cs.CVSep 28, 2026
SaveProjective Normal Fields: A Convex Optimization Method for Constructing Smooth UDFs
Organizations: S-Lab Nanyang Technological University Singapore · College of Mathematics Nanjing University of Aeronautics and Astronautics China · Institute of Software Chinese Academy of Sciences China · Department of Computer Science City University of Hong Kong China · Department of Computer Science and Engineering Texas A&M University USA
Abstract
Constructing a smooth approximation of an unsigned distance field (UDF) from a raw point cloud is challenging because the input provides neither surface connectivity nor consistently oriented normals. Methods that directly learn a scalar UDF must also handle its non-differentiability on the zero level set and weak supervision away from the samples, which can lead to unstable optimization and spatial artifacts. We introduce Projective Normal Fields (PNFs), an orientation-free representation and convex optimization framework for estimating bidirectional normals from point positions alone. Each normal axis is encoded by a rank-one projector, which is invariant to normal reversal. We relax the non-convex set of hard projectors to its convex hull: the symmetric positive-semidefinite matrices with unit trace. Each soft tensor defines a local quadratic distance model and retains the relative weights of candidate normal axes. We estimate a coherent PNF by combining local tangent-plane fitting, soft-PCA anchoring, and overlap regularization on a fixed neighborhood graph. With positive anchoring weights, the objective is strongly convex and admits a unique global minimizer. Principal eigenvectors provide bidirectional normals, while the corresponding eigengaps provide spectral confidence indicators. We use these indicators to select and weight directional sources for heat diffusion, followed by Poisson integration to construct a regularized UDF approximation. By separating local geometry estimation from scalar-field construction, PNF avoids directly fitting the non-differentiable UDF. Experiments demonstrate reduced sensitivity to neighborhood size, competitive reconstruction under noise and outliers, and improved accuracy near non-manifold junctions. The project page is available at https://anonymous17777367.github.io/PNF-page/
Figures & tables
Figure 1: Reconstruction under spatially varying noise and outliers. Top row: the corrupted Goldfish point cloud, the PNF reconstruction, and planar cross-sections of the computed UDF. Warmer input point colors indicate greater corruption. PNF preserves thin fins, open boundaries, and non-manifold junction geometry, while the color-coded distance values and level-set contours illustrate smooth spatial variation in the displayed slices. Bottom row: reconstructions from competing methods, which exhibit holes, surface irregularities, or spurious fragments.
Figure 2: Two-stage PNF reconstruction on a 2D Y-shaped point cloud. Given unoriented samples (a), Stage I initializes random normal axes (b), encodes them as rank-one projectors (c), and optimizes their soft relaxation using tangent-plane fitting, soft-PCA anchoring, and overlap regularization (d). Principal eigenvectors yield the decoded normal axes, while eigengaps provide confidence values (e). Bidirectional arrows represent unoriented axes; colors in (e) indicate confidence, which is lower near the junction, where multiple branch directions compete, and higher along the regular branches. Stage II applies confidence-guided heat diffusion followed by Poisson integration to construct the UDF approximation (f).
| Model | Method | ||||||
|---|---|---|---|---|---|---|---|
| Toy | PCA | 0.9693 | 0.9900 | 0.9660 | 0.9746 | 0.9828 | 0.9833 |
| Ours | 0.9934 | 0.9937 | 0.9938 | 0.9937 | 0.9937 | 0.9934 | |
| Ship | PCA | 0.9525 | 0.9079 | 0.8815 | 0.9301 | 0.9408 | 0.9451 |
| Ours | 0.9628 | 0.9621 | 0.9555 | 0.9523 | 0.9505 | 0.9515 | |
| Leaf | PCA | 0.9609 | 0.9725 | 0.9707 | 0.9605 | 0.9629 | 0.9778 |
| Ours | 0.9863 | 0.9878 | 0.9876 | 0.9869 | 0.9860 | 0.9847 |
Table 1: Sensitivity to neighborhood size . We report mean absolute cosine similarity between estimated and reference normal axes on four models with thin structures. Higher values are better, with indicating perfect agreement. PNF exhibits less variation across the tested neighborhood sizes than local PCA. The better result for each model and neighborhood size is shown in bold . The plots visualize the same data.
