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SpiderMatch: 3D Shape Matching with Global Optimality and Geometric Consistency
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Represents the source 3D shape as a self-intersecting SpiderCurve and solves a cyclic shortest-path ILP in the product graph with the target shape, yielding globally optimal, geometrically consistent 3D shape matching.
Problem
Dense 3D-3D non-rigid shape matching that is both globally optimal and geometrically consistent; prior global formulations (QAP, product-space 3D-3D matching) are NP-hard or intractable, and learning-based methods lack consistency guarantees.
Approach
Discretise the source surface as a long self-intersecting SpiderCurve (TSP-based cycle covering all vertices), then match it to the target as a cyclic shortest path in the curve-by-shape product graph, using ILP constraints that preserve existing self-intersections and forbid new ones.
Key results
- On FAUST, mean geodesic error 0.029 for the proposed method vs 0.031 for Cao et al. and 0.042 for Roetzer et al.
- On SMAL, mean geodesic error 0.044 vs 0.048 for Cao et al.; DT4D-H intra-class 0.027 vs 0.033 for Cao et al.
- Gurobi solves instances with 1000 triangles in about 100 s, all instances solved to global optimality in under 200 s; all 400 experimental instances were feasible.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add PES and ANS constraints (ablation) | SpiderMatch · FAUST | mean geodesic error: 0.036 -> 0.034 | Helped |
| add avoidance of new self-intersections (ANS) (ablation) | SpiderMatch · FAUST | mean geodesic error: 0.035 -> 0.034 | Helped |
| add preservation of existing self-intersections (PES) (ablation) | SpiderMatch · FAUST | mean geodesic error: 0.035 -> 0.034 | Helped |
| use the robust loss instead of L2 loss (ablation) | SpiderMatch · FAUST | mean geodesic error: 0.036 -> 0.034 | Helped |
| use learned features instead of WKS features (ablation) | SpiderMatch · FAUST | mean geodesic error: 0.213 -> 0.034 | Helped |
| use SpiderMatch instead of Cao et al. | 3D shape matching · FAUST | mean geodesic error: 0.031 -> 0.029 | Helped |
| use SpiderMatch instead of Cao et al. | 3D shape matching · SMAL | mean geodesic error: 0.048 -> 0.044 | Helped |
| use SpiderMatch instead of Cao et al. | 3D shape matching · DT4D-H intra-class | mean geodesic error: 0.033 -> 0.027 | Helped |
| use SpiderMatch instead of Cao et al. | 3D shape matching · DT4D-H inter-class | mean geodesic error: 0.041 -> 0.041 | No clear effect |
Surprising findings
- Constraint matrix is not totally unimodular, so worst-case runtime may be exponential, yet off-the-shelf ILP solving is fast in practice on the tested instances.
Limitations
- Worst-case exponential runtime since constraints are not totally unimodular; no theoretical complexity analysis
- SpiderCurve extraction via TSP/Christofides is heuristic and alternative curve discretisations were not explored
- ILP may be infeasible in low-resolution settings due to discretisation artefacts
Experimental setup
Datasets: FAUST SMAL DT4D-H
Benchmarks: FAUST SMAL DT4D-H intra-class DT4D-H inter-class
Metrics: mean geodesic error percentage of correct matchings vs geodesic error threshold runtime
Baselines: Cao et al. Ren et al. Eisenberger et al. Roetzer et al.
Builds on: Lähner et al. curve-to-3D shape matching Cao et al. (DiffusionNet with point maps) Roetzer et al. triangle-matching ILP Christofides algorithm
Closest papers by meaning
- Partial-to-Partial Shape Matching with Geometric Consistency (CVPR 2024 · similarity 0.88)
- Higher-Order Ratio Cycles for Fast and Globally Optimal Shape Matching (CVPR 2025 · similarity 0.87)
- Multi-Shape Matching with Cycle Consistency Basis via Functional Maps (AAAI 2025 · similarity 0.87)
- HOPE: Shape Matching Via Aligning Different K-hop Neighbourhoods (NeurIPS 2024 · similarity 0.85)
- 3D Geometric Shape Assembly via Efficient Point Cloud Matching (ICML 2024 · similarity 0.85)
- Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation (CVPR 2024 · similarity 0.84)
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