Convolution Restriction Fig. 1 . A new framework for applying convolution to functions defined over point clouds: First, a function over the point cloud (in this case the constant one) is extended to a continuous volumetric function over the ambient space; second, a continuous volumetric convolution is applied to this function (without any discretization or approximation); and lastly, the result is restricted back to the point cloud. 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 de ned by pull-back of the Euclidean volumetric convolution via an extension-restriction mechanism. The point cloud convolution is computationally e cient, invariant to the order of points in the point cloud, robust to di erent 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.
REF ] finds a way to apply convolutional neural nets on point-clouds as well by mapping point cloud functions to volumetric functions and vice-versa.
Point convolutional neural networks by extension operators
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Computer Science