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Contour Boxplots: A Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles
"... Fig. 1. Contour boxplot for an ensemble of the pressure field of a fluid flow simulation with a LIC background image for context. Abstract — Ensembles of numerical simulations are used in a variety of applications, such as meteorology or computational solid mechanics, in order to quantify the uncert ..."
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Fig. 1. Contour boxplot for an ensemble of the pressure field of a fluid flow simulation with a LIC background image for context. Abstract — 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 centeroutward 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 twodimensional simulation data in weather forecasting and computational fluid dynamics. Index Terms—Uncertainty visualization, Boxplots, band depth, ensemble visualization, order statistics. file:///Users/mahsa/Documents/Misc/paper_pics/envelop_ensemble_may.svg Page 1 of 1 1
APPROXIMATE LEVELCROSSING PROBABILITIES FOR INTERACTIVE VISUALIZATION OF UNCERTAIN ISOCONTOURS
, 2012
"... A major method for quantitative visualization of a scalar field is depiction of its isocontours. If the scalar field is afflicted with uncertainties, uncertain counterparts to isocontours have to be extracted and depicted. We consider the case where the input data is modeled as a discretized Gaussia ..."
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Cited by 5 (1 self)
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A major method for quantitative visualization of a scalar field is depiction of its isocontours. If the scalar field is afflicted with uncertainties, uncertain counterparts to isocontours have to be extracted and depicted. We consider the case where the input data is modeled as a discretized Gaussian field with spatial correlations. For this situation we want to compute levelcrossing probabilities that are associated to grid cells. To avoid the high computational cost of Monte Carlo integrations and directiondependencies of raycasting methods, we formulate two approximations for these probabilities that can be utilized during rendering by lookingup precomputed univariate and bivariate distribution functions. The first method, called maximum edge crossing probability, considers only pairwise correlations at a time. The second method, called linkedpairs method, considers joint and conditional probabilities between vertices along paths of a spanning tree over the n vertices of the grid cell; with each possible tree an ndimensional approximate distribution is associated; the choice of the distribution is guided by minimizing its Bhattacharyya distance to the original distribution. We perform a quantitative and qualitative evaluation of the approximation errors on synthetic data and show the utility of both approximations on the example of climate simulation data.
Visualizing the Variability of Gradients in Uncertain 2D Scalar Fields
"... Abstract—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 variab ..."
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Abstract—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 realworld 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. Index Terms—Uncertainty visualization, gradient variability, structural uncertainty, glyphs. 1
MultiCharts for Comparative 3D Ensemble Visualization
"... operational ensemble weather forecast system [30]. The ensemble consists of 51 members of resolution 256×128×64 each. Each bar in the multichart is associated with a distinct 3D subdomain, and encodes the distribution of the ensemble members in this subdomain by means of a histogram. In addition, a ..."
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operational ensemble weather forecast system [30]. The ensemble consists of 51 members of resolution 256×128×64 each. Each bar in the multichart is associated with a distinct 3D subdomain, and encodes the distribution of the ensemble members in this subdomain by means of a histogram. In addition, a few userselected ensemble members are depicted using polylines. By means of brushing in the multichart view (indicated by yellow background color), the user has selected regions where the range over the ensemble members and thus the uncertainty is high. The selected regions are instantly emphasized in the 3D view. Abstract—A comparative visualization of multiple volume data sets is challenging due to the inherent occlusion effects, yet it is important to effectively reveal uncertainties, correlations and reliable trends in 3D ensemble fields. In this paper we present bidirectional linking of multicharts and volume visualization as a means to analyze visually 3D scalar ensemble fields at the data level. Multicharts are an extension of conventional bar and line charts: They linearize the 3D data points along a spacefilling curve and draw them as multiple charts in the same plot area. The bar charts encode statistical information on ensemble members, such as histograms and probability densities, and line charts are overlayed to allow comparing members against the ensemble. Alternative linearizations based on histogram similarities or ensemble variation allow clustering of spatial locations depending on data distribution. Multicharts organize the data at multiple scales to quickly provide overviews and enable users to select regions exhibiting interesting behavior interactively. They are further put into a spatial context by allowing the user to brush or query value intervals and specific distributions, and to simultaneously visualize the corresponding spatial points via volume rendering. By providing a picking mechanism in 3D and
Predictabilitybased adaptive mouse interaction and zooming for visual flow exploration
 International Journal for Uncertainty Quantification
"... Abstract—Flow fields are often investigated by adopting a Lagrangian view, for example, by particle tracing of integral curves such as streamlines and path lines or by computing delocalized quantities. For visual exploration, mouse interaction is predominantly used to define starting points for time ..."
