Uncertainty Visualization of Critical Points of 2D Scalar Fields for Parametric and Nonparametric Probabilistic Models

TL;DR

This work addresses uncertain 2D scalar fields by providing a closed-form end-to-end framework for computing and visualizing critical-point uncertainty under independent parametric and nonparametric noise with finite support. It derives exact probabilities for local minima, maxima, and saddles in both two- and four-pixel neighborhoods, using parametric kernels (Uniform, Epanechnikov) and histogram-based nonparametric models, and accelerates computation with a VTK-m parallel backend integrated into ParaView. Key innovations include a piecewise integration strategy that avoids Monte Carlo sampling, a linear-time algorithm for 1D subproblems, and a semianalytical option for nonparametric models that balances accuracy and speed. The framework demonstrates substantial speedups over MC, robustness to outliers with nonparametric models, and practical applicability to climate and oceanography datasets through production-capable visualization pipelines.

Abstract

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.

Figures (7)

• Figure 1: Depiction of a problem setting. The probability distributions at a grid vertex $p$ and its neighbors $e, n, w$, and $s$ represent uncertainty in data. Our aim is to compute the probability of point $p$ to be critical when distributions are represented with parametric and nonparametic models.
• Figure 2: Illustration of our three-step approach for critical point uncertainty computation for the 1D case. (1) Determine the range of random variable $X_1$ for which critical point probability is nonzero (i.e., $[a_1, b_{min} = b_2]$) and determine its pieces. The range for which critical point probability is zero is shown in brown in (a). Each new start point $a_i \in [a_1, b_{min}]$ creates a new piece. Here, $a_3$ results in two pieces $P_1 = [a_1,a_3]$ and $P_2 = [a_3,b_{min}]$. (2) Compute integration for piece $P_1$ (i.e., $I_{P_1}$) depending on which intervals overlap it. Since the interval $[a_2,b_2]$ overlaps with $P_1$, $I_{P_1}$ corresponds to integral over a joint distribution of $X_1$ and $X_2$, as depicted in (b). (3) Update integrals for next pieces depending on observed start points (e.g., inclusion of random variable $X_3$ in the integral $I_{P_2}$ in (c) based on the start point $a_3$) and sum all piecewise integrals to compute the local minimum probability.
• Figure 3: Epanechnikov distribution (b) gives more weight to the mean unlike the uniform distribution (a), and hence, can provide enhanced visualization compared to the uniform noise model.
• Figure 4: The qualitative and quantitative proof of correctness and enhanced performance of our proposed closed-form computations (column c) with the MC sampling approach (columns a and b) as the baseline. The results are shown for the uniform noise model. The solution obtained with 2000 MC samples converges to closed-form computations with our algorithms (see the difference images in column d and convergence curves in column e), thereby confirming their correctness. Our methods provide $64 \times$ and $119 \times$ speed-up with respect to the MC sampling approach with $2000$ samples.
• Figure 5: Gaussian mixture data for computation of critical point probability.