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Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation
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Unsupervised framework that jointly learns 3D shape matching and interpolation by fusing deep functional maps (spectral) with ARAP-based spatial deformation, plus a test-time adaptation for shape-dominant deformations.
Problem
Shape matching and shape interpolation between non-rigid 3D shapes are usually studied separately and applied sequentially, giving sub-optimal correspondences and interpolations; existing functional map methods yield non-smooth point-wise maps and geodesic-based interpolation methods are costly and
Approach
Combine spectral (deep functional map) regularisation with spatial (ARAP deformation, alignment, symmetry and temporal variance) losses in one unsupervised model, and add a test-time optimisation of a shape-dominant deformation field under Chamfer distance to capture shape-dominant deformations and refine correspondences.
Key results
- Achieves 1.4 geodesic error on FAUST, 1.8 on SCAPE, and 3.2 on SHREC'19 cross-dataset, beating URSSM (1.6, 1.9, 4.6).
- On SMAL non-isometric matching, geodesic error 1.9 vs URSSM 3.9; DT4D-H inter-class 3.3 vs URSSM 4.1.
- On LUNA16 SSM, generalisation 2.44 mm and specificity 5.03 vs S3M 2.92 and 6.23.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| combine spectral and spatial maps | Ours · FAUST | mean geodesic error: 1.6 -> 1.4 | Helped |
| combine spectral and spatial maps | Ours · SCAPE | mean geodesic error: 1.9 -> 1.8 | Helped |
| combine spectral and spatial maps | Ours · SHREC’19 trained on FAUST + SCAPE | mean geodesic error: 4.6 -> 3.2 | Helped |
| combine spectral and spatial maps | Ours · SMAL | geodesic error (×100): 3.9 -> 1.9 | Helped |
| combine spectral and spatial maps | Ours · DT4D-H inter-class | geodesic error (×100): 4.1 -> 3.3 | Helped |
| add spatial regularisation to spectral-only matching (ablation) | Ours · SMAL | geodesic error (×100): 3.9 -> 1.9 | Helped |
| add spatial regularisation to spectral-only matching (ablation) | Ours · DT4D-H inter-class | geodesic error (×100): 4.1 -> 3.3 | Helped |
| add test-time adaptation (ablation) | Ours · SMAL | geodesic error (×100): 2.6 -> 1.9 | Helped |
| add test-time adaptation (ablation) | Ours · DT4D-H inter-class | geodesic error (×100): 3.4 -> 3.3 | Helped |
| use Ours instead of S3M | LUNA16 statistical shape model · LUNA16 SSM | generalisation: 2.92 -> 2.44 | Helped |
| use Ours instead of S3M | LUNA16 statistical shape model · LUNA16 SSM | specificity: 6.23 -> 5.03 | Helped |
| use Ours instead of Hamiltonian | shape interpolation · FAUST | AUC (Chamfer distance): 0.91 -> 0.95 | Helped |
Surprising findings
- Interpolation-based test-time adaptation improves matching; ablation shows removing it raises SMAL error from 1.9 to 2.6.
Limitations
- Separation between pose-dominant and shape-dominant deformation is not always perfect; the interpolation module sometimes alters shape.
- Test-time adaptation adds per-pair optimisation cost.
Experimental setup
Datasets: FAUST SCAPE SHREC'19 SMAL DT4D-H MANO LUNA16
Benchmarks: FAUST SCAPE SHREC'19 SMAL DT4D-H MANO LUNA16
Metrics: mean geodesic error PCK AUC conformal distortion Chamfer distance SSM generalisation SSM specificity
Baselines: URSSM AttnFMaps DUO-FMNet NeuroMorph Deep Shells GeomFMaps ZoomOut Smooth Shells DiscreteOp LIMP Hamiltonian S3M
Builds on: deep functional maps NeuroMorph ARAP DiffusionNet URSSM
Closest papers by meaning
- Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching (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 0.88)
- Memory-Scalable and Simplified Functional Map Learning (CVPR 2024 · similarity 0.88)
- Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning (CVPR 2024 · similarity 0.87)
- Multi-Shape Matching with Cycle Consistency Basis via Functional Maps (AAAI 2025 · similarity 0.87)
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