text: Convolution plays a crucial role in various applications in signal and image processing, analysis and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processing point cloud data. However, one of the major challenges is to define a proper way to "sweep" filters through the point cloud as a natural generalization of the planar convolution and to reflect the point cloud's geometry at the same time. In this paper, we consider generalizing convolution by adapting parallel transport on the point cloud. Inspired by a triangulated surface based method [Stefan C. Schonsheck, Bin Dong, and Rongjie Lai, arXiv:1805.07857.], we propose the Narrow-Band Parallel Transport Convolution (NPTC) using a specifically defined connection on a voxelized narrow-band approximation of point cloud data. With that, we further propose a deep convolutional neural network based on NPTC (called NPTC-net) for point cloud classification and segmentation. Comprehensive experiments show that the proposed NPTC-net achieves similar or better results than current state-of-the-art methods on point clouds classification and segmentation.
  @cite_15: Standard convolutional neural networks assume a grid structured input is available and exploit discrete convolutions as their fundamental building blocks. This limits their applicability to many real-world applications. In this paper we propose Parametric Continuous Convolution, a new learnable operator that operates over non-grid structured data. The key idea is to exploit parameterized kernel functions that span the full continuous vector space. This generalization allows us to learn over arbitrary data structures as long as their support relationship is computable. Our experiments show significant improvement over the state-of-the-art in point cloud segmentation of indoor and outdoor scenes, and lidar motion estimation of driving scenes.
summary: The newly proposed PTC @cite_15 defines convolution directly on the manifold, while uses tangent planes to transport kernels by a properly chosen parallel transport. PTC can be equivalently cast into the form of ) using the inverse exponential map, and implementation of the proposed parallel transported is realized through choosing specific vector fields @math guided by a Eikonal equation for transforming vectors along geodesic curves on manifolds.