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Polyhedral Complex Derivation from Piecewise Trilinear Networks
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Shows that under the eikonal constraint, hypersurfaces inside a piecewise trilinear region (HashGrid + ReLU MLP SDF) become planes, enabling analytical polyhedral mesh extraction via curved edge subdivision.
Problem
Analytically extracting exact polyhedral meshes from SDF networks whose positional encoding (trilinear interpolation such as HashGrid) makes the network non-piecewise-affine, so CPWA-based mesh extraction methods do not directly apply.
Approach
Views ReLU networks with a trilinear positional encoding as piecewise trilinear, proves that under the eikonal constraint the zero-level hypersurface within a trilinear cell reduces to a plane, and extends edge subdivision to curved edges by approximating the intersection of two hypersurfaces with a diagonal plane.
Key results
- On the Stanford bunny with the Small model, the method reaches chamfer efficiency 25.6 versus 19.0 for marching cubes at 32 samples per axis, using about 4.3K vertices.
- On the Drill Bit, chamfer efficiency is 47.8 versus 19.2 for marching cubes at 64 samples per axis.
- Mesh extraction for the Large bunny model takes 4.15 ± 0.61 seconds for about 140K vertices, versus 1.31 seconds for marching cubes at 256 samples per axis.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed extraction method instead of marching cubes | Mesh extraction on Drill · Drill | CE: 19.2 -> 47.8 | Helped |
| Remove the eikonal loss (ablation) | Piecewise trilinear networks and mesh extraction · Unit sphere | flatness error | Hurt |
| Increase the eikonal-loss weight (ablation) | Piecewise trilinear networks · Unit sphere | flatness error | Helped |
| Use the Large model's smaller grid size instead of the Small model's grid size | Proposed mesh extraction · Stanford 3D Scanning objects | CD | Helped |
Surprising findings
- Marching cubes at 256 samples per axis is faster (1.31 s) than the proposed method on Large models, so the method is slower in wall-clock time at that scale.
Limitations
- Intermediate vertices and edges during edge subdivision are numerous (over 11K vertices for the Small bunny) before reduction, creating overhead.
- Exact planarity holds only approximately; the diagonal-plane approximation introduces error controlled by the eikonal loss weight and epsilon.
- Analysis is limited to HashGrid and ReLU networks; generalization to TensoRF is argued informally rather than empirically validated.
Experimental setup
Datasets: Stanford 3D Scanning repository (Bunny, Dragon, Happy Buddha, Armadillo, Drill Bit, Lucy)
Benchmarks: Stanford 3D Scanning repository
Metrics: chamfer distance chamfer efficiency angular distance flatness error time
Baselines: Marching Cubes Marching Tetrahedra Neural Dual Contour
Builds on: edge subdivision InstantNGP HashGrid NeuS Marching Cubes
Closest papers by meaning
- PoNQ: a Neural QEM-based Mesh Representation (CVPR 2024 · similarity 0.86)
- DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling (CVPR 2024 · similarity 0.84)
- PolyhedronNet: Representation Learning for Polyhedra with Surface-attributed Graph (ICLR 2025 · similarity 0.84)
- Flatten Anything: Unsupervised Neural Surface Parameterization (NeurIPS 2024 · similarity 0.84)
- MIND: Material Interface Generation from UDFs for Non-Manifold Surface Reconstruction (NeurIPS 2025 · similarity 0.83)
- Dynamic Neural Surfaces for Elastic 4D Shape Representation and Analysis (CVPR 2025 · similarity 0.83)
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