拓扑信号处理用于3D点云数据
Topological Signal Processing for 3D Point Cloud Data
AI总结：
本文提出基于拓扑信号处理的3D点云分析方法，通过高阶拉普拉斯算子实现对点云颜色和几何属性的高效处理与重构。
摘要：
本文的目标是将拓扑信号处理（TSP）框架应用于基于单纯复形表示的3D点云（PCs）分析。基于离散外微分学（DEC）理论中的向量场理论，我们引入了高阶拉普拉斯算子，以实现对三角网格上的信号处理。与传统方法不同，所提出的方法允许我们同时表征颜色属性（建模为节点上的3D向量）和几何结构（建模为每个三角形重心上的3D向量）。然后，我们展示TSP工具可以高效地用于采样、恢复和过滤PCs属性，将其视为边信号进行处理。在合成PCs上的数值结果表明，该方法在稀疏数据下具有鲁棒性，并能有效进行几何细化。所提出的方法提供了一种基于拓扑的表示，以表征PCs的几何和属性。
英文摘要：
Our goal in this paper is to apply the topological signal processing (TSP) framework to the analysis of 3D Point Clouds (PCs) represented on simplicial complexes. Building on Discrete Exterior Calculus (DEC) theory for vector fields, we introduce higher-order Laplacian operators that enable the processing of signals over triangular meshes. Unlike traditional approaches, the proposed approach allows us to characterize both color attributes, modeled as 3D vectors on nodes, and geometry, modeled as 3D vectors on the barycenter of each triangle. Then, we show as TSP tools may efficiently be used to sample, recover and filter PCs attributes treating them as edge signals. Numerical results on synthetic PCs demonstrate accurate color reconstruction with robustness to sparse data and geometry refinement in the case of noisy PC coordinates. The proposed approach provides a topology-based representation to characterize the geometry and attributes of PCs.
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