Rotation invariance is an important requirement for the analysis of 3D point clouds. In this paper, we present a learnable descriptor for rotation- and reflection-invariant 3D point cloud classification based on recently introduced steerable 3D spherical neurons and vector neurons. Specifically, we show that the two approaches are compatible, and we show how to apply steerable neurons in an end-to-end method for the first time. In our approach, we perform TetraTransform -- which lifts the 3D input to an equivariant 4D representation, constructed by the steerable neurons -- and extract deeper rotation-equivariant features using vector neurons, subsequently computing pair-wise O(3)-invariant inner products of these features. This integration of the TetraTransform into the VN-DGCNN framework, termed TetraSphere, is used to classify synthetic and real-world data in arbitrary orientations. Taking only 3D coordinates as input, TetraSphere sets a new state-of-the-art classification performance on randomly rotated objects of the hardest subset of ScanObjectNN, even when trained on data without additional rotation augmentation. Our results reveal the practical value of spherical decision surfaces for learning in 3D Euclidean space.

翻译：旋转不变性是三维点云分析的重要需求。本文基于近期提出的可操控三维球面神经元与向量神经元，提出一种可学习的旋转与反射不变性三维点云分类描述符。具体而言，我们证明了两种方法的兼容性，并首次展示了如何以端到端方式应用可操控神经元。该方法执行TetraTransform——通过可操控神经元将三维输入提升至等变四维表示——并利用向量神经元提取更深层的旋转等变特征，随后计算这些特征之间的配对O(3)不变内积。将TetraTransform整合至VN-DGCNN框架（称为TetraSphere），用于分类任意姿态下的合成数据与真实世界数据。仅以三维坐标为输入，即使训练数据未采用旋转增强，TetraSphere在ScanObjectNN最难子集的随机旋转物体分类任务中仍达到新的最优性能。我们的结果揭示了球形决策曲面在三维欧氏空间学习中的实用价值。