arXiv:2507.00725 · v2
PL Morse-Cerf Theory for Topological Analysis of Time-Varying Scalar Fields
摘要
Morse-Cerf theory studies the evolution of critical points in one-parameter families of smooth functions defined on a manifold. This paper describes an adaptation of Morse-Cerf theory of smooth functions to one-parameter families of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as topological descriptors that represent evolution of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. Algorithms and methods for computing, querying, and comparing Cerf diagrams support its application towards the analysis of time-varying scalar fields. Stability theorems on the comparison measure provide a guarantee of robustness. Experimental results on multiple scientific datasets demonstrate its practical utility.
作者备注：Extended journal version of the previous IEEE VIS 2025 Short Paper, PrePrints https://doi.org/10.1109/VIS60296.2025.00024
期刊信息：IEEE Transactions on Visualization and Computer Graphics, vol. , no. 01, pp. 1-14, 2026