Research map / 3D vision / Face and shape
Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching
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Proposes hybrid functional maps that combine LBO eigenbasis with non-orthogonal elastic thin-shell Hessian eigenbasis, with HS-norm formulation, to improve non-isometric and topologically noisy shape matching.
Problem
Non-isometric 3D shape correspondence struggles to capture high-frequency extrinsic deformations such as bending and creases, since LBO eigenfunctions only capture low-frequency intrinsic information.
Approach
Construct functional maps in a block-diagonal hybrid spectral space that joins LBO eigenfunctions (coarse isometric alignment) with non-orthogonal elastic thin-shell Hessian eigenfunctions (creases and details), using a Hilbert-Schmidt/mass-matrix-aware formulation of the data and regularization terms so the non-orthogonal basis is handled correctly; the hybrid basis is a drop-in replacement in ex
Key results
- Hybrid ULRSSM reduces TOPKIDS geodesic error from 9.2 to 5.0 (about 45% improvement) and SMAL from 3.9 to 3.3.
- Hybrid GeomFMaps improves SHREC'19 error from 7.9 to 5.6 and SMAL from 8.4 to 7.6 under supervised settings.
- Hybrid Smooth Shells improves TOPKIDS inter-class error from 10.8 to 7.5 and SMAL from 30.0 to 28.4.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · FAUST | geodesic error (×100): 1.6 -> 1.5 | Helped |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · SCAPE | geodesic error (×100): 1.9 -> 1.8 | Helped |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · SHREC’19 | geodesic error (×100): 4.6 -> 3.6 | Helped |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · SMAL | geodesic error (×100): 3.9 -> 3.3 | Helped |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · DT4D-H intra-class | geodesic error (×100): 0.9 -> 1.0 | Hurt |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · DT4D-H inter-class | geodesic error (×100): 4.1 -> 3.5 | Helped |
| Replace the LBO basis with the hybrid LBO-elastic basis | ULRSSM · TOPKIDS | geodesic error (×100): 9.2 -> 5.0 | Helped |
| Adapt the elastic-basis solver to non-orthogonality using the HS norm instead of the standard solver (ablation) | ULRSSM with elastic basis · SMAL | geodesic error (×100): 40.2 ± 0.80 -> 5.15 ± 0.99 | Helped |
| Orthogonalize the elastic basis (ablation) | ULRSSM with elastic basis · SMAL | geodesic error (×100): 40.2 ± 0.80 -> 5.75 ± 1.20 | Helped |
| Use the hybrid basis instead of only the LBO basis (ablation) | ULRSSM · SMAL | geodesic error (×100): 5.15 ± 0.99 -> 3.83 ± 0.74 | Helped |
| Add HS-norm adaptation to the hybrid-basis solver (ablation) | ULRSSM · SMAL | geodesic error (×100): 4.33 ± 0.56 -> 3.83 ± 0.74 | Helped |
Surprising findings
- Naively replacing the LBO basis with the elastic basis does not improve, and can degrade, learning-based pipelines (elastic-only ULRSSM with standard solver gives 40.2 geodesic error on SMAL versus 5.15 for LBO).
Limitations
- The k^2 x k^2 expanded linear system introduces computational overhead, limiting elastic basis size (60 in ULRSSM).
- Elastic basis lacks isometric invariance and underperforms when used alone in learned settings.
- Inter-basis maps are assumed zero (block-diagonal), an assumption rather than learned.
Experimental setup
Datasets: FAUST SCAPE SHREC'19 SMAL DeformingThings4D (DT4D-H) TOPKIDS
Benchmarks: FAUST SCAPE SHREC'19 SMAL DT4D-H intra-class DT4D-H inter-class TOPKIDS
Metrics: mean geodesic error percentage of correct keypoints
Baselines: ZoomOut DiscreteOp Smooth Shells FMNet GeomFMaps Deep Shells DUO-FMNet AttentiveFMaps AttentiveFMaps-Fast SSCDFM ULRSSM
Builds on: functional maps ZoomOut DiffusionNet elastic thin-shell basis ULRSSM GeomFMaps Smooth Shells
Closest papers by meaning
- Memory-Scalable and Simplified Functional Map Learning (CVPR 2024 · similarity 0.89)
- Efficient Deformable Shape Correspondence via Multiscale Spectral Manifold Wavelets Preservation (CVPR 2021 · similarity 0.89)
- Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation (CVPR 2024 · similarity 0.88)
- Denoising Functional Maps: Diffusion Models for Shape Correspondence (CVPR 2025 · similarity 0.88)
- Locality Preserving Refinement for Shape Matching with Functional Maps (AAAI 2024 · similarity 0.86)
- Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning (CVPR 2024 · similarity 0.86)
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