Feature tracking is a common task in visualization applications, where methods based on topological data analysis (TDA) have successfully been applied in the past for feature definition as well as tracking. In this work, we focus on tracking extrema of temporal scalar fields. A family of TDA approaches address this task by establishing one-to-one correspondences between extrema based on discrete gradient vector fields. More specifically, two extrema of subsequent time steps are matched if they fall into their respective ascending and descending manifolds. However, due to this one-to-one assignment, these approaches are prone to fail where, e.g., extrema are located in regions with low gradient magnitude, or are located close to boundaries of the manifolds. Therefore, we propose a probabilistic matching that captures a larger set of possible correspondences via neighborhood sampling, or by computing the overlap of the manifolds. We illustrate the usefulness of the approach with two application cases.

翻译：特征追踪是可视化应用中的常见任务，基于拓扑数据分析的方法在特征定义及追踪方面已有成功应用。本研究聚焦于时变标量场中极值点的追踪问题。当前有一类拓扑数据分析方法通过离散梯度向量场建立极值点间的一一对应关系来实现该任务——具体而言，当相邻时间步的两个极值点分别落入彼此的上升流形与下降流形时，则判定二者匹配。然而这种一一对应机制存在局限性：例如当极值点位于梯度幅值较低区域或流形边界附近时，匹配容易失效。为此，我们提出一种概率匹配方法，通过邻域采样或计算流形重叠度来捕获更大范围的可能对应关系。最后通过两个应用案例验证了该方法的有效性。