We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images.

中文翻译：

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欧拉特征曲面

我们研究了使用Euler特性进行多参数拓扑数据分析的方法。欧拉特征是一种经典的，易于理解的拓扑不变量，已出现在许多应用中，包括在随机字段的情况下。本文的目的是提出在高维参数空间中使用欧拉特征的扩展。尽管使用更强的不变性（例如同源性）对高维参数空间进行拓扑数据分析仍然是研究的主题，但欧拉特征在理论和计算上都更易于管理，并且该分析可以看作是多参数拓扑中重要的中介步骤数据分析。我们使用人工生成的示例展示了该技术的有用性，