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.

本文介绍了点卷积神经网络（PCNN）的框架和两个算子：扩展和限制。此外，还定义了点云卷积的概念，其具有高效性，不受各种因素的影响，并可将卷积核用于任何点云中。通过三项常见的点云学习基准测试，证明了PCNN方法的有效性。

通过扩展算子的点卷积神经网络