无网格方法的点云质量
Point Cloud Quality for Meshfree Methods
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中文总结 AI 辅助
本研究针对无网格点云质量展开系统研究，对比分析现有指标并引入新指标，经大量数值测试后确定六项可靠指标，同时指出部分常用指标预测精度的能力较差。
中文摘要 AI 辅助
网格质量已得到充分研究，并被广泛用于衡量优质网格。相比之下，针对无网格点云质量的系统研究仍有所欠缺。这一缺口使得难以证实“生成高质量点云比生成优质网格更简单”这一普遍说法。目前已提出多种点云质量的定义，其中部分定义对收敛性和误差界的证明具有理论意义，其余则用于计算研究。本研究中，我们对比分析现有点云质量度量指标，并引入若干新指标。我们采用无网格配点法开展大量数值测试，测试场景涵盖椭圆型与双曲型方程、二维与三维域，以及数值方法参数的各类变化。基于这些测试，我们评估哪些质量度量指标与数值误差的相关性最佳。研究结果显示，有六项指标可始终作为点云质量的可靠指示器，同时也表明若干广泛使用的指标对精度的预测能力较差。
英文摘要
Mesh quality is very well studied and widely used to quantify a good mesh. In contrast, a systematic study of quality of meshfree point clouds is lacking. This gap makes it difficult to substantiate the common claim that generating a good-quality point cloud is easier than generating a good mesh. Various definitions of point cloud quality have been proposed, some of which have theoretical significance for proving convergence and error bounds, while others are used in computational studies. In this work, we compare and contrast existing point cloud quality metrics and introduce a few new ones. We conduct extensive numerical tests with a meshfree collocation method across a wide range of scenarios, including both elliptic and hyperbolic equations, 2D and 3D domains, and variations in parameters of the numerical method. Based on these tests, we assess which quality metrics best correlate with numerical error. Our findings reveal six metrics that consistently serve as reliable indicators of point cloud quality, while also demonstrating that several widely used metrics are poor predictors of accuracy.
