Reeb空间曲面的时间跟踪
Temporal Tracking of Reeb-Space Sheets
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中文总结 AI 辅助
本研究针对时变双变量场，提出基于空间域与值域互补相似性度量的Reeb空间曲面跟踪框架，经合成圆环与分子电子结构数据集验证，可揭示持久结构并支撑时变双变量数据可视化分析。
中文摘要 AI 辅助
时变双变量场出现在许多科学应用中，其中两个标量量之间的关系随时间演化。虽然合并树等拓扑方法为识别和跟踪单变量数据中的特征提供了有效框架，但针对双变量场的类似方法仍相对未被充分探索。Reeb空间通过一组相互连接的曲面表示纤维连通性，将拓扑分析扩展到多变量数据，这些曲面成为描述双变量结构的自然候选。然而，由于Reeb空间的结构复杂性、对噪声的敏感性以及跨时间步定义有意义相似性度量的难度，建立曲面间的时间对应关系具有挑战性。我们提出了一种用于跟踪时变双变量场中Reeb空间曲面的框架，该方法利用在空间域和值域中定义的互补相似性度量，建立连续时间步中曲面间的对应关系。我们在合成圆环数据集和两个时变分子电子结构数据集上评估了该方法，结果表明Reeb空间曲面跟踪可揭示持久结构并突出时间变化的有趣区间。总体而言，结果证明Reeb空间曲面可作为可跟踪的拓扑结构，并为时变双变量数据的可视化分析提供基础。
英文摘要
Time-varying bivariate fields arise in many scientific applications, where the relationship between two scalar quantities evolves over time. While topological methods such as merge trees provide an effective framework for identifying and tracking features in univariate data, analogous approaches for bivariate fields remain comparatively underexplored. Reeb spaces extend topological analysis to multivariate data by representing fiber connectivity through a collection of interconnected sheets, making these sheets natural candidates for describing bivariate structures. However, establishing temporal correspondences between sheets is challenging due to the structural complexity of Reeb spaces, sensitivity to noise, and the difficulty of defining meaningful similarity measures across timesteps. We present a framework for tracking Reeb space sheets in time-varying bivariate fields. The method establishes correspondences between sheets in consecutive timesteps using complementary similarity measures defined in the spatial domain and the range space. We evaluate the method on a synthetic torus dataset and two time-varying molecular electronic structure datasets. The results show that Reeb space sheet tracking reveals persistent structures and highlights interesting intervals of temporal change. Overall, the results demonstrate that Reeb space sheets can serve as trackable topological structures and provide a foundation for the visual analysis of time-varying bivariate data.
