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Critical points mark locations in the domain where the level-set topology ofa scalar function undergoes fundamental changes and thus indicate potentiallyinteresting features in the data. Established methods exist to locate andrelate such points in a deterministic setting, but it is less well understoodhow the concept of critical points can be extended to the analysis of uncertaindata. Most methods for this task aim at finding likely locations of criticalpoints or estimate the probability of their occurrence locally but do notindicate if critical points at potentially different locations in differentrealizations 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 realizationsreported challenges for interpreting the resulting spatial distribution ofcritical points but did not investigate the causes. In this work, we provide amathematical formulation of the problem of finding critical points with spatialuncertainty and computing their spatial distribution, which leads us to thenotion of uncertain critical points. We analyze the theoretical properties ofthese structures and highlight connections to existing works for specialclasses of uncertain fields. We derive conditions under whichwell-interpretable results can be obtained and discuss the implications ofthose restrictions for the field of visualization. We demonstrate that thediscussed limitations are not purely academic but also arise in real-worlddata.

A Mathematical Foundation for the Spatial Uncertainty of Critical Points in Probabilistic Scalar Fields