  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.
