Fiber Uncertainty Visualization for Bivariate Data With Parametric and Nonparametric Noise Models

TL;DR

This work advances fiber-surface uncertainty visualization by generalizing uncertainty models from Gaussian to independent parametric (uniform, Epanechnikov, Gaussian) and nonparametric (histograms, KDE) distributions, and by enabling arbitrary polygonal FSCPs. It introduces closed-form Green's theorem-based integration for independent parametric and nonparametric noise, and a nonparametric, numerically integrated framework for correlated noise, together with a vertex-based method to extract the most probable fiber surface. The authors provide comprehensive memory and computational analyses and demonstrate substantial accuracy gains in synthetic and simulation datasets, with nonparametric models offering closer alignment to ground truth than parametric or mean-field approaches. The approach supports direct volume rendering and probabilistic segmentation to visualize fiber-position uncertainty, with practical implications for risk-aware interpretation of multivariate simulations and experiments. Applications include ocean eddies and atmospheric/tensor-field contexts, and the work points to future extensions to higher-dimensional data and topology-aware analyses of fiber surfaces.

Abstract

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

Figures (10)

• Figure 1: Fiber surface visualization of the ethanediol dataset. (a) Visualization of a continuous scatterplot of the electron density (Rho) and reduced gradient (s) at a logarithmic scale. (b) Fiber surfaces for four traits $\{\mathcal{T}_1, \mathcal{T}_2, \mathcal{T}_3, \mathcal{T}_4\}$ denoted by polygons in image (a).
• Figure 2: Illustration of the interior probability computation for a single point of a spatial domain when FSCP is rectangular. $X$ and $Y$ denote the uncertain data ranges of a point, with blue denoting the joint probability distribution over the uncertain range. The black rectangle denotes trait $\mathcal{T}$. The probability of a point being in the interior of a fiber surface can be computed by integrating the probability density function (blue) over trait $\mathcal{T}$ or the region enclosed by the dotted orange rectangle.
• Figure 3: Illustration of interior probability computation for a single point of a spatial domain when FSCP has an arbitrary shape. $X$ and $Y$ denote the uncertain data ranges of a point, with blue denoting the joint probability distribution over the uncertain range. The polygon with black boundaries denotes trait $\mathcal{T}$. The intersection of the two regions is indicated by dotted orange lines, and the coordinates of the intersection are indicated by pairs ($p_i, q_i$). The probability of a point being in the interior of a fiber surface can be computed with Green's theorem by summing the line integrals of the probability density function (blue) along the edges of the intersection polygon in a counterclockwise direction (see arrow heads).
• Figure 4: Fiber visualizations for the synthetic tangle-sphere dataset. A continuous scatterplot for the dataset is visualized in image (a) with trait $\mathcal{T}$ indicated by a cyan polygon. Image (b) visualizes the ground truth fiber surface that corresponds to trait $\mathcal{T}$. The ground truth dataset is mixed with random samples drawn from a bimodal noise distribution to generate an ensemble. Image (c) visualizes the mean-field fiber surface with a color-mapped signed distance from the ground truth fiber surface. The volume rendering of the interior probability volume, most probable fiber surface, and probabilistic segmentation are visualized for parametric (d)--(f) and nonparametric (g)--(i) statistical models. The volume rendering for nonparametric noise models visually represents the ground truth fiber positions more accurately than the volume rendering for parametric models (see \ref{['fig:errorAnalysisTangleSphereIntersection']} for quantitative analysis). Similarly, the most probable fiber surface for nonparametric models is spatially closer to the ground truth fiber surface than the surfaces for the mean-field and parametric models, as is evident by the color-mapped signed distance. The probabilistic segmentation provides insight into the positions with relatively high (opaque) or low (translucent) probability of fiber existence.
• Figure 5: Quantitative error analysis of interior probability computations for independent parametric and nonparametric noise models with respect to the ground truth interior probabilities. The solid lines plot the error as a function of the number of Monte Carlo samples. The magenta dotted line denotes the error for the mean-field, and the dashed lines denote the error for our proposed analytical solutions.