Critical points mark locations in the domain where the level-set topology of a scalar function undergoes fundamental changes and thus indicate potentially interesting features in the data. Established methods exist to locate and relate such points in a deterministic setting, but it is less well understood how the concept of critical points can be extended to the analysis of uncertain data. Most methods for this task aim at finding likely locations of critical points or estimate the probability of their occurrence locally but do not indicate if critical points at potentially different locations in different realizations of a stochastic process are manifestations of the same feature, which is required to characterize the spatial uncertainty of critical points. Previous work on relating critical points across different realizations reported challenges for interpreting the resulting spatial distribution of critical points but did not investigate the causes. In this work, we provide a mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution, which leads us to the notion of uncertain critical points. We analyze the theoretical properties of these structures and highlight connections to existing works for special classes of uncertain fields. We derive conditions under which well-interpretable results can be obtained and discuss the implications of those restrictions for the field of visualization. We demonstrate that the discussed limitations are not purely academic but also arise in real-world data.

翻译：临界点标量函数域中水平集拓扑发生根本变化的位点，因而指示数据中可能存在的有趣特征。已有成熟方法可在确定性情境下定位和关联此类点，但如何将临界点概念扩展至不确定数据分析尚缺乏深入理解。现有方法大多旨在寻找临界点的可能位置或局部估算其发生概率，却未能指明随机过程不同实现中可能处于不同位置的临界点是否为同一特征的体现——而这正是刻画临界点空间不确定性所需的关键。此前关于跨实现间临界点关联的研究，虽报告了阐释临界点空间分布结果的困难，却未探究其成因。本研究提出包含空间不确定性的临界点寻找及其空间分布计算的数学形式化描述，由此引出非确定临界点概念。我们分析这些结构的理论性质，并阐明其与现有不确定场特殊类别研究的关联。推导出可获取可解释性结果的充分条件，并讨论这些约束条件对可视化领域的启示。实验证明所述局限性并非纯理论问题，亦存在于实际数据中。