Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.
To visualize the effect of uncertainty on contours, envelopes within a gridded domain are extracted to indicate in which volume the contour will lie (with a certain confidence) @cite_11 @cite_26 . Uncertainty associated with a contour can also be rendered via animation @cite_33 or as a collection of points where each point is displayed from its original location along the surface normal by an amount proportional to the uncertainty at that point" @cite_41 . Positional and geometrical variations of contours are captured by variability in gradients for uncertain scalar fields @cite_5 . Positional uncertainty of contours could also be encoded by spatial correlation @cite_32 @cite_63 or numerical sensitivity @cite_38 . Building on the notions of functional boxplots and data depth, contour boxplots @cite_36 display statistical quantities analogous to the mean, median, and order statistics for ensembles of contours.
@cite_38 @cite_26 @cite_33 @cite_41 @cite_36 @cite_32 @cite_63 @cite_5 @cite_11
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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. Visualized data often have dubious origins and quality. Different forms of uncertainty and errors are also introduced as the data are derived, transformed, interpolated, and finally rendered. In the absence of integrated presentation of data and uncertainty, the analysis of the visualization is incomplete at best and often leads to inaccurate or incorrect conclusions. This paper surveys techniques for presenting data together with uncertainty. These uncertainty visualization techniques present data in such a manner that users are made aware of the locations and degree of uncertainties in their data so as to make more informed analyses and decisions. The techniques include adding glyphs, adding geometry, modifying geometry, modifying attributes, animation, sonification, and psycho-visual approaches. We present our results in uncertainty visualization for environmental visualization, surface interpolation, global illumination with radiosity, flow visualization, and figure animation. We also present a classification of the possibilities in uncertainty visualization, and locate our contributions within this classification. Research into the visualisation of imprecise data is a relatively new field in visualisation. Work is beginning to appear detailing the process of visualising uncertainty in data. Continuing previous work by the author, this paper seeks to extend techniques used to visualise uncertainty from the spatial to the temporal domain, by using visual vibrations to indicate the level of imprecision at a visualised data point. The paper contains an analysis of the present visual features used to indicate imprecision, and then details a methodology for using visual vibrations to display the uncertainty contained in visualised data. Novel additions include addressing chart junk issues outlined by Tufte, additions of perceptual factors and extension to stereo vision applications. Efficient and informative visualization of surfaces with uncertainties is an important topic with many applications in science and engineering. In these applications, the correct course of action may depend not only on the location of a boundary, but on the precision with which that location is known. Examples include environmental pollution borderline detection, oil basin edge characterization, or discrimination between cancerous and healthy tissue in medicine. We present a method for producing visualizations of surfaces with uncertainties using points as display primitives. Our approach is to render the surface as a collection of points and to displace each point from its original location along the surface normal by an amount proportional to the uncertainty at that point. This approach can be used in combination with other techniques such as pseudocoloring to produce efficient and revealing visualizations. The basic approach is sufficiently flexible to allow natural extensions; we show incorporation of expressive modulation of opacity, change of the stroke primitive, and addition of an underlying polygonal model. The method is used to visualize real and simulated tumor formations with uncertainty of tumor boundaries. The point-based technique is compared to pseudocoloring for a position estimation task in a preliminary user study. Ensembles of numerical simulations are used in a variety of applications, such as meteorology or computational solid mechanics, in order to quantify the uncertainty or possible error in a model or simulation. Deriving robust statistics and visualizing the variability of an ensemble is a challenging task and is usually accomplished through direct visualization of ensemble members or by providing aggregate representations such as an average or pointwise probabilities. In many cases, the interesting quantities in a simulation are not dense fields, but are sets of features that are often represented as thresholds on physical or derived quantities. In this paper, we introduce a generalization of boxplots, called contour boxplots, for visualization and exploration of ensembles of contours or level sets of functions. Conventional boxplots have been widely used as an exploratory or communicative tool for data analysis, and they typically