We present a 3D capsule module for processing point clouds that is
equivariant to 3D rotations and translations, as well as invariant to
permutations of the input points. The operator receives a sparse set of local
reference frames, computed from an input point cloud and establishes end-to-end
transformation equivariance through a novel dynamic routing procedure on
quaternions. Further, we theoretically connect dynamic routing between capsules
to the well-known Weiszfeld algorithm, a scheme for solving \emph{iterative
re-weighted least squares} (IRLS) problems with provable convergence
properties. It is shown that such group dynamic routing can be interpreted as
robust IRLS rotation averaging on capsule votes, where information is routed
based on the final inlier scores. Based on our operator, we build a capsule
network that disentangles geometry from pose, paving the way for more
informative descriptors and a structured latent space. Our architecture allows
joint object classification and orientation estimation without explicit
supervision of rotations. We validate our algorithm empirically on common
benchmark datasets.

提出一种能够处理点云的三维胶囊模块，它对于三维旋转和移动具有等变性，并且对于输入点的排列具有不变性；该模块通过四元数上的新颖动态路由过程来建立端到端的变换等变性；此外，作者在理论上将胶囊之间的动态路由与欧几里德旋转平均算法相连接，通过此连接，证明了这种路由方式能够被解释为对胶囊投票进行鲁棒的旋转平均，进而在大型数据集上进行了实验验证。