Neural varifolds: an aggregate representation for quantifying the geometry of point clouds

Point clouds are popular 3D representations for real-life objects (such as in LiDAR and Kinect) due to their detailed and compact representation of surface-based geometry. Recent approaches characterise the geometry of point clouds by bringing deep learning based techniques together with geometric fidelity metrics such as optimal transportation costs (e.g., Chamfer and Wasserstein metrics). In this paper, we propose a new surface geometry characterisation within this realm, namely a neural varifold representation of point clouds. Here the surface is represented as a measure/distribution over both point positions and tangent spaces of point clouds. The varifold representation quantifies not only the surface geometry of point clouds through the manifold-based discrimination, but also subtle geometric consistencies on the surface due to the combined product space. This study proposes neural varifold algorithms to compute the varifold norm between two point clouds using neural networks on point clouds and their neural tangent kernel representations. The proposed neural varifold is evaluated on three different sought-after tasks -- shape matching, few-shot shape classification and shape reconstruction. Detailed evaluation and comparison to the state-of-the-art methods demonstrate that the proposed versatile neural varifold is superior in shape matching and few-shot shape classification, and is competitive for shape reconstruction.