  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
voxel-based 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 cloud classification and
segmentation.
