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Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning
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Unsupervised non-rigid 3D shape matching that combines deep functional maps with a sliced-Wasserstein OT feature-alignment loss and an entropic-OT adaptive refinement step.
Problem
Establishing point-to-point correspondences between non-rigid 3D shapes; explicit feature alignment is hard to learn, and entropic OT-based approaches are costly (quadratic time and memory).
Approach
Treat extracted per-vertex features as probability measures and align them with an efficient sliced Wasserstein (and energy-based sliced Wasserstein) OT loss, jointly with functional map regularizers, then refine soft correspondences with a one-time entropic OT step at inference.
Key results
- On SHREC'19 (trained on FAUST+SCAPE), the method reaches 4.7 mean geodesic error versus 4.6 for ULRSSM on FAUST+SCAPE+SHREC'19 combined and 7.6 for ULRSSM on SCAPE-trained SHREC'19 setting, per Table 1.
- On SMAL the method achieves 4.0 error vs 4.2 for ULRSSM; on DT4D-H 4.2 vs 4.5 for ULRSSM (Table 2).
- On FAUST segmentation transfer, coarse mIoU 87.8 and fine-grained 60.5, above ULRSSM (84.2, 58.2) (Table 3).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed method instead of ULRSSM | Non-isometric shape matching · SMAL | mean geodesic error (×100): 4.2 -> 4.0 | Helped |
| Use the proposed method instead of ULRSSM | Non-isometric shape matching · DT4D-H | mean geodesic error (×100): 4.5 -> 4.2 | Helped |
| Use the proposed correspondence method instead of UDMSM | FAUST segmentation transfer · Coarse annotations | mIOU: 85.3 -> 87.8 | Helped |
| Use the proposed correspondence method instead of ULRSSM | FAUST segmentation transfer · Fine-grained annotations | mIOU: 58.2 -> 60.5 | Helped |
| Replace LOT with LMSE (ablation) | Proposed method · SHREC’19 | mean geodesic error (×100): 4.7 -> 34.3 | Hurt |
| Replace energy-based sliced Wasserstein loss with uniform sliced Wasserstein loss (ablation) | Proposed method · SHREC’19 | mean geodesic error (×100): 4.7 -> 4.9 | Hurt |
| Use LbiSW instead of the energy-based sliced Wasserstein loss (ablation) | Proposed method · SHREC’19 | mean geodesic error (×100): 4.7 -> 4.8 | Hurt |
| Add adaptive refinement (ablation) | Proposed method · SHREC’19 | mean geodesic error (×100): 7.2 -> 4.7 | Helped |
Surprising findings
- A plain MSE feature-alignment loss in place of the OT loss fails badly (SHREC'19 error 34.3), showing OT-based alignment is essential.
Limitations
- Gains over ULRSSM are often marginal (e.g., 1.5 vs 1.6 on FAUST, 4.0 vs 4.2 on SMAL).
- Adaptive refinement is applied per pair and relies on entropic OT with a Sinkhorn step.
Experimental setup
Datasets: FAUST SCAPE SHREC'19 SMAL DeformingThings4D FAUST segmentation benchmark
Benchmarks: FAUST SCAPE SHREC'19 SMAL DT4D-H FAUST segmentation transfer
Metrics: mean geodesic error (x100) segmentation mIoU
Baselines: ZoomOut BCICP Smooth Shells FMNet GeomFMaps TransMatch SURFMNet Deep Shells AFMap SSLMSM UDMSM ULRSSM
Builds on: deep functional maps DiffusionNet Deep Shells energy-based sliced Wasserstein regularized functional maps
Closest papers by meaning
- Memory-Scalable and Simplified Functional Map Learning (CVPR 2024 · similarity 0.88)
- Denoising Functional Maps: Diffusion Models for Shape Correspondence (CVPR 2025 · similarity 0.87)
- Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation (CVPR 2024 · similarity 0.87)
- Locality Preserving Refinement for Shape Matching with Functional Maps (AAAI 2024 · similarity 0.87)
- Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching (CVPR 2024 · similarity 0.86)
- Multi-Shape Matching with Cycle Consistency Basis via Functional Maps (AAAI 2025 · similarity 0.86)
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