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SE(3)-bi-equivariant Transformers for Point Cloud Assembly
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Proposes BITR, a correspondence-free SE(3)-bi-equivariant transformer that assembles point clouds by projecting learned SE(3)×SE(3) features to SE(3) via an Arun-type SVD step, with swap and scale equivariance incorporated via weight constraints.
Problem
Point cloud assembly: recovering the rigid transformation aligning a source point cloud to a reference when the clouds may be non-overlapping and start from arbitrary poses; correspondence-based registration methods fail in these settings.
Approach
Extend SE(3)-transformers to an SE(3)×SE(3) equivariant transformer acting on a 6-D merged point cloud of learned key points, then project the learned degree-(1,1) feature and offsets to SE(3) with an Arun-style SVD projection, giving SE(3)-bi-equivariance without correspondences; swap and scale equivariance are added by weight constraints.
Key results
- On 2-fragment WineBottle, BITR achieves Δr 8.4 (0.9) and Δt 0.07 versus 98–101 Δr for DGL, NSM and LEV.
- On 7Scenes, BITR+OT reaches Δr 9.5 and Δt 0.3, comparable to GEO (9.2) and ROI (9.0); plain BITR gets Δr 26.7.
- BITR outperforms baselines on ShapeNet airplane assembly when overlap is small (s ≤ 0.5).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BITR instead of LEV | 2-fragment WineBottle reassembly · 2-fragment WineBottle (BB) | rotation error (Δr): 98.3 -> 8.4 (0.9) | Helped |
| Use BITR instead of DGL | 2-fragment WineBottle reassembly · 2-fragment WineBottle (BB) | translation error (Δt): 0.09 -> 0.07 (0.008) | Helped |
| Add ICP refinement | BITR · 7Scenes | rotation error (Δr): 26.7 (0.0) -> 11.1 (0.0) | Helped |
| Add OT refinement | BITR · 7Scenes | rotation error (Δr): 26.7 (0.0) -> 9.5 (0.0) | Helped |
| Add ICP refinement | BITR · 7Scenes | translation error (Δt): 0.8 (0.0) -> 0.3 (0.0) | Helped |
| Add OT refinement | BITR · 7Scenes | translation error (Δt): 0.8 (0.0) -> 0.3 (0.0) | Helped |
Limitations
- Computationally inefficient since each feature degree is computed independently without parallelism.
- Deterministic, so it cannot handle symmetric point clouds and predicts only one result.
- Can produce unrealistic outputs such as colliding point clouds in manipulation tasks.
Experimental setup
Datasets: ShapeNet BB dataset 7Scenes ASL
Benchmarks: ShapeNet airplane BB WineBottle 7Scenes
Metrics: rotation error (degrees) translation error
Baselines: GEO ROI NSM LEV DGL ICP OMN
Builds on: SE(3)-transformer Arun's method tensor field networks Vector Neurons Elu
Closest papers by meaning
- SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks (NeurIPS 2020 · similarity 0.89)
- Approximately Piecewise E(3) Equivariant Point Networks (ICLR 2024 · similarity 0.87)
- Not-So-Optimal Transport Flows for 3D Point Cloud Generation (ICLR 2025 · similarity 0.86)
- SIEVE: Overcoming Topological Obstruction in Equivariant Self-Supervised Learning (NeurIPS 2026 · similarity 0.85)
- Fast, Expressive $\mathrm{SE}(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space (ICLR 2024 · similarity 0.85)
- Point Transformer V3: Simpler Faster Stronger (CVPR 2024 · similarity 0.85)
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