Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
Visualizing robustness of critical points for 2D time-varying vector fields
EuroVis '13 Proceedings of the 15th Eurographics Conference on Visualization
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Scalar functions defined on a topological space $\Omega$ are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sub-level sets $\{x \in \Omega\colon \ f(x) \le c\}$ persist as c changes. Recently it was shown that the critical values defining a topological structure with relatively large persistence remain almost unaffected by small perturbations. This result suggests that topological persistence is a good measure for matching and comparing scalar functions. We extend these results to critical points in the domain by redefining persistence and critical points and replacing sub-level sets $\{x \in \Omega\colon \ f(x) \le c\}$ with interval sets $\{x \in \Omega\colon \ a \le f(x)