The extraction of singular patterns is a fundamental problem in theoretical and practical domains due to the ability of such patterns to detect the intrinsic characteristics of vector fields. In this study, we propose an approach for extracting singular patterns from discrete planar vector fields. Our method involves converting the planar discrete vector field into a specialized digraph and computing its one-dimensional persistent path homology. By analyzing the persistence diagram, we can determine the location of singularity and segment a region of the singular pattern, which is referred to as a singular polygon. Moreover, the variations of singular patterns can also be analyzed. The experimental results demonstrate the effectiveness of our method in analyzing the centers and impact areas of tropical cyclones, positioning the dip poles from geomagnetic fields, and measuring variations of singular patterns between vector fields.

翻译：奇异模式的提取是理论与应用领域的一个基本问题，因其能够揭示向量场的内在特征。本研究提出一种从离散平面向量场中提取奇异模式的方法。该方法将平面离散向量场转化为特定有向图，并计算其一维持续路径同调。通过分析持续图，我们可以确定奇点的位置并分割出奇异模式的区域（称为奇异多边形）。此外，奇异模式的变化亦可被分析。实验结果表明，我们的方法在分析热带气旋的中心与影响范围、定位地磁场的磁极位置以及测量向量场间奇异模式变化等方面具有有效性。