show the median, mean, confidence intervals, and outliers of a population. The proposed contour boxplots are a generalization of functional boxplots, which build on the notion of data depth. Data depth approximates the extent to which a particular sample is centrally located within its density function. This produces a center-outward ordering that gives rise to the statistical quantities that are essential to boxplots. Here we present a generalization of functional data depth to contours and demonstrate methods for displaying the resulting boxplots for two-dimensional simulation data in weather forecasting and computational fluid dynamics. Visualizing correlations, i.e., the tendency of uncertain data values at different spatial positions to change contrarily or according to each other, allows inferring on the possible variations of structures in the data. Visualizing global correlation structures, however, is extremely challenging, since it is not clear how the visualization of complicated long-range dependencies can be integrated into standard visualizations of spatial data. Furthermore, storing correlation information imposes a memory requirement that is quadratic in the number of spatial sample positions. This paper presents a novel approach for visualizing both positive and inverse global correlation structures in uncertain 2D scalar fields, where the uncertainty is modeled via a multivariate Gaussian distribution. We introduce a new measure for the degree of dependency of a random variable on its local and global surroundings, and we propose a spatial clustering approach based on this measure to classify regions of a particular correlation strength. The clustering performs a correlation filtering, which results in a representation that is only linear in the number of spatial sample points. Via cluster coloring the correlation information can be embedded into visualizations of other statistical quantities, such as the mean and the standard deviation. We finally propose a hierarchical cluster subdivision scheme to further allow for the simultaneous visualization of local and global correlations. © 2012 Wiley Periodicals, Inc. In uncertain scalar fields where data values vary with a certain probability, the strength of this variability indicates the confidence in the data. It does not, however, allow inferring on the effect of uncertainty on differential quantities such as the gradient, which depend on the variability of the rate of change of the data. Analyzing the variability of gradients is nonetheless more complicated, since, unlike scalars, gradients vary in both strength and direction. This requires initially the mathematical derivation of their respective value ranges, and then the development of effective analysis techniques for these ranges. This paper takes a first step into this direction: Based on the stochastic modeling of uncertainty via multivariate random variables, we start by deriving uncertainty parameters, such as the mean and the covariance matrix, for gradients in uncertain discrete scalar fields. We do not make any assumption about the distribution of the random variables. Then, for the first time to our best knowledge, we develop a mathematical framework for computing confidence intervals for both the gradient orientation and the strength of the derivative in any prescribed direction, for instance, the mean gradient direction. While this framework generalizes to 3D uncertain scalar fields, we concentrate on the visualization of the resulting intervals in 2D fields. We propose a novel color diffusion scheme to visualize both the absolute variability of the derivative strength and its magnitude relative to the mean values. A special family of circular glyphs is introduced to convey the uncertainty in gradient orientation. For a number of synthetic and real-world data sets, we demonstrate the use of our approach for analyzing the stability of certain features in uncertain 2D scalar fields, with respect to both local derivatives and feature orientation. Characterization of the earth's subsurface involves the construction of 3D models from sparse data and so leads to simulation results that involve some degree of uncertainty. This uncertainty is often neglected in the subsequent visualization, due to the fact that no established methods or available software exist. We describe a visualization method to render scalar fields with a probability density function at each data point. We render these data as isosurfaces and make use of a colour scheme, which intuitively gives the viewer an idea of which parts of the surface are more reliable than others. We further show how to extract an envelope that indicates within which volume the isosurface will lie with a certain confidence, and augment the isosurfaces with additional geometry in order to show this information. The resulting visualization is easy and intuitive to understand and is suitable for rendering multiple distinguishable isosurfaces at a time. It can moreover be easily used together with other visualized objects, such as the geological context. Finally we show how we have integrated this into a visualization pipeline that is based on the Visualization Toolkit (VTK) and the open source scenegraph OpenSG, allowing us to render the results on a desktop and in different kinds of virtual environments.