Positional Uncertainty of Isocontours: Condition Analysis and Probabilistic Measures

Abstract

Uncertainty is ubiquitous in science, engineering and medicine. Drawing conclusions from uncertain data is the normal case, not an exception. While the field of statistical graphics is well established, only a few 2D and 3D visualization and feature extraction methods have been devised that consider uncertainty. We present mathematical formulations for uncertain equivalents of isocontours based on standard probability theory and statistics and employ them in interactive visualization methods. As input data, we consider discretized uncertain scalar fields and model these as random fields. To create a continuous representation suitable for visualization we introduce interpolated probability density functions. Furthermore, we introduce numerical condition as a general means in feature-based visualization. The condition number-which potentially diverges in the isocontour problem-describes how errors in the input data are amplified in feature computation. We show how the average numerical condition of isocontours aids the selection of thresholds that correspond to robust isocontours. Additionally, we introduce the isocontour density and the level crossing probability field; these two measures for the spatial distribution of uncertain isocontours are directly based on the probabilistic model of the input data. Finally, we adapt interactive visualization methods to evaluate and display these measures and apply them to 2D and 3D data sets.

Uncertainty-Aware Multidimensional Ensemble Data Visualization and Exploration

“…Glyph based methods encode uncertainties into well-designed glyphs (e.g., the flow radar glyph [4], the circular glyph [19], and the summary plot [5]) and place them into the original data field. Similarly, visual variables such as color [10], [20], brightness [8], [21], blurriness [22], and texture [7], [23] can also be employed to encode uncertainty. The geometry based approaches are a family of techniques that adapt the basic geometry to represent uncertainty, including point [20], line [24], cube [25], and surrounding volume [10], [26].…”

“…However, most of them are only designed for 1D or 2D datasets and have limited capabilities to reveal the intrinsic structures in the ensemble dataset. Recent methods can effectively characterize and analyze the uncertainty structures [9], [10] and forms [11], [12] but are designed for data objects with one variable.…”

Feature-Based Uncertainty Visualization

“…Pfaffelmoser et al [19] presented a method for visualizing the positional variability around a mean iso-surface using direct volume rendering. A method to compute and visualize the positional uncertainty of contours in uncertain input data has been suggested by Pöthkow and Hege [21]. Assuming certain probability density functions, they modeled a discretely sampled uncertain scalar field by a discrete random field.…”

“…Then, we calculate the differences between them and use the mean and variance of the differences to represent the contour-level uncertainty. Our method is different from previous methods of studying uncertain contours which visualize probability field through color-mapping or volume rendering around a contour [19], [21], [22] since it is visualized through contour trees and measures the uncertainty as contour deviations in different ensemble members directly.…”

Application of Uncertainty Modeling Frameworks to Uncertain Isosurface Extraction

“…Most relevant one to the current work is isosurface extraction in the presence of uncertainty and hence, we focus only on visualization of uncertain isosurfaces. In order to quantify and visualize the uncertainty in isosurfaces extracted from uncertain scalar fields, parametric probabilistic models have been used to approximate Level-Crossing Probabilities (LCP) [15,16]. The concept of levelcrossing probabilities has been deployed to extend the conventional marching cubes algorithm, the predominant isosurface visualization scheme, for probabilistic modeling of uncertainty in scalar fields [17,18].…”