| Abstract: |
| Persistent homology is a crucial tool in topological data analysis (TDA), and the key to its applications lies in constructing a suitable finite-length ascending sequence of simplicial complexes (i.e., a filtration). Unlike most other approaches that directly apply persistence diagrams to downstream tasks, our core insight is that changes in the topological structure of simplicial complexes within a filtration are necessarily driven by specific vertices. We thus propose the concept of homology-generating/vanishing critical points, which aims to locate the vertices in the original data that induce topological changes in images. Subsequently, taking a novel geodesic filtration as an example, we investigate the relationships between 0-dim, 1-dim, and 2-dim PH points in persistence diagrams and the vertices in tubular structures of images. Experiments conducted on multiple datasets demonstrate that our method can accurately locate the homology-generating/vanishing critical points (e.g., branch points, endpoints, adhesions, etc.) and exhibits significant applications potential in relevant fields. |
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