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Point Convolutional Neural Networks by Extension Operators
Matan Atzmon, Haggai Maron, Yaron Lipman
TL;DR
PCNN addresses the challenge of extending convolutional neural networks to irregular point clouds. It translates volumetric convolution through extension and restriction operators, achieving strong benchmark performance with robustness to sampling and point order.
Problem
Deep-learning methods for 3D shapes must handle point clouds while preserving useful convolutional flexibility and accommodating irregular representations.
Method
PCNN maps point-cloud functions to volumetric functions with extension, applies Euclidean convolution, and maps results back with restriction.
Results
PCNN achieved state-of-the-art results on standard point-cloud benchmarks and outperformed other point-cloud methods and most methods using surface connectivity.
Takeaways & Limitations
The framework adapts standard image-based CNN architectures to point clouds while using convolution kernels robust to sampling and invariant to point order.
Takeaways & Limitations
PCNN incurs extra computational burden from farthest-point sampling and computing the translating tensor q.
Abstract
from arXiv · showhide
This paper presents Point Convolutional Neural Networks (PCNN): a novel framework for applying convolutional neural networks to point clouds. The framework consists of two operators: extension and restriction, mapping point cloud functions to volumetric functions and vise-versa. A point cloud convolution is defined by pull-back of the Euclidean volumetric convolution via an extension-restriction mechanism. The point cloud convolution is computationally efficient, invariant to the order of points in the point cloud, robust to different samplings and varying densities, and translation invariant, that is the same convolution kernel is used at all points. PCNN generalizes image CNNs and allows readily adapting their architectures to the point cloud setting. Evaluation of PCNN on three central point cloud learning benchmarks convincingly outperform competing point cloud learning methods, and the vast majority of methods working with more informative shape representations such as surfaces and/or normals.
1 Introduction
PCNN extends CNNs to point clouds by translating volumetric operators through extension and restriction, targeting point-order invariance, sampling robustness, and translation-invariant convolutions. The framework adapts image CNNs while retaining point-cloud data and achieves strong benchmark performance.
- Motivation: Point-cloud networks must handle point-order invariance, varying sampling density and distribution, and translation-invariant convolution kernels.These challenges arise because point clouds approximate underlying shapes rather than directly encoding their full structure.
- Method: PCNN uses extension and restriction operators to map point-cloud functions to volumetric functions and pull volumetric operators back to point clouds.Extension maps functions into ambient R3, while restriction performs the inverse action.
- Method: The extension operator is an RBF approximation, restriction samples volumetric functions at point locations, and the resulting convolution is computed with a sparse linear tensor.The tensor combines learnable kernel weights, point-cloud function values, and connections defined directly from the point cloud.
- Properties: Point-order invariance follows from the extension operator, while its approximation power addresses robustness to sampling density and distribution.The construction also provides equivariant convolutional layers with more degrees of freedom than earlier permutation-equivariant approaches.
- Properties: Different samplings of the same surface produce the same extended volumetric function up to approximation error, supporting translation invariance and sampling robustness.The gradient of the extension of the constant function approximates the surface’s mean curvature normal field.
- Evaluation: PCNN outperformed all other point-cloud methods and most methods using richer representations on classification, segmentation, and normal-estimation benchmarks.The framework maintains data over the point cloud while learning sampling-robust convolution kernels.
2 Previous Work
Previous work represents point clouds through symmetric or equivariant networks, volumetric grids, graphs, and surfaces. PCNN relates most directly to volumetric methods while replacing rigid grid-cell bases with more general basis functions.
- Point clouds: PointNet established permutation-invariant point-cloud learning using per-point networks followed by symmetric aggregation.Hierarchical extensions capture structure at multiple resolutions through subsets and suitable aggregation mechanisms.
- Point clouds: Other point-cloud methods use equivariant layers with symmetric functions, kd-tree convolutions, or continuous filters for molecular interactions.These approaches differ in how they encode point relationships and define convolution-like operations.
- Volumetric methods: Volumetric methods place shapes on regular occupancy or distance-function grids, but low resolution limits approximation quality.Sparse structures such as octrees were proposed to reduce the limitations of regular grids.
- Volumetric methods: PCNN generalizes volumetric methods by replacing grid-cell indicator functions with more general basis functions such as radial basis functions.This connects the proposed framework to grid-based shape representations while broadening the function and kernel basis.
- Graphs: Graph convolution in spectral methods uses graph-Laplacian eigenbases, but different graphs have different spectral bases, making correspondences challenging.Functional maps were introduced to target this correspondence problem.
- Surfaces: Surface methods exploit connectivity and normal information through rendering, projections, local representations, or global parameterizations.These methods reduce geometric learning to image or planar-domain processing in different ways.
3 Method
PCNN defines point-cloud operators by extending point-cloud functions into volumetric functions, applying Euclidean convolution, and restricting the result back to sampled points. The construction targets efficiency, point-order equivariance, sampling robustness, and translation-invariant kernels while supporting pooling and upsampling.
- Extension and restriction: PCNN maps point-cloud functions to volumetric functions with extension, applies a volumetric operator, and maps results back with restriction.The restriction operator samples volumetric functions at points in X.
- Operator properties: The point-cloud operator is computationally efficient and equivariant to the ordering of points in X.Its construction provides more degrees of freedom than earlier equivariant layers described in the paper.
