Research map / Point clouds / Shape and surface completion
Consistent Normal Orientation for 3D Point Clouds via Least Squares on Delaunay Graph
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Proposes a global, non-learning normal orientation method that relaxes binary sign assignment on a Delaunay graph built from a shape-diameter-guided level set into a sparse least-squares problem.
Problem
Globally consistent orientation of surface normals in dense 3D point clouds, which is hard under noise, outliers, holes, thin structures and complex topology.
Approach
Relax the binary normal orientation problem to a least-squares sign field on a Delaunay graph of a shape-diameter-chosen level set of an unsigned implicit function; solve a sparse linear system with convex hull vertices fixed to +1, then flip normals along the sign-field gradient.
Key results
- On clean fifteen-shape benchmarks, the proposed method achieves NION 0 on many shapes (e.g., torus, holes, bunny, kitten) with runtime of seconds, versus GCNO taking up to over 12 hours on large clouds.
- Under 0.25% noise and 5% outliers, the method keeps NION at 0 on most small and medium shapes where MST, SNO, Dipole and GCNO fail on some.
- Compared with learning-based methods (PCPNet, DPGO, SHSNet, NGL, NeuralGF), it yields lower NION on large complex shapes such as dragon (493) and filigree (426) and runs without a GPU.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed Delaunay-graph least-squares method instead of SHSNet | normal orientation · bird (7k) | NION: 252 -> 3 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · fandisk (11k) | NION: 4 -> 2 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of SHSNet | normal orientation · bull (25k) | NION: 16 -> 15 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · feline (50k) | NION: 114 -> 7 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · metatron (69k) | NION: 1065 -> 11 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · elephant (93k) | NION: 4 -> 0 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · honeycomb (129k) | NION: 4136 -> 61 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NGL | normal orientation · hilbert cube (144k) | NION: 47k -> 0 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NGL | normal orientation · dragon (296k) | NION: 1873 -> 493 | Helped |
| Use the proposed Delaunay-graph least-squares method instead of NeuralGF | normal orientation · filigree (300k) | NION: 9k -> 426 | Helped |
Surprising findings
- Learning-based supervised methods such as PCPNet and DPGO are reported to struggle on holes (e.g., bird), possibly because they were trained on closed point clouds.
Limitations
- Assumes the underlying surface is closed so space splits into inner and outer regions.
- Sensitive to the user-specified ratio alpha; overly large values cause incorrect orientation in thin structures.
Experimental setup
Datasets: PCPNet FamousShapes Thingi10K AIM@Shape
Benchmarks: PCPNet FamousShapes Thingi10K AIM@Shape real scans
Metrics: NION (number of incorrectly oriented normals) running time
Baselines: MST SNO Dipole GCNO PCPNet DPGO SHSNet NGL NeuralGF
Builds on: shape diameter function extended point-based shape diameter CGALMesh winding/unsigned implicit function (Iu)
Closest papers by meaning
- NeuralGF: Unsupervised Point Normal Estimation by Learning Neural Gradient Function (NeurIPS 2023 · similarity 0.88)
- An Explicit Frame Construction for Normalizing 3D Point Clouds (ICML 2024 · similarity 0.87)
- High-quality Point Cloud Oriented Normal Estimation via Hybrid Angular and Euclidean Distance Encoding (CVPR 2025 · similarity 0.87)
- Symmetry-Robust 3D Orientation Estimation (ICML 2025 · similarity 0.86)
- Digging into Intrinsic Contextual Information for High-fidelity 3D Point Cloud Completion (AAAI 2025 · similarity 0.86)
- CMG-Net: Robust Normal Estimation for Point Clouds via Chamfer Normal Distance and Multi-Scale Geometry (AAAI 2024 · similarity 0.85)
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