Geometry-Aware Merge Tree Comparisons for Time-Varying Data With Interleaving Distances
Geometry-Aware Merge Tree Comparisons for Time-Varying Data With Interleaving Distances
复制标题
具有交错距离的时变数据的几何感知合并树比较
DOI：
10.1109/tvcg.2022.3163349
复制
发表时间：
2021
影响因子：
5.2
通讯作者：
Bei Wang
中科院分区：
文献类型：
--
作者：
Lin Yan;Talha Bin Masood;F. Rasheed;I. Hotz;Bei Wang
合并树是一种拓扑描述符，用于识别和总结标量场的拓扑特征。它们在分析和可视化时变数据方面具有巨大的潜力。首先，它们给出了数据实例的压缩和拓扑保持表示。其次，它们的比较为研究数据实例之间的关系提供了基础，例如它们的分布，聚类，离群值和周期性。已经为合并树开发了许多比较措施。然而，这些措施往往是计算昂贵的，因为他们隐含地考虑所有可能的合并树的临界点之间的对应关系。在本文中，我们使用标记的交织距离进行合并树的几何感知比较。其主要思想是将比较度量的计算解耦为两个步骤：标记步骤，生成两个合并树的临界点之间的对应关系;以及比较步骤，通过将它们编码为矩阵来计算一对标记合并树之间的距离。我们表明，我们的方法是一般的，计算效率高，实际上是有用的。我们的框架使得有可能在标记过程中集成数据域的几何信息。同时，该框架降低了计算复杂度，因为不是所有可能的对应关系都必须考虑。我们通过实验证明，这种几何感知的合并树比较有助于检测时变数据集的转换，聚类和周期性，以及诊断和突出相邻数据实例之间的拓扑变化。
Merge trees, a type of topological descriptors, serve to identify and summarize the topological characteristics associated with scalar fields. They have great potential for analyzing and visualizing time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expensive since they implicitly consider all possible correspondences between critical points of the merge trees. In this paper, we perform geometry-aware comparisons of merge trees using labeled interleaving distances. The main idea is to decouple the computation of a comparative measure into two steps: a labeling step that generates a correspondence between the critical points of two merge trees, and a comparison step that computes distances between a pair of labeled merge trees by encoding them as matrices. We show that our approach is general, computationally efficient, and practically useful. Our framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, the framework reduces the computational complexity since not all possible correspondences have to be considered. We demonstrate via experiments that such geometry-aware merge tree comparisons help to detect transitions, clusters, and periodicities of time-varying datasets, as well as to diagnose and highlight the topological changes between adjacent data instances.
用于计算 Gromov-Hausdorff 和树间交错距离的 FPT 算法
DOI：
--
发表时间：
2019
期刊：
European Symposium on Algorithms
影响因子：
--
作者：
Farahbakhsh Touli, Elena;Wang, Yusu
通讯作者：
Wang, Yusu