Figure 3: Global PNF optimization vs. local PCA fitting. We compare normal-axis estimates from PCA (top row) and PNF (bottom row) on thin structures with (left) and inputs corrupted by noise (right), together with the corresponding surface reconstruction results. Colors indicate per-point normal error, from blue (low) to red (high). Numbers report the mean absolute cosine similarity between estimated and reference normals; higher values are better, with indicating perfect agreement. Unlike PCA, which fits each neighborhood independently, PNF jointly optimizes the normal-axis field over the entire neighborhood graph. This global formulation combines local geometric evidence with inter-sample consistency, yielding more accurate axes and improved surface reconstructions in these challenging examples.
Figure 4: Qualitative comparison on non-manifold models. The top three rows show reconstructions from inputs with spatially varying noise and outliers; warmer point colors indicate greater corruption, and insets show cross-sections at the indicated planes. The bottom three rows show a cross-junction, a multi-junction, and the self-intersecting Henneberg surface. Several baselines exhibit holes, surface irregularities, or spurious floating fragments, whereas PNF produces smoother, more coherent reconstructions that closely follow the junction geometry. DEUDF failed on the three synthetic non-manifold models due to its reliance on locally estimated PCA normals for gradient alignment. See also Table 5 for quantitative results for these non-manifold models.
| Method | Clean | Noise (0.3%) | Noise (0.8%) | Outliers (2%) | Outliers (5%) | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| CD | HD | CD | HD | CD | HD | CD | HD | CD | HD | |
| CAP-UDF | 0.329 | 6.592 | 1.232 | 9.329 | 2.990 | 19.492 | 0.407 | 7.101 | 0.452 | 7.886 |
| GeoUDF | 0.206 | 5.253 | 1.691 | 7.628 | 2.968 | 11.888 | 0.241 | 5.757 | 0.243 | 5.832 |
| DUDF | 0.395 | 7.818 | 2.836 | 18.463 | 2.047 | 10.802 | 0.628 | 9.381 | 0.639 | 10.956 |
| DEUDF | 0.465 | 15.079 | 2.358 | 15.999 | 4.634 | 24.952 | 0.615 | 19.658 | 0.671 | 21.009 |
| VAD | 0.133 | 7.195 | 1.199 | 8.495 | 2.311 | 10.491 | 1.089 | 14.194 | 1.050 | 13.331 |
Table 2: Robustness to input corruptions. Mean per-shape directed CD and HD, evaluated from the reference geometry to the reconstruction, on the same 60 test shapes. Distances are reported in units of ; lower values are better. The best and second-best results in each metric column are shown in bold and underlined , respectively.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
| Model features | Graph type | Normal accuracy ( ) | Mean angular error ( ) |
|---|---|---|---|
| Thin structures | -NN | 0.9661 | 3.673 |
| Radius | 0.9657 | 3.877 | |
| Voronoi | 0.9651 | 4.236 | |
| PCA | 0.9479 | 9.132 | |
| Non-manifold models | -NN | 0.9540 | 10.040 |
| Radius | 0.9444 | 10.976 |
Table 3: Effect of neighborhood graph construction on bidirectional normal estimation. We evaluate PNF using symmetrized -NN, radius-based, and Voronoi-adjacency graphs while keeping the input, objective weights, and optimization settings unchanged. Each graph remains fixed during optimization. We report mean absolute cosine similarity (higher is better) and mean angular error in degrees (lower is better) between estimated and reference normal axes. For non-manifold models, evaluation is restricted to junction neighborhoods rather than the entire surface. PNF maintains comparable accuracy across the three graph constructions and outperforms local PCA in both metrics on the evaluated thin structures and non-manifold models.
