DeltaConv: Anisotropic Point Cloud Learning with Exterior Calculus

Learning from 3D point-cloud data has rapidly gainedmomentum, motivated by the success of deep learning onimages and the increased availability of 3D data. In thispaper, we aim to construct anisotropic convolutions thatwork directly on the surface derived from a point cloud.This is challenging because of the lack of a global coordi-nate system for tangential directions on surfaces. We intro-duce a new convolution operator called DeltaConv, whichcombines geometric operators from exterior calculus to en-able the construction of anisotropic filters on point clouds.Because these operators are defined on scalar- and vector-fields, we separate the network into a scalar- and a vector-stream, which are connected by the operators. The vectorstream enables the network to explicitly represent, evalu-ate, and process directional information. Our convolutionsare robust and simple to implement and show improved ac-curacy compared to state-of-the-art approaches on severalbenchmarks, while also speeding up training and inference.