# NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration

arXiv: https://arxiv.org/abs/2108.03443 · HTML: https://mlsanity.com/paper/2108.03443/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 7 Aug 2021.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019)
- [An Unsupervised Learning Model for Deformable Medical Image Registration](https://mlsanity.com/paper/1802.02604/) (2018)
- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [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)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Learning Conditional Deformable Templates with Convolutional Networks](https://mlsanity.com/paper/1908.02738/) (2019)
- [Conditional Deformable Image Registration with Convolutional Neural Network](https://mlsanity.com/paper/2106.12673/) (2021)
- [Networks for Joint Affine and Non-parametric Image Registration](https://mlsanity.com/paper/1903.08811/) (2019)
- [Generative Adversarial Registration for Improved Conditional Deformable Templates](https://mlsanity.com/paper/2105.04349/) (2021)
- [ShapeFlow: Learnable Deformations Among 3D Shapes](https://mlsanity.com/paper/2006.07982/) (2020)
- [ICON: Learning Regular Maps Through Inverse Consistency](https://mlsanity.com/paper/2105.04459/) (2021)
- [Region-specific Diffeomorphic Metric Mapping](https://mlsanity.com/paper/1906.00139/) (2019)
- [Multi-scale Neural ODEs for 3D Medical Image Registration](https://mlsanity.com/paper/2106.08493/) (2021)
- [Knowledge-Based Learning of Nonlinear Dynamics and Chaos](https://mlsanity.com/paper/2010.03415/) (2020)
- [Anatomical Data Augmentation via Fluid-based Image Registration](https://mlsanity.com/paper/2007.02447/) (2020)
- [Multiple-shooting adjoint method for whole-brain dynamic causal modeling](https://mlsanity.com/paper/2102.11013/) (2021)

## Built on by (most cited first, 28 of 28)

- [A survey on deep learning in medical image registration: new technologies, uncertainty, evaluation metrics, and beyond](https://mlsanity.com/paper/2307.15615/) (2023)
- [Deep Learning in Medical Image Registration: Magic or Mirage?](https://mlsanity.com/paper/2408.05839/) (2024)
- [Diffeomorphic Image Registration with Neural Velocity Field](https://mlsanity.com/paper/2202.12498/) (2022)
- [Medical Image Registration via Neural Fields](https://mlsanity.com/paper/2206.03111/) (2022)
- [SITReg: Multi-resolution architecture for symmetric, inverse consistent, and topology preserving image registration](https://mlsanity.com/paper/2303.10211/) (2023)
- [NODER: Image Sequence Regression Based on Neural Ordinary Differential Equations](https://mlsanity.com/paper/2407.13241/) (2024)
- [Deep Implicit Optimization enables Robust Learnable Features for Deformable Image Registration](https://mlsanity.com/paper/2406.07361/) (2024)
- [ORRN: An ODE-based Recursive Registration Network for Deformable Respiratory Motion Estimation with Lung 4DCT Images](https://mlsanity.com/paper/2305.14673/) (2023)
- [CPT-Interp: Continuous sPatial and Temporal Motion Modeling for 4D Medical Image Interpolation](https://mlsanity.com/paper/2405.15385/) (2024)
- [FLOWING: Implicit Neural Flows for Structure-Preserving Morphing](https://mlsanity.com/paper/2510.09537/) (2025)
- [Echocardiography Segmentation Using Neural ODE-based Diffeomorphic Registration Field](https://mlsanity.com/paper/2306.09687/) (2023)
- [Co-Learning Semantic-aware Unsupervised Segmentation for Pathological Image Registration](https://mlsanity.com/paper/2310.11040/) (2023)
- [Towards Establishing Dense Correspondence on Multiview Coronary Angiography: From Point-to-Point to Curve-to-Curve Query Matching](https://mlsanity.com/paper/2312.11593/) (2023)
- [Gaussian Primitives for Deformable Image Registration](https://mlsanity.com/paper/2406.03394/) (2024)
- [Learning Geodesics of Geometric Shape Deformations From Images](https://mlsanity.com/paper/2410.18797/) (2024)
- [VoxelOpt: Voxel-Adaptive Message Passing for Discrete Optimization in Deformable Abdominal CT Registration](https://mlsanity.com/paper/2506.19975/) (2025)
- [Deep learning in medical image registration: introduction and survey](https://mlsanity.com/paper/2309.00727/) (2023)
- [Learning Diffeomorphism for Image Registration with Time-Continuous Networks using Semigroup Regularization](https://mlsanity.com/paper/2405.18684/) (2024)
- [Stationary Velocity Fields on Matrix Groups for Deformable Image Registration](https://mlsanity.com/paper/2410.10997/) (2024)
- [PointODE: Lightweight Point Cloud Learning with Neural Ordinary Differential Equations on Edge](https://mlsanity.com/paper/2506.00438/) (2025)
- [A Scalable Distributed Framework for Multimodal GigaVoxel Image Registration](https://mlsanity.com/paper/2509.25044/) (2025)
- [Leveraging Predictive Models for Adaptive Sampling of Spatiotemporal Fluid Processes](https://mlsanity.com/paper/2304.00732/) (2023)
- [Continuous sPatial-Temporal Deformable Image Registration (CPT-DIR) for motion modelling in radiotherapy: beyond classic voxel-based methods](https://mlsanity.com/paper/2405.00430/) (2024)
- [Modeling and Reversing Brain Lesions Using Diffusion Models](https://mlsanity.com/paper/2507.05670/) (2025)
- [Deformable Medical Image Registration with KAN-based Implicit Neural Representations](https://mlsanity.com/paper/2509.22874/) (2025)
- [Multimodal Diffeomorphic Registration with Neural ODEs and Structural Descriptors](https://mlsanity.com/paper/2512.22689/) (2025)
- [Two-Parameter Flow Map Learning for Continuous-Time Diffeomorphic Image Registration](https://mlsanity.com/paper/2609.10789/) (2026)
- [NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures](https://mlsanity.com/paper/2609.31539/) (2026)

## Abstract (arXiv)

Deformable image registration (DIR), aiming to find spatial correspondence between images, is one of the most critical problems in the domain of medical image analysis. In this paper, we present a novel, generic, and accurate diffeomorphic image registration framework that utilizes neural ordinary differential equations (NODEs). We model each voxel as a moving particle and consider the set of all voxels in a 3D image as a high-dimensional dynamical system whose trajectory determines the targeted deformation field. Our method leverages deep neural networks for their expressive power in modeling dynamical systems, and simultaneously optimizes for a dynamical system between the image pairs and the corresponding transformation. Our formulation allows various constraints to be imposed along the transformation to maintain desired regularities. Our experiment results show that our method outperforms the benchmarks under various metrics. Additionally, we demonstrate the feasibility to expand our framework to register multiple image sets using a unified form of transformation,which could possibly serve a wider range of applications.