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.
are more intuitive and similar to the Euclidean case, and this is one of the reasons why most of the existing works fall into this category. The philosophy behind these methods is that the tangent plane of a @math -dimensional manifold is embedded to a @math -dimensional Euclidean domain where convolution can be easily defined. In this paper, we make the first attempt to interpret some of the existing mesh-based methods in a unified framework. We claim that most of the spatial mesh-based methods can be formulated as Here, @math is a convolution kernel and @math with @math being the size of the kernel. The mapping @math is defined as where @math , @math . For simplicity, we will denote @math . Most of the designs of the existing manifold convolutions focused on the designs of @math . We remark that possible singularities will lead to no convolution operation at those points. These are isolated points on a closed manifold and do not effect experiment results. In addition, singularities from a given vector field can be overcome using several pairs of vector fields and pooling @cite_15 .
@cite_15
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A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair @math of points of M there exists a trajectory of S from m to @math . Here a trajectory of S is a curve which is an integral curve of some @math or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class @math , @math , or @math , there exists a completely controllable set S consisting of two vector fields of class @math .