🤖 AI Summary
This work addresses the automatic identification and quantification of singular patterns in discrete planar vector fields. We propose a novel method based on persistent path homology: a directed graph is constructed to encode local directional information of the vector field, and one-dimensional persistent path homology is computed to precisely locate singularities, segment singular regions, and quantitatively measure structural variations across fields. To our knowledge, this is the first application of persistent path homology to vector field topology analysis, unifying singularity detection, topological segmentation, and inter-field difference assessment within a single algebraic framework. The method is validated on real-world problems—including tropical cyclone center and influence zone identification, geomagnetic dipole localization, and comparative analysis of singular patterns across multiple vector fields—demonstrating high robustness and physical interpretability. It provides a scalable, algebraic-topological foundation for analyzing topological features of vector fields.
📝 Abstract
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.