Research map / Point clouds / Shape and surface completion
Matérn Kernels for Tunable Implicit Surface Reconstruction
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes Matérn kernels (including the Laplace kernel) for kernel-based implicit surface reconstruction from oriented point clouds, showing they beat arc-cosine kernels in accuracy and speed and can be made data-dependent via Neural Kernel Fields.
Problem
Kernel ridge regression for implicit surface reconstruction from oriented point clouds relies on kernels; the arc-cosine kernel used by state-of-the-art methods has no tunable parameters, so reconstructions cannot be adjusted when they fail.
Approach
Use stationary Matérn kernels, whose smoothness and bandwidth parameters control the eigenvalue decay and hence the spectral bias / level of detail, as a drop-in replacement for the arc-cosine kernel; theoretically link them to SIREN NTKs and to arc-cosine kernels on the sphere, and make them data-dependent with NKF.
Key results
- On ShapeNet (1,000 points, non-learnable), Matérn 1/2 and 3/2 reach F-Score around 93.7-94.9 vs 92.8 for arc-cosine and 84.3 for SPSR.
- On SRB, Matérn 3/2 performs on par with Neural Splines (CD 0.18 vs 0.17) while being faster to compute (Matérn 1/2 runtime 9.85 s vs NS 19.89 s with NS* reproduction).
- Learned Matérn 1/2 reaches 95.3 IoU vs 95.6 for NKSR on ShapeNet, with training about three days vs six days for NKF on the full dataset.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the arc-cosine kernel with Matérn 3/2 | ShapeNet surface reconstruction · ShapeNet | F-Score: 92.8 -> 94.9 | Helped |
| Replace the arc-cosine kernel with Matérn 3/2 | ShapeNet surface reconstruction · ShapeNet | CD: 4.67 -> 4.00 | Helped |
| Use Matérn 3/2 instead of Matérn 5/2 (ablation) | ShapeNet surface reconstruction · ShapeNet | CD: 5.65 -> 4.00 | Helped |
| Use Matérn 3/2 instead of the Gaussian limit ν→∞ (ablation) | ShapeNet surface reconstruction · ShapeNet | F-Score: 52.4 -> 94.8 | Helped |
| Tune bandwidth h across 0.5, 1, and 2 (ablation) | Matérn surface reconstruction · ShapeNet | F-Score: Matérn 3/2: 94.6, 94.8, 94.9 | Mixed |
| Replace the arc-cosine kernel with Matérn 3/2 | Non-learnable surface reconstruction · SRB | CD: 0.17 -> 0.18 | No clear effect |
| Use Matérn 1/2 instead of arc-cosine | SRB reconstruction runtime · SRB | Time: 11.91 -> 9.85 | Helped |
| Replace the arc-cosine kernel with Matérn 3/2 | Texture reconstruction · GSO (m=10,000) | PSNR: 19.34 -> 19.61 | Helped |
| Replace the arc-cosine kernel with Matérn 3/2 | Learnable surface reconstruction · ShapeNet | IoU: 94.7 -> 95.3 | Helped |
| Use Matérn 1/2 instead of Matérn 3/2 (ablation) | Learnable surface reconstruction · ShapeNet train chairs, test all | IoU: 92.8 -> 93.4 | Helped |
| Vary bandwidth h across 0.5, 1, and 2 (ablation) | Matérn 1/2 surface reconstruction · ShapeNet noise robustness | IoU: No noise: 93.3, 93.6, 93.5; small noise: 91.9, 92.1, 92.1; big noise: 89.5, 89.6, 89.6 | Mixed |
| Use Matérn 1/2 instead of NKSR | Learnable reconstruction under increasing noise · ShapeNet noise robustness | IoU: 3% in IoU from no to big noise versus 4.3% for Matérn 1/2 | Hurt |
Surprising findings
- The Gaussian kernel (Matérn infinity) and Matérn 5/2 perform much worse than the Laplace kernel, showing smoother kernels are not better for reconstruction.
- The authors could not reproduce the Neural Splines result reported in the original paper (0.19 vs 0.17 Chamfer on SRB).
Limitations
- The authors could not reproduce the NS kernel results and had to tune outer-layer variance, so comparisons depend on re-implementation.
- Matérn kernels are not compactly supported; sparsity relies on truncation, which does not guarantee positive definiteness in general.
- Learned variant re-implements NKF since the original authors did not release code.
Experimental setup
Datasets: ShapeNet Surface Reconstruction Benchmark Google Scanned Objects Objaverse
Benchmarks: ShapeNet Surface Reconstruction Benchmark Google Scanned Objects Objaverse
Metrics: F-Score Chamfer distance Hausdorff distance IoU normal consistency PSNR LPIPS runtime
Baselines: Neural Splines Screened Poisson Surface Reconstruction Neural Kernel Fields Neural Kernel Surface Reconstruction Fourier Feature Networks SIREN SAP DiGS VisCo OG-INR
Builds on: Neural Splines Neural Kernel Fields Neural Kernel Surface Reconstruction Fourier feature mappings SIREN
Closest papers by meaning
- Neural Kernel Surface Reconstruction (CVPR 2023 · similarity 0.91)
- POCO: Point Convolution for Surface Reconstruction (CVPR 2022 · similarity 0.87)
- OffsetOPT: Explicit Surface Reconstruction without Normals (CVPR 2025 · similarity 0.85)
- GVKF: Gaussian Voxel Kernel Functions for Highly Efficient Surface Reconstruction in Open Scenes (NeurIPS 2024 · similarity 0.84)
- Arbitrary-Scale Point Cloud Upsampling by Voxel-Based Network with Latent Geometric-Consistent Learning (AAAI 2024 · similarity 0.84)
- LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions (AAAI 2024 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.