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Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction
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Proposes DDOT, a diffeomorphic neural-ODE mesh deformation network trained with a sliced Wasserstein loss on meshes represented as oriented varifolds, and applies it to cortical white-matter surface reconstruction from MRI.
Problem
Chamfer-based set comparisons for learning mesh deformation lack guarantees on point-cloud sample size, are only pseudo-metrics, have quadratic cost, and get trapped in local minima, which hurts cortical surface reconstruction.
Approach
Represent meshes as probability measures (continuous, empirical, or discrete oriented varifolds) and compare them with sliced Wasserstein distance, which is a valid metric with linear memory and O(m log m) cost, inside a neural-ODE diffeomorphic deformation model.
Key results
- On ADNI left WM, the method reports EMD 0.728 mm vs 0.803 for CortexODE, and self-intersection below 10^-4 %, versus 0.013% for CortexODE.
- On OASIS left WM, EMD 0.418 mm vs 0.425 for CortexODE and ASSD 0.161 mm vs 0.183 mm.
- On the TRT consistency test, the method achieves the best EMD (0.780) and ASSD (0.229), with FreeSurfer better on SWD (0.358 vs 0.403).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use oriented varifold instead of point sampling with SWD (ablation) | DDOT · ADNI Left WM | EMD: .850 -> .728 | Helped |
| use oriented varifold instead of point sampling with SWD (ablation) | DDOT · ADNI Right WM | EMD: .832 -> .702 | Helped |
| replace CD with SWD on oriented varifold (ablation) | DDOT · ADNI Left WM | SWD: .499 -> .420 | Helped |
| replace CD with SWD on oriented varifold (ablation) | DDOT · ADNI Right WM | SWD: .529 -> .365 | Helped |
| replace Sinkhorn with SWD on oriented varifold (ablation) | DDOT · ADNI Left WM | EMD: 1.023 -> .728 | Helped |
| replace Sinkhorn with SWD on oriented varifold (ablation) | DDOT · ADNI Right WM | EMD: 1.002 -> .702 | Helped |
| use DDOT instead of CortexODE | cortical surface reconstruction · ADNI and OASIS WM surfaces | SI (%): .013 $\pm .011$ -> $\boldsymbol{<10^{-4}}$ (ADNI Left WM); .004 $\pm .005$ -> $\boldsymbol{<10^{-4}}$ (ADNI Right WM); . | Helped |
| use DDOT instead of CortexODE | cortical surface reconstruction · TRT | SWD: .358 $\pm .275$ -> .403 $\pm .184$ | Hurt |
| use DDOT instead of CortexODE | cortical surface reconstruction · TRT | EMD: .799 $\pm .038$ -> .780 $\pm .032$ | Helped |
| use DDOT instead of CortexODE | cortical surface reconstruction · TRT | ASSD: .241 $\pm .040$ -> .229 $\pm .010$ | Helped |
Surprising findings
- Sinkhorn divergence on oriented varifolds performed worse than SWD on the same representation, despite approximating Wasserstein distance.
- Optimizing SWD on sampled point clouds underperformed optimizing on oriented varifold representation.
Limitations
- Performance depends on mesh resolution (deterministic supports tied to face count), introducing stochastic memory during training.
- Evaluated only on cortical surface reconstruction; applicability to other deformation tasks untested.
- Uses FreeSurfer pseudo-ground-truth surfaces as references.
Experimental setup
Datasets: ADNI OASIS TRT
Benchmarks: ADNI OASIS TRT
Metrics: EMD SWD ASSD Chamfer normals self-intersection face ratio
Baselines: DeepCSR Vox2Cortex CFPP CortexODE FreeSurfer Sinkhorn divergence
Builds on: Neural ODE CortexODE CorticalFlow Sliced Wasserstein distance oriented varifold
Closest papers by meaning
- 4Deform: Neural Surface Deformation for Robust Shape Interpolation (CVPR 2025 · similarity 0.89)
- CorticalFlow: A Diffeomorphic Mesh Transformer Network for Cortical Surface Reconstruction (NeurIPS 2021 · similarity 0.89)
- Small Steps and Level Sets: Fitting Neural Surface Models with Point Guidance (CVPR 2024 · similarity 0.88)
- Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing (NeurIPS 2025 · similarity 0.87)
- Learning Bijective Surface Parameterization for Inferring Signed Distance Functions from Sparse Point Clouds with Grid Deformation (CVPR 2025 · similarity 0.87)
- Motion2VecSets: 4D Latent Vector Set Diffusion for Non-rigid Shape Reconstruction and Tracking (CVPR 2024 · similarity 0.86)
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