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Positional uncertainty of isocontours: Condition analysis and probabilistic measures
 IEEE Transactions on Visualization and Computer Graphics
"... 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 ..."
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Cited by 28 (5 self)
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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 featurebased 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.
A user study to compare four uncertainty visualization methods for 1d and 2d datasets
 IEEE Transactions on Visualization and Computer Graphics
"... AbstractMany techniques have been proposed to show uncertainty in data visualizations. However, very little is known about their effectiveness in conveying meaningful information. In this paper, we present a user study that evaluates the perception of uncertainty amongst four of the most commonly us ..."
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Cited by 25 (1 self)
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AbstractMany techniques have been proposed to show uncertainty in data visualizations. However, very little is known about their effectiveness in conveying meaningful information. In this paper, we present a user study that evaluates the perception of uncertainty amongst four of the most commonly used techniques for visualizing uncertainty in onedimensional and twodimensional data. The techniques evaluated are traditional errorbars, scaled size of glyphs, colormapping on glyphs, and colormapping of uncertainty on the data surface. The study uses generated data that was designed to represent the systematic and random uncertainty components. Twentyseven users performed two types of search tasks and two types of counting tasks on 1D and 2D datasets. The search tasks involved finding data points that were least or most uncertain. The counting tasks involved counting data features or uncertainty features. A 44u fullfactorial ANOVA indicated a significant interaction between the techniques used and the type of tasks assigned for both datasets indicating that differences in performance between the four techniques depended on the type of task performed. Several oneway ANOVAs were computed to explore the simple main effects. Bonferronnis correction was used to control for the familywise error rate for alphainflation. Although we did not find a consistent order among the four techniques for all the tasks, there are several findings from the study that we think are useful for uncertainty visualization design. We found a significant difference in user performance between searching for locations of high and searching for locations of low uncertainty. Errorbars consistently underperformed throughout the experiment. Scaling the size of glyphs and colormapping of the surface performed reasonably well. The efficiency of most of these techniques were highly
Visualizing the positional and geometrical variability of isosurfaces in uncertain scalar fields
 in Computer Graphics Forum
, 2011
"... We present a novel approach for visualizing the positional and geometrical variability of isosurfaces in uncertain 3D scalar fields. Our approach extends recent work by Pöthkow and Hege [PH10] in that it accounts for correlations in the data to determine more reliable isosurface crossing probabiliti ..."
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Cited by 22 (5 self)
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We present a novel approach for visualizing the positional and geometrical variability of isosurfaces in uncertain 3D scalar fields. Our approach extends recent work by Pöthkow and Hege [PH10] in that it accounts for correlations in the data to determine more reliable isosurface crossing probabilities. We introduce an incremental updatescheme that allows integrating the probability computation into fronttoback volume raycasting efficiently. Our method accounts for homogeneous and anisotropic correlations, and it determines for each sampling interval along a ray the probability of crossing an isosurface for the first time. To visualize the positional and geometrical uncertainty even under viewing directions parallel to the surface normal, we propose a new color mapping scheme based on the approximate spatial deviation of possible surface points from the mean surface. The additional use of saturation enables to distinguish between areas of high and low statistical dependence. Experimental results confirm the effectiveness of our approach for the visualization of uncertainty related to position and shape of convex and concave isosurface structures. Categories and Subject Descriptors (according to ACM CCS): Generation—Display algorithms, Viewing algorithms
J.D.: Evaluating Sketchiness as a Visual Variable for the Depiction of Qualitative Uncertainty. IEEE Transactions on Visualization and Computer Graphics 18 (2012), 2769–2778. 6 [BPC∗10
 BW08] BYRON L., WATTENBERG M.: Stacked Graphs — Geometry & Aesthetics. IEEE Transactions on Visualization and Computer Graphics 14 (2008), 1245–1252. 2 [CJ10] CHEN M., JAENICKE
"... Abstract—We report on results of a series of user studies on the perception of four visual variables that are commonly used in the literature to depict uncertainty. To the best of our knowledge, we provide the first formal evaluation of the use of these variables to facilitate an easier reading of ..."
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Cited by 13 (5 self)
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Abstract—We report on results of a series of user studies on the perception of four visual variables that are commonly used in the literature to depict uncertainty. To the best of our knowledge, we provide the first formal evaluation of the use of these variables to facilitate an easier reading of uncertainty in visualizations that rely on line graphical primitives. In addition to blur, dashing and grayscale, we investigate the use of ‘sketchiness ’ as a visual variable because it conveys visual impreciseness that may be associated with data quality. Inspired by work in nonphotorealistic rendering and by the features of handdrawn lines, we generate line trajectories that resemble handdrawn strokes of various levels of proficiency—ranging from child to adult strokes—where the amount of perturbations in the line corresponds to the level of uncertainty in the data. Our results show that sketchiness is a viable alternative for the visualization of uncertainty in lines and is as intuitive as blur; although people subjectively prefer dashing style over blur, grayscale and sketchiness. We discuss advantages and limitations of each technique and conclude with design considerations on how to deploy these visual variables to effectively depict various levels of uncertainty for line marks. Index Terms—Uncertainty visualization, qualitative evaluation, quantitative evaluation, perception. 1
Exploring the Millennium Run  Scalable Rendering of LargeScale Cosmological Datasets
"... In this paper we investigate scalability limitations in the visualization of largescale particlebased cosmological simulations, and we present methods to reduce these limitations on current PC architectures. To minimize the amount of data to be streamed from disk to the graphics subsystem, we prop ..."
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Cited by 12 (1 self)
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In this paper we investigate scalability limitations in the visualization of largescale particlebased cosmological simulations, and we present methods to reduce these limitations on current PC architectures. To minimize the amount of data to be streamed from disk to the graphics subsystem, we propose a visually continuous levelofdetail (LOD) particle representation based on a hierarchical quantization scheme for particle coordinates and rules for generating coarse particle distributions. Given the maximal world space error per level, our LOD selection technique guarantees a subpixel screen space error during rendering. A brickbased pagetree allows to further reduce the number of disk seek operations to be performed. Additional particle quantities like density, velocity dispersion, and radius are compressed at no visible loss using vector quantization of logarithmically encoded floating point values. By finegrain viewfrustum culling and presence acceleration in a geometry shader the required geometry throughput on the GPU can be significantly reduced. We validate the quality and scalability of our method by presenting visualizations of a particlebased cosmological darkmatter simulation exceeding 10 billion elements.
From quantification to visualization: A taxonomy of uncertainty visualization approaches
 In Uncertainty Quantification in Scientific Computing
, 2012
"... Abstract. Quantifying uncertainty is an increasingly important topic across many domains. The uncertainties present in data come with many diverse representations having originated from a wide variety of disciplines. Communicating these uncertainties is a task often left to visualization without c ..."
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Cited by 11 (0 self)
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Abstract. Quantifying uncertainty is an increasingly important topic across many domains. The uncertainties present in data come with many diverse representations having originated from a wide variety of disciplines. Communicating these uncertainties is a task often left to visualization without clear connection between the quantification and visualization. In this paper, we first identify frequently occurring types of uncertainty. Second, we connect those uncertainty representations to ones commonly used in visualization. We then look at various approaches to visualizing this uncertainty by partitioning the work based on the dimensionality of the data and the dimensionality of the uncertainty. We also discuss noteworthy exceptions to our taxonomy along with future research directions for the uncertainty visualization community.
Visualization for the Physical Sciences
 EUROGRAPHICS
"... Close collaboration with other scientific fields is