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Abstract—Flow fields are often investigated by adopting a Lagrangian view, for example, by particle tracing of integral curves such as streamlines and path lines or by computing delocalized quantities. For visual exploration, mouse interaction is predominantly used to define starting points for timedependent Lagrangian methods. This paper focuses on the uncertainty of mouse input and its impact on the visualization process. In typical cases, the interaction is achieved by mouse motion, exhibiting uncertainty in the range of a screen pixel. From the perspective of dynamical systems theory, an integral curve represents an initial value problem, the uncertainty a perturbation of its initial condition, and the uncertainty of the visualization procedure a predictability problem. Predictability analysis is concerned with the growth of perturbations under the action of flow. In our case, it is not unusual that the perturbations grow from single pixels to substantial deviations. We therefore present an interaction scheme based on the largest finitetime Lyapunov exponent and the flow map gradient, providing accurate, smooth, and easytouse flow exploration. This scheme employs datadriven adaptation of mouse speed and direction as well as optional augmentation by an adaptive zoom lens with consistent magnification. We compare our approach to nonadaptive mouse interaction and demonstrate it for several examples of datasets. Furthermore, we present results from a user study with nine domain experts. Index Terms—Interactive flow visualization, predictability, finitetime Lyapunov exponent, adaptive mouse speed, adaptive zoom. 1
Ovis: A framework for visual analysis of ocean forecast ensembles
 IEEE TVCG
"... Abstract—We present a novel integrated visualization system that enables interactive visual analysis of ensemble simulations of the sea surface height that is used in ocean forecasting. The position of eddies can be derived directly from the sea surface height and our visualization approach enables ..."
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Abstract—We present a novel integrated visualization system that enables interactive visual analysis of ensemble simulations of the sea surface height that is used in ocean forecasting. The position of eddies can be derived directly from the sea surface height and our visualization approach enables their interactive exploration and analysis. The behavior of eddies is important in different application settings of which we present two in this paper. First, we show an application for interactive planning of placement as well as operation of offshore structures using realworld ensemble simulation data of the Gulf of Mexico. Offshore structures, such as those used for oil exploration, are vulnerable to hazards caused by eddies, and the oil and gas industry relies on ocean forecasts for efficient operations. We enable analysis of the spatial domain, as well as the temporal evolution, for planning the placement and operation of structures. Eddies are also important for marine life. They transport water over large distances and with it also heat and other physical properties as well as biological organisms. In the second application we present the usefulness of our tool, which could be used for planning the paths of autonomous underwater vehicles, so called gliders, for marine scientists to study simulation data of the largely unexplored Red Sea.
CORRELATION VISUALIZATION FOR STRUCTURAL UNCERTAINTY ANALYSIS
 INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION, 3 (2): 171–186 (2013)
, 2013
"... In uncertain scalar fields, where the values at every point can be assumed as realizations of a random variable, standard deviations indicate the strength of possible variations of these values from their mean values, independently of the values at any other point in the domain. To infer the possibl ..."
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Cited by 2 (1 self)
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In uncertain scalar fields, where the values at every point can be assumed as realizations of a random variable, standard deviations indicate the strength of possible variations of these values from their mean values, independently of the values at any other point in the domain. To infer the possible variations at different points relative to each other, and thus to predict the possible structural occurrences, i.e., the structural variability, of particular features in the data, the correlation between the values at these points has to be considered. The purpose of this paper is to shed light on the use of correlation as an indicator for the structural variability of isosurfaces in uncertain threedimensional scalar fields. In a number of examples, we first demonstrate some general conclusions one can draw from the correlations in uncertain data regarding its structural variability. We will further explain, why an adequate correlation visualization is crucial for a comprehensive uncertainty analysis. Then, our focus is on the visualization of local and usually anisotropic correlation structures in the vicinity of uncertain isosurfaces. Therefore, we propose a model that can represent anisotropic correlation structures on isosurfaces and allows visual distinguishing of the local correlations between points on the surface and along the surface’s normal directions. A glyphbased approach is used to simultaneously visualize these dependencies. The practical relevance of our work is demonstrated in artificial and realworld examples using standard random distributions and ensemble simulations.
Uncertaintyaware Multidimensional Ensemble Data Visualization and Exploration
"... Abstract—This paper presents an efficient visualization and exploration approach for modeling and characterizing the relationships and uncertainties in the context of multidimensional ensemble datasets. Its core is a novel dissimilaritypreserving projection technique that characterizes not only the ..."
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Abstract—This paper presents an efficient visualization and exploration approach for modeling and characterizing the relationships and uncertainties in the context of multidimensional ensemble datasets. Its core is a novel dissimilaritypreserving projection technique that characterizes not only the relationships among the mean values of the ensemble data objects but also the relationships among the distributions of ensemble members. This uncertaintyaware projection scheme leads to an improved understanding of the intrinsic structure in an ensemble dataset. The analysis of the ensemble dataset is further augmented by a suite of visual encoding and exploration tools. Experimental results on both artificial and realworld datasets demonstrate the effectiveness of our approach. Index Terms—Ensemble visualization, uncertainty quantification, uncertainty visualization, multidimensional data visualization F 1
Visualization for Understanding Uncertainty in the Simulation of Myocardial Ischemia
"... We have created the Myocardial Uncertainty Viewer (muView or μView) tool for exploring data stemming from the forward simulation of cardiac ischemia. The simulation uses a collection of conductivity values to understand how ischemic regions effect the undamaged anisotropic heart tissue. The data res ..."
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We have created the Myocardial Uncertainty Viewer (muView or μView) tool for exploring data stemming from the forward simulation of cardiac ischemia. The simulation uses a collection of conductivity values to understand how ischemic regions effect the undamaged anisotropic heart tissue. The data resulting from the simulation is multivalued and volumetric and thus, for every data point, we have a collection of samples describing cardiac electrical properties. μView combines a suite of visual analysis methods to explore the area surrounding the ischemic zone and identify how perturbations of variables changes the propagation of their effects.