- Operator properties: For samples of an underlying surface, extension approximates continuous surface functions and makes operators approximately consistent across different samplings.Extending the constant-one function approximates the surface indicator, while its gradient approximates the mean curvature normal field.
- Operator properties: The operator is translation invariant because it uses a stationary, location-independent convolution kernel.The same kernel is therefore used across point locations.
- RBF construction: The extension remains similar across airplane point clouds sampled with 2048, 1024, and 256 points.The extended functions resemble the airplane indicator function across these sampling densities.
- RBF construction: RBF bases are selected to provide approximation properties and an efficient closed-form convolution computation, with Gaussian bases used in practice.The extension and convolution requirements motivate the Gaussian choice.
- Spatial operators: PCNN also defines upsampling from a point set to a superset and pooling to a subset, with pooling based on nearby-point groups.The next-layer subset is calculated using farthest point sampling.
4 Properties
PCNN’s extension and restriction operators provide point-cloud operators with permutation equivariance and robustness to different samplings. The framework also connects surface-function extensions to geometric quantities and recovers standard image convolutions on regular grids.
- Invariance and equivariance: Extension operators are permutation-invariant, while restriction operators are permutation-equivariant, yielding equivariant point-cloud operators.This construction applies to volumetric operators and, in particular, to point-cloud convolutions.
- Robustness: The extension operator is designed to produce approximately consistent volumetric functions from different samplings of the same smooth surface.Its approximation quality improves as the point-cloud sample becomes denser.
- Robustness: The continuous extension operator approximates surface functions on the surface and decays toward zero away from it.This behavior underlies the extension operator’s approximation properties.
- Robustness: Extending the constant-one surface function approximates the surface indicator, while its gradient approximates mean curvature vectors.These geometric quantities can be revealed from all-ones input data through suitable linear operators.
- Revisiting image CNNs: On regular grid point clouds, the extension-restriction construction reproduces standard discrete image convolution.Images are represented as point clouds in a regular grid configuration with intensities defined over the points.
- Revisiting image CNNs: The paper uses separate standard convolution architectures for classification and segmentation tasks.The segmentation architecture includes upsampling and skip-layer connections to preserve fine output details.
5 Experiments
PCNN was evaluated on classification, segmentation, and normal estimation, including robustness, design variants, feature representations, and computational size. The experiments show sampling robustness, increasingly semantic learned features, competitive benchmark performance, and compact models.
- PCNN was tested on point cloud classification, segmentation, and normal estimation, alongside evaluations of design choices and network variations.
- Point cloud classification: The classification experiments used ModelNet40 and ModelNet10, comparing PCNN with point-cloud methods that use only 3D point coordinates.ModelNet40 contains 9843 training and 2468 test models across 40 classes; ModelNet10 contains 3991 training and 908 test models across 10 classes.
- Method variants: PCNN was evaluated with varying point counts, kernel translations, input tensors, interpolation bases, RBF scale, and numbers of learnable parameters.
- Feature visualizations: The learned first-layer features remain consistent across shapes and sampling conditions, while third-layer features capture more semantically meaningful shape parts.Figure 5 varies sampling from 10K to 1K points; Figure 6 visualizes nine third-layer features.
- Point cloud segmentation: PCNN also supports ShapeNet part segmentation, where it outperforms the compared point-cloud methods and the reported 3DCNN results.The ShapeNet part benchmark contains 16,881 shapes across 16 categories and 50 part labels.
- Model size: The smaller classification network achieves a state-of-the-art result with 1.4M parameters and a total model size of 17 MB.
6 Conclusions
PCNN defines point-cloud convolutions by translating volumetric convolution through extension and restriction operators. Standard benchmark tests report state-of-the-art results with compact networks, while the framework incurs additional computational costs.
- PCNN translates volumetric convolution to arbitrary point clouds using extension and restriction operators.
- Testing on standard point-cloud benchmarks shows state-of-the-art results using compact networks.
- Limitations and future work: The framework’s main limitation is extra computational burden from farthest-point sampling and computing the point-cloud-dependent translating tensor q.The paper identifies sparse efficient implementation as a possible future way to alleviate this limitation.
A.1 Multiplication law for Gaussians
The appendix notes that the convolution of two normal distributions is again a normal distribution, with the result following from convolution linearity.
- The appendix states that convolving two normal distributions produces another normal distribution.
- The stated result follows from the linearity of convolution.
A.2 Theoretical properties of the extension operator
The extension-operator analysis establishes approximation and convergence properties using local surface parameterizations, uniform continuity, and shrinking kernel scales.
- Extension behavior: As σ→0, the extension ES[f](x) tends to zero for x outside S, while convergence on S follows from uniform continuity of f.The proof controls |f(x)−f(x′)| by choosing sufficiently small surface neighborhoods and σ.
- Local geometry: A local parameterization y(u) maps the tangent plane TxS to the surface S and supports the geometric approximation analysis.The argument uses smoothness and compactness of S within a neighborhood Υδ.
- Error control: Choosing δ=σ^(1−τ), with 0<τ<1, makes multiple terms in the decomposition O(σ), while the remaining term is handled separately.The proof explicitly identifies terms (i) and (iv) as O(σ) under this scale choice.
- Equicontinuity: For continuous f on a compact surface, the functions gx(y)=f(y)Φσ(|x−y|) form a uniformly equicontinuous family.Uniform continuity of f and the distance function, together with boundedness of f, yields equicontinuity.