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.

中文翻译：

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标量数据的局部拓扑简化

本文描述了一种用于标量数据拓扑简化的局部算法，这是拓扑数据分析 (TDA) 的重要预处理步骤。给定一个标量场 f 和一组要保留的极值，所提出的局部拓扑简化 (LTS) 推导了一个接近 f 的函数 g，并且只展示了所选的极值集。具体来说，与不需要的极值相关的子集和超级集组件首先局部展平，然后正确嵌入到全局标量场中，这样从组合的角度来看，保证这些区域不再包含任何不需要的极值。与之前的全局方法相比，LTS 仅独立处理实际需要简化的域区域，这已经导致明显的加速。而且，由于算法的局部性，LTS 可以利用共享内存并行性以高并行效率（70%）同时简化区域。因此，LTS 显着提高了探索简化参数及其对后续拓扑分析的影响的交互性。对于此类探索任务，LTS 将大量 TDA 管道的整体执行时间从几分钟缩短到几秒钟，观察到的平均加速比最先进的技术高达 x36。此外，在基于拓扑持久性选择保留极值的特殊情况下，LTS 的适应版本部分计算持久性图，同时简化低于预定义持久性阈值的特征。LTS 的有效性，它的并行效率，