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GSO-Net: Grid Surface Optimization via Learning Geometric Constraints
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Encodes grid surfaces as geometric images and trains a residual image-denoising-style network (GSO-Net) with geometric-constraint losses to optimize developable surfaces, flatten surfaces, and denoise surfaces, far faster than numerical optimization.
Problem
Discrete surface optimization (developable surface optimization, surface flattening, surface denoising) is slow and hard to scale with numerical optimization, and existing mesh deep learning methods are not suited to precise surface optimization.
Approach
Map grid vertex positions to RGB pixel values so a grid surface becomes a geometric image, then learn vertex displacements with a residual image network trained self-supervised using discrete geometric losses (Gaussian curvature, fairness, isometry, fidelity) implemented as fixed convolution kernels.
Key results
- Net-CF reduces dH-a from 0.823% (TNO) to 0.517% (about 37% relative reduction) on developable surface optimization at 0.026s versus 38.256s for TNO.
- On surface flattening, Net-W reduces lossiso-cell from 3.30e-4 (TNO) to 2.46e-4 (about 25% reduction) with 0.015s runtime versus 4.235s.
- Surface denoising achieves theta_opt-r-a of 83.84% to 94.42% across noise levels sigma 0.001 to 0.015.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace TNO with GSO-Net Net-S | Developable surface optimization · Developable surface optimization | dH−a: 0.823% -> 0.608% | Helped |
| Replace TNO with GSO-Net Net-S | Developable surface optimization · Developable surface optimization | time: 38.256s -> 0.013s | Helped |
| Use the coarse-to-fine strategy instead of a single model (ablation) | GSO-Net · Developable surface optimization | dH−a: 0.608% -> 0.517% | Helped |
| Use the coarse-to-fine strategy instead of a single model (ablation) | GSO-Net · Developable surface optimization | time: 0.013s -> 0.026s | Hurt |
| Use the coarse model rather than the single model (ablation) | GSO-Net · Developable surface optimization | dH−a: 0.608% -> 0.414% | Helped |
| Use the coarse model rather than the single model (ablation) | GSO-Net · Developable surface optimization | KA−r−a: 0.8776 -> 0.8300 | Hurt |
| Add dynamic isometry-loss weights (ablation) | GSO-Net Net-W · Surface flattening | lossiso−cell: 2.94e-4 -> 2.46e-4 | Helped |
| Replace TNO with GSO-Net | Surface flattening · Surface flattening | lossiso−cell: 3.30e-4 -> 2.94e-4 | Helped |
| Increase noise level from σ = 0.001 to σ = 0.015 | GSO-Net · Surface denoising | θopt−r−a: 83.84% -> 94.30% | Mixed |
Surprising findings
- The Net-C intermediate model lowers dH-a more (0.414%) than the full coarse-to-fine model but sacrifices developability (KA-r-a drops about 6%).
Limitations
- Developable optimization trades off developability (KA-r-a) against proximity; TNO achieves higher KA-r-a on low-curvature or simple surfaces.
- Denoising experiments are described as simple and limited in scope.
Experimental setup
Datasets: Bezier-surface-generated grid surface dataset (10401 surfaces)
Benchmarks: Grid Surface Dataset
Metrics: dH-a (average Hausdorff distance) KA-r-a (Gaussian curvature reduction rate) lossiso-cell normal angle difference time
Baselines: TNO (traditional numerical optimization) Flatten-init
Builds on: IMDB (image multi-distillation block) DnCNN residual learning Jiang et al. 2020 isometry-based flattening Qi et al. 2017 point cloud normalization Meyer et al. 2003 discrete curvature
Closest papers by meaning
- GridFormer: Point-Grid Transformer for Surface Reconstruction (AAAI 2024 · similarity 0.85)
- Learning Bijective Surface Parameterization for Inferring Signed Distance Functions from Sparse Point Clouds with Grid Deformation (CVPR 2025 · similarity 0.85)
- FNIN: A Fourier Neural Operator-based Numerical Integration Network for Surface-from-gradients (AAAI 2025 · similarity 0.84)
- Flatten Anything: Unsupervised Neural Surface Parameterization (NeurIPS 2024 · similarity 0.83)
- Improving Neural Network Surface Processing with Principal Curvatures (NeurIPS 2024 · similarity 0.83)
- PoNQ: a Neural QEM-based Mesh Representation (CVPR 2024 · similarity 0.83)
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