# Point Convolutional Neural Networks by Extension Operators

arXiv: https://arxiv.org/abs/1803.10091 · HTML: https://mlsanity.com/paper/1803.10091/

## Key facts

- Cited by 154 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2018.

## Built on by (most cited first, 30 of 154)

- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [OpenMask3D: Open-Vocabulary 3D Instance Segmentation](https://mlsanity.com/paper/2306.13631/) (2023)
- [Self-Supervised Pretraining of 3D Features on any Point-Cloud](https://mlsanity.com/paper/2101.02691/) (2021)
- [Group-Free 3D Object Detection via Transformers](https://mlsanity.com/paper/2104.00678/) (2021)
- [PAConv: Position Adaptive Convolution with Dynamic Kernel Assembling on Point Clouds](https://mlsanity.com/paper/2103.14635/) (2021)
- [RPVNet: A Deep and Efficient Range-Point-Voxel Fusion Network for LiDAR Point Cloud Segmentation](https://mlsanity.com/paper/2103.12978/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Stratified Transformer for 3D Point Cloud Segmentation](https://mlsanity.com/paper/2203.14508/) (2022)
- [Revisiting Point Cloud Shape Classification with a Simple and Effective Baseline](https://mlsanity.com/paper/2106.05304/) (2021)
- [Beyond Self-attention: External Attention using Two Linear Layers for Visual Tasks](https://mlsanity.com/paper/2105.02358/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [OctFormer: Octree-based Transformers for 3D Point Clouds](https://mlsanity.com/paper/2305.03045/) (2023)
- [Point Mamba: A Novel Point Cloud Backbone Based on State Space Model with Octree-Based Ordering Strategy](https://mlsanity.com/paper/2403.06467/) (2024)
- [Deep Learning on Implicit Neural Representations of Shapes](https://mlsanity.com/paper/2302.05438/) (2023)
- [Adaptive Graph Convolution for Point Cloud Analysis](https://mlsanity.com/paper/2108.08035/) (2021)
- [Open-Vocabulary Functional 3D Scene Graphs for Real-World Indoor Spaces](https://mlsanity.com/paper/2503.19199/) (2025)
- [Self-Contrastive Learning with Hard Negative Sampling for Self-supervised Point Cloud Learning](https://mlsanity.com/paper/2107.01886/) (2021)
- [ReDAL: Region-based and Diversity-aware Active Learning for Point Cloud Semantic Segmentation](https://mlsanity.com/paper/2107.11769/) (2021)
- [Style-based Point Generator with Adversarial Rendering for Point Cloud Completion](https://mlsanity.com/paper/2103.02535/) (2021)
- [Self-positioning Point-based Transformer for Point Cloud Understanding](https://mlsanity.com/paper/2303.16450/) (2023)
- [So3krates: Equivariant attention for interactions on arbitrary length-scales in molecular systems](https://mlsanity.com/paper/2205.14276/) (2022)
- [PointBA: Towards Backdoor Attacks in 3D Point Cloud](https://mlsanity.com/paper/2103.16074/) (2021)
- [A Unified Query-based Paradigm for Point Cloud Understanding](https://mlsanity.com/paper/2203.01252/) (2022)
- [Frame Mining: a Free Lunch for Learning Robotic Manipulation from 3D Point Clouds](https://mlsanity.com/paper/2210.07442/) (2022)
- [Dual Transformer for Point Cloud Analysis](https://mlsanity.com/paper/2104.13044/) (2021)
- [3DCTN: 3D Convolution-Transformer Network for Point Cloud Classification](https://mlsanity.com/paper/2203.00828/) (2022)
- [Isometric 3D Adversarial Examples in the Physical World](https://mlsanity.com/paper/2210.15291/) (2022)
- [3D Segmentation of Humans in Point Clouds with Synthetic Data](https://mlsanity.com/paper/2212.00786/) (2022)
- [Deep Learning-based 3D Point Cloud Classification: A Systematic Survey and Outlook](https://mlsanity.com/paper/2311.02608/) (2023)
- [Improved Generalization of Weight Space Networks via Augmentations](https://mlsanity.com/paper/2402.04081/) (2024)

## Abstract (arXiv)

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.