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Locality Preserving Refinement for Shape Matching with Functional Maps
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Proposes Locality Preserving Refinement (LOPR), a two-step pointwise map refinement for functional maps that identifies inliers via neighborhood consensus and re-matches outliers with LLE in a spectral space.
Problem
Recovering accurate pointwise correspondences from functional maps for nonrigid shape matching with outliers, where nearest-neighbor spectral matching lacks continuity and expensive assignment solvers are costly.
Approach
Identify inliers with a closed-form neighborhood-consensus score (linear time), then re-match outliers by locally linear embedding over K nearest spectral neighbors, embedding geometric locality constraints into the functional space with linearithmic cost.
Key results
- LOPR achieves average relative geodesic error of 0.0070 on FAUST versus 0.0084 for GCPD and 0.0100 for MWP.
- On TOPKIDS LOPR reports 0.0466 average error, lower than Sinkhorn (0.0515) and MWP (0.0502).
- LOPR runs faster than Sinkhorn and GCPD (e.g., 9.47 s vs 63.87 s and 14.18 s on TOPKIDS), though slower than MWP alone.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply LOPR pointwise map refinement to MWP instead of using GCPD | MWP pointwise map recovery · FAUST | average relative geodesic error: 0.0084 -> 0.0070 | Helped |
| Apply LOPR pointwise map refinement to MWP instead of using GCPD | MWP pointwise map recovery · SCAPE | average relative geodesic error: 0.0087 -> 0.0091 | Hurt |
| Apply LOPR pointwise map refinement to MWP instead of using GCPD | MWP pointwise map recovery · TOPKIDS | average relative geodesic error: 0.0709 -> 0.0466 | Helped |
| Apply LOPR pointwise map refinement to MWP instead of using GCPD | MWP pointwise map recovery · TOSCA | average relative geodesic error: 0.0159 -> 0.0157 | Helped |
Surprising findings
- On SCAPE, LOPR's average error (0.0091) is slightly higher than MWP (0.0096) and GCPD (0.0087) is better, so LOPR does not win on every dataset.
- GCPD has the highest error on TOPKIDS (0.0709) despite strong results elsewhere.
Limitations
- Hyperparameter lambda must be tuned (set to 0.2 on FAUST pairs), and K1/K2 are set heuristically.
- Slower than the MWP baseline alone due to the added refinement step.
- Gains on SCAPE are marginal relative to MWP and GCPD.
Experimental setup
Datasets: FAUST SCAPE TOSCA TOPKIDS Partial Functional Correspondences cuts dataset
Benchmarks: FAUST SCAPE TOSCA TOPKIDS
Metrics: average relative geodesic error correspondence quality characteristics (CQC) curves runtime
Baselines: ICP BCICP ZoomOut DiscreteOp SmoothShells MWP Sinkhorn GCPD SHOT
Builds on: functional maps ZoomOut Locally Linear Embedding Coherent Point Drift
Closest papers by meaning
- Memory-Scalable and Simplified Functional Map Learning (CVPR 2024 · 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)
- 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 0.87)
- Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning (CVPR 2024 · similarity 0.87)
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