标量数据的局部拓扑简化
Localized Topological Simplification of Scalar Data
摘要：
本文描述了一种局部化的标量数据拓扑简化算法，这是拓扑数据分析（TDA）的一个重要的预处理步骤。给定一个标量场f和一个选择的极值保存，建议的局部拓扑简化（LTS）得出一个函数g，是接近f，只表现出所选择的极值集。具体而言，子和超水平集组件与不希望的极值首先局部平坦化，然后正确地嵌入到全球标量场，使这些地区得到保证-从组合的角度来看-不再包含任何不希望的极值。与以前的全局方法相比，LTS仅独立处理实际需要简化的域区域，这已经导致了明显的加速。此外，由于本地化的性质的算法，LTS可以利用共享内存并行简化区域同时具有较高的并行效率（70%）。因此，LTS显着提高了交互性的探索简化参数及其对后续拓扑分析的影响。对于这样的探索任务，LTS将大量TDA管道的总体执行时间从几分钟降低到几秒，与最先进的技术相比，平均观察到的加速高达x36。此外，在保存极值的拓扑持久性的基础上选择的特殊情况下，适应版本的LTS部分计算持久性图，同时简化功能低于预定义的持久性阈值。LTS的有效性，它的并行效率，并由此带来的好处TDA上的几个模拟和采集的数据集从不同的应用领域，包括物理，化学和生物医学成像证明。
英文摘要：
This paper describes a localized algorithm for the topological simplification of scalar data, an essential pre-processing step of topological data analysis (TDA). Given a scalar field f and a selection of extrema to preserve, the proposed localized topological simplification (LTS) derives a function g that is close to f and only exhibits the selected set of extrema. Specifically, sub- and superlevel set components associated with undesired extrema are first locally flattened and then correctly embedded into the global scalar field, such that these regions are guaranteed -- from a combinatorial perspective -- to no longer contain any undesired extrema. In contrast to previous global approaches, LTS only and independently processes regions of the domain that actually need to be simplified, which already results in a noticeable speedup. Moreover, due to the localized nature of the algorithm, LTS can utilize shared-memory parallelism to simplify regions simultaneously with a high parallel efficiency (70%). Hence, LTS significantly improves interactivity for the exploration of simplification parameters and their effect on subsequent topological analysis. For such exploration tasks, LTS brings the overall execution time of a plethora of TDA pipelines from minutes down to seconds, with an average observed speedup over state-of-the-art techniques of up to x36. Furthermore, in the special case where preserved extrema are selected based on topological persistence, an adapted version of LTS partially computes the persistence diagram and simultaneously simplifies features below a predefined persistence threshold. The effectiveness of LTS, its parallel efficiency, and its resulting benefits for TDA are demonstrated on several simulated and acquired datasets from different application domains, including physics, chemistry, and biomedical imaging.
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