In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the evolution of connected components in the sublevel sets of scalar fields. We present a new technique for modeling and comparing merge trees using tools from partial optimal transport. In particular, we model a merge tree as a measure network, that is, a network equipped with a probability distribution, and define a notion of distance on the space of merge trees inspired by partial optimal transport. Such a distance offers a new and flexible perspective for encoding intrinsic and extrinsic information in the comparative measures of merge trees. More importantly, it gives rise to a partial matching between topological features in time-varying data, thus enabling flexible topology tracking for scientific simulations. Furthermore, such partial matching may be interpreted as probabilistic coupling between features at adjacent time steps, which gives rise to probabilistic tracking graphs. We derive a stability result for our distance and provide numerous experiments indicating the efficacy of distance in extracting meaningful feature tracks.

翻译：本文提出了一种灵活的概率框架，利用合并树与部分最优传输技术追踪时变标量场中的拓扑特征。合并树作为拓扑描述符，记录了标量场子水平集中连通分量的演化过程。我们提出了一种基于部分最优传输工具对合并树进行建模与比较的新技术。具体而言，将合并树建模为测度网络（即带有概率分布的赋权网络），并借鉴部分最优传输思想定义了合并树空间中的距离度量。该距离为编码合并树比较测度中的内在与外在信息提供了全新的灵活视角。更重要的是，它能在时变数据中建立拓扑特征间的部分匹配，从而为科学模拟中的灵活拓扑追踪奠定基础。此外，这种部分匹配可被解释为相邻时间步特征之间的概率耦合，进而生成概率追踪图。我们推导了该距离的稳定性结果，并通过大量实验验证了该距离在提取有效特征轨迹方面的有效性。