基于部分最优传输的灵活概率拓扑追踪
Flexible and Probabilistic Topology Tracking with Partial Optimal Transport
AI总结：
本文提出一种基于合并树与部分最优传输的灵活概率框架，通过测度网络建模和部分匹配实现时变标量场中拓扑特征的追踪，并验证了其稳定性与有效性。
摘要：
在本文中，我们提出了一种灵活且概率性的框架，利用合并树（merge trees）和部分最优传输（partial optimal transport）来追踪时变标量场中的拓扑特征。合并树是一种拓扑描述符，用于记录标量场子水平集（sublevel sets）中连通分量的演化过程。我们提出了一种利用部分最优传输工具对合并树进行建模和比较的新技术。具体而言，我们将合并树建模为测度网络（measure network），即一个配备概率分布的网络，并受部分最优传输启发，在合并树空间上定义了一种距离概念。这种距离为在合并树的比较测度中编码内在和外在信息提供了一种新的灵活视角。更重要的是，它能够在时变数据中的拓扑特征之间产生部分匹配，从而为科学模拟提供灵活的拓扑追踪能力。此外，这种部分匹配可以被解释为相邻时间步特征之间的概率耦合，进而产生概率追踪图。我们推导了该距离的稳定性结果，并提供了大量实验，表明我们的框架在提取有意义的特征轨迹方面具有有效性。
英文摘要：
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 our framework in extracting meaningful feature tracks.
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