🤖 AI Summary
研究针对无网格点云质量缺乏系统性研究的问题，通过比较现有及新提出的质量度量，并经广泛数值测试，确定了六个可靠的点云质量指标。
📝 Abstract
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.