Temporal Tracking of Reeb-Space Sheets

2026-08-06

AI Summary

This study addresses the challenge of establishing temporal correspondences among Reeb space sheets in time-varying bivariate scalar fields to enable dynamic tracking of multivariate topological structures. For the first time, Reeb space sheets are treated as trackable topological entities, and a complementary similarity measure integrating both spatial and range-domain information is proposed to effectively mitigate issues arising from structural complexity and noise sensitivity. The approach encompasses key steps including Reeb space construction, sheet extraction, and spatiotemporal matching. Experiments on synthetic toroidal and molecular electron density datasets demonstrate the method’s efficacy in identifying persistent topological features and critical evolution intervals, highlighting its strong potential for topological analysis of time-dependent multivariate data.

Abstract

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.

Problem

• Reeb space
• bivariate fields
• temporal tracking
• topological structures
• time-varying data

Innovation

• Reeb space
• bivariate fields
• temporal tracking
• topological data analysis
• similarity measures