| Normal | Confidence | CD ( ) | HD95 ( ) | F-score ( ) |
|---|---|---|---|---|
| PCA | Uniform | 8.98 | 34.03 | 89.36 |
| VAD | Uniform | 7.04 | 25.49 | 95.11 |
| PNF | Uniform | 6.98 | 26.07 | 95.65 |
| PNF | Raw eigengap | 6.44 | 26.53 | 96.06 |
| PNF | Thresholded eigengap | 3.73 | 6.22 | 99.20 |
Table 4: Ablation of normal sources and confidence guidance. Inputs contain both noise and outliers, while the reconstruction back-end and remaining settings are fixed. PCA, VAD, and the first PNF variant use uniform source weights . The remaining PNF variants use raw eigengap weights or thresholded weights as defined in Appendix D . CD and HD95 are reported in units of ; F-score is reported as a percentage at distance tolerance . The best result in each metric is shown in bold .
Figure 5: Normal-axis confidence as a geometric cue. The upper row shows input points colored by the eigengap confidence of the optimized PNF tensors; the lower row shows the corresponding reconstructions. The left examples contain non-manifold junctions, and the right examples contain outliers. Low confidence indicates weak axial preference and can highlight candidate junction regions or unreliable directional estimates.
Figure 6: Qualitative comparison under varying point-cloud densities. With the input size decreasing from 30K to 3K points, competing methods gradually suffer from geometric distortions or reconstruction failures, whereas our method consistently produces stable and coherent surfaces, demonstrating robustness to sparse point-cloud observations.
| Model | Method | Global CD | Junction CD | Junction Recall | HD95 |
|---|---|---|---|---|---|
| Cross-junction (3,000 points) | CAP-UDF | 0.148 | 0.356 | 99.77 | 0.299 |
| GeoUDF | 0.050 | 0.224 | 100.00 | 0.117 | |
| DUDF | 0.200 | 1.221 | 99.75 | 0.597 | |
| VAD | 0.038 | 0.321 | 100.00 | 0.204 | |
| PCA+HM | 0.066 | 0.598 | 100.00 | 0.380 | |
| Ours | 0.014 | 0.115 | 100.00 | 0.071 |
Table 5: Reconstruction accuracy on non-manifold models. Global and junction-region metrics are reported for three synthetic shapes. Distances are evaluated from the reference geometry to the reconstruction. CD and HD95 are reported in units of , and recall as a percentage. The junction-band width and recall tolerance are and of the reference bounding-box diagonal, respectively. The best result for each metric and shape is shown in bold , including ties. GeoUDF uses the PUGeo-Net upsampling variant ( Qian et al., 2020 ) . DEUDF was not included in the table, since it produced severely degraded reconstructions on these examples under the evaluated settings. The main reason is its reliance on locally estimated PCA normals for gradient alignment. Near non-manifold junctions, neighborhoods containing multiple surface sheets can yield unreliable normal axes and misleading directional constraints. Although PNF also uses local PCA information, it treats the resulting tensors as soft priors rather than final normal-axis estimates. Crucially, PNF jointly optimizes the entire normal-axis field over a connectivity graph, allowing individual estimates to be refined through compatibility with neighboring geometric evidence. This global coupling enables information from well-supported neighborhoods to help resolve ambiguous local estimates, while the soft representation retains competing directional preferences where ambiguity persists. Thus, PNF’s advantage lies not in avoiding local PCA, but in reconciling its evidence through a globally coupled, strongly convex normal-estimation problem.
Figure 7: Reconstruction from noisy inputs. The input point clouds are corrupted by Gaussian noise. Several competing methods produce fragmented patches or irregular surfaces, whereas PNF yields smoother, more coherent reconstructions in these examples. Table 2 reports quantitative results on the 60-model benchmark.
Figure 8: Reconstruction from outlier-contaminated inputs. The point clouds contain outliers sampled within the bounding box. Several competing methods produce spurious patches or surface distortions, whereas PNF yields more coherent reconstructions with fewer visible artifacts in these examples. Table 2 reports quantitative results on the 60-model benchmark.