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LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions
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Proposes LISR, a learned linear implicit surface representation using compactly supported RBFs with a query-point selection scheme that makes the least-squares SDF problem full-rank, enabling a network to predict coefficients from point clouds.
Problem
Learning implicit 3D surfaces from partial, noisy point clouds with linear RBF representations fails to converge because the basis-by-query matrix is not full rank, so gradient descent does not reach the global least-squares solution.
Approach
Replace globally supported RBFs with compactly supported gradient-of-cubic RBFs whose local supports come from Voronoi partitions of kernel points, and sample three axis-offset query points per support so that VV^T becomes full rank (identity-like), giving well-conditioned training of the linear coefficients by a two-network (kernel prediction and coefficient prediction) architecture.
Key results
- LISR achieves mean Chamfer distance 0.027 on five ShapeNet classes, the best among compared methods (OccNet 0.046, ConvNet 0.040, IF-Net 0.030, 3DILG 0.028).
- Mean F-score of LISR is 0.987, comparable to IF-Net and 3DILG (0.988).
- On single-shape overfitting with 1000 kernel points, VV^T rank is 3000/3000 for CSRBF versus 13/3000 for HRBF, 954/1000 for tri-harmonic and 997/1000 for mono-harmonic.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LISR instead of 3DILG | 3D surface reconstruction · ShapeNet (mean, five classes) | Chamfer distance (CD): 0.028 -> 0.027 | Helped |
| Use LISR instead of 3DILG | 3D surface reconstruction · ShapeNet (mean, five classes) | F-Score: 0.988 -> 0.987 | Hurt |
| Use compactly supported CSRBF basis instead of Tri-Harmonic basis (ablation) | LISR single-shape learning · Single-shape learning | rank of VVᵀ: 954 -> 3000 | Helped |
| Use compactly supported CSRBF basis instead of Mono-Harmonic basis (ablation) | LISR single-shape learning · Single-shape learning | rank of VVᵀ: 997 -> 3000 | Helped |
| Use compactly supported CSRBF basis instead of HRBF (ablation) | LISR single-shape learning · Single-shape learning | rank of VVᵀ: 13 -> 3000 | Helped |
Surprising findings
- Global-support RBFs (HRBF, tri-harmonic, mono-harmonic) fail to reach the least-squares solution when trained with gradient descent, as shown by rank deficiency of VV^T.
- LISR's F-score is marginally lower than IF-Net and 3DILG on average despite better Chamfer distance.
Limitations
- Supports mainly single-object tasks trained on ShapeNet, not scene-level reconstruction.
- Requires ground-truth signed distance field supervision.
- Shape is highly sensitive to coefficient perturbations in global-support bases; F-score is only comparable, not best, to state of the art.
Experimental setup
Datasets: ShapeNet MVP
Benchmarks: ShapeNet MVP
Metrics: Chamfer distance F-score rank of VV^T
Baselines: OccNet ConvNet IF-Net 3DILG Tri-harmonic RBF Mono-harmonic RBF HRBF
Builds on: Hermite radial basis functions Occupancy Networks IF-Net 3DILG Tri-harmonic RBF
Closest papers by meaning
- Iso-Points: Optimizing Neural Implicit Surfaces With Hybrid Representations (CVPR 2021 · similarity 0.90)
- Learning to Infer Implicit Surfaces without 3D Supervision (NeurIPS 2019 · similarity 0.89)
- Local Deep Implicit Functions for 3D Shape (CVPR 2020 · similarity 0.89)
- ARO-Net: Learning Implicit Fields From Anchored Radial Observations (CVPR 2023 · similarity 0.88)
- A Lightweight UDF Learning Framework for 3D Reconstruction Based on Local Shape Functions (CVPR 2025 · similarity 0.88)
- Small Steps and Level Sets: Fitting Neural Surface Models with Point Guidance (CVPR 2024 · similarity 0.86)
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