# CortexODE: Learning Cortical Surface Reconstruction by Neural ODEs

arXiv: https://arxiv.org/abs/2202.08329 · HTML: https://mlsanity.com/paper/2202.08329/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 16 Feb 2022.

## Builds on (most cited first, 11 of 11)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Accelerating 3D Deep Learning with PyTorch3D](https://mlsanity.com/paper/2007.08501/) (2020)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019)
- [FastSurfer -- A fast and accurate deep learning based neuroimaging pipeline](https://mlsanity.com/paper/1910.03866/) (2019)
- [Neural Mesh Flow: 3D Manifold Mesh Generation via Diffeomorphic Flows](https://mlsanity.com/paper/2007.10973/) (2020)
- [Voxel2Mesh: 3D Mesh Model Generation from Volumetric Data](https://mlsanity.com/paper/1912.03681/) (2019)
- [QuickNAT: A Fully Convolutional Network for Quick and Accurate Segmentation of Neuroanatomy](https://mlsanity.com/paper/1801.04161/) (2018)
- [DeepCSR: A 3D Deep Learning Approach for Cortical Surface Reconstruction](https://mlsanity.com/paper/2010.11423/) (2020)
- [PialNN: A Fast Deep Learning Framework for Cortical Pial Surface Reconstruction](https://mlsanity.com/paper/2109.03693/) (2021)

## Built on by (most cited first, 22 of 22)

- [SDF4CHD: Generative Modeling of Cardiac Anatomies with Congenital Heart Defects](https://mlsanity.com/paper/2311.00332/) (2023)
- [Neural deformation fields for template-based reconstruction of cortical surfaces from MRI](https://mlsanity.com/paper/2401.12938/) (2024)
- [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023)
- [SUGAR: Spherical Ultrafast Graph Attention Framework for Cortical Surface Registration](https://mlsanity.com/paper/2307.00511/) (2023)
- [Hybrid-CSR: Coupling Explicit and Implicit Shape Representation for Cortical Surface Reconstruction](https://mlsanity.com/paper/2307.12299/) (2023)
- [LinFlo-Net: A two-stage deep learning method to generate simulation ready meshes of the heart](https://mlsanity.com/paper/2310.20065/) (2023)
- [From Pixels to Polygons: A Survey of Deep Learning Approaches for Medical Image-to-Mesh Reconstruction](https://mlsanity.com/paper/2505.03599/) (2025)
- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEs](https://mlsanity.com/paper/2506.00947/) (2025)
- [MedGS: Gaussian Splatting for Multi-Modal 3D Medical Imaging](https://mlsanity.com/paper/2509.16806/) (2025)
- [SurfNN: Joint Reconstruction of Multiple Cortical Surfaces from Magnetic Resonance Images](https://mlsanity.com/paper/2303.02922/) (2023)
- [From Low Field to High Value: Robust Cortical Mapping from Low-Field MRI](https://mlsanity.com/paper/2505.12228/) (2025)
- [NeuroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders](https://mlsanity.com/paper/2502.18786/) (2025)
- [Reconsider the Template Mesh in Deep Learning-based Mesh Reconstruction](https://mlsanity.com/paper/2505.15285/) (2025)
- [UNSURF: Uncertainty Quantification for Cortical Surface Reconstruction of Clinical Brain MRIs](https://mlsanity.com/paper/2506.00498/) (2025)
- [Reconstruction of Cortical Surfaces with Spherical Topology from Infant Brain MRI via Recurrent Deformation Learning](https://mlsanity.com/paper/2312.05986/) (2023)
- [V2C-Long: Longitudinal Cortex Reconstruction with Spatiotemporal Correspondence](https://mlsanity.com/paper/2402.17438/) (2024)
- [Self-Supervised Masked Mesh Learning for Unsupervised Anomaly Detection on 3D Cortical Surfaces](https://mlsanity.com/paper/2412.05580/) (2024)
- [SimCortex: Collision-free Simultaneous Cortical Surfaces Reconstruction](https://mlsanity.com/paper/2507.06955/) (2025)
- [Template-Based Cortical Surface Reconstruction with Minimal Energy Deformation](https://mlsanity.com/paper/2509.14827/) (2025)
- [KidMesh: Computational Mesh Reconstruction for Pediatric Congenital Hydronephrosis Using Deep Neural Networks](https://mlsanity.com/paper/2602.13299/) (2026)
- [High-Fidelity 3D Geometric Reconstruction of Pelvic Organs from MRI: A Hybrid Deep Learning and Iterative Optimization Approach](https://mlsanity.com/paper/2606.17836/) (2026)

## Abstract (arXiv)

We present CortexODE, a deep learning framework for cortical surface reconstruction. CortexODE leverages neural ordinary differential equations (ODEs) to deform an input surface into a target shape by learning a diffeomorphic flow. The trajectories of the points on the surface are modeled as ODEs, where the derivatives of their coordinates are parameterized via a learnable Lipschitz-continuous deformation network. This provides theoretical guarantees for the prevention of self-intersections. CortexODE can be integrated to an automatic learning-based pipeline, which reconstructs cortical surfaces efficiently in less than 5 seconds. The pipeline utilizes a 3D U-Net to predict a white matter segmentation from brain Magnetic Resonance Imaging (MRI) scans, and further generates a signed distance function that represents an initial surface. Fast topology correction is introduced to guarantee homeomorphism to a sphere. Following the isosurface extraction step, two CortexODE models are trained to deform the initial surface to white matter and pial surfaces respectively. The proposed pipeline is evaluated on large-scale neuroimage datasets in various age groups including neonates (25-45 weeks), young adults (22-36 years) and elderly subjects (55-90 years). Our experiments demonstrate that the CortexODE-based pipeline can achieve less than 0.2mm average geometric error while being orders of magnitude faster compared to conventional processing pipelines.