via scaling-and-squaring. While non-autonomous ODEs with time-dependent velocities increase expressiveness, existing approaches rely on numerical integration to implicitly enforce flow structure that entangles model expressiveness with discretization accuracy. We propose a framework to directly learn the continuous-time solution of a non-autonomous ODE formulated as a two-parameterflow map. By enforcing cocycle consistency, a fundamental structural property of time-varying flows, we learn the flow maps without time discretization and velocity integration during training. The framework recovers diffeomorphic mappings at inference using a small number of compositions. Our proposed framework seamlessly incorporates standard registration backbones and improves alignment accuracy consistently across nine datasets while preserving diffeomorphic structure. Notably, the proposed method achieves an average Dice improvement of 2.1% on brain MRI benchmarks, a 12% TRE reduction on lung CT, and a 2.6% Dice gain on cardiac MRI and ultrasound datasets.