  This paper presents a novel end-to-end framework for closed-form computation
and visualization of critical point uncertainty in 2D uncertain scalar fields.
Critical points are fundamental topological descriptors used in the
visualization and analysis of scalar fields. The uncertainty inherent in data
(e.g., observational and experimental data, approximations in simulations, and
compression), however, creates uncertainty regarding critical point positions.
Uncertainty in critical point positions, therefore, cannot be ignored, given
their impact on downstream data analysis tasks. In this work, we study
uncertainty in critical points as a function of uncertainty in data modeled
with probability distributions. Although Monte Carlo (MC) sampling techniques
have been used in prior studies to quantify critical point uncertainty, they
are often expensive and are infrequently used in production-quality
visualization software. We, therefore, propose a new end-to-end framework to
address these challenges that comprises a threefold contribution. First, we
derive the critical point uncertainty in closed form, which is more accurate
and efficient than the conventional MC sampling methods. Specifically, we
provide the closed-form and semianalytical (a mix of closed-form and MC
methods) solutions for parametric (e.g., uniform, Epanechnikov) and
nonparametric models (e.g., histograms) with finite support. Second, we
accelerate critical point probability computations using a parallel
implementation with the VTK-m library, which is platform portable. Finally, we
demonstrate the integration of our implementation with the ParaView software
system to demonstrate near-real-time results for real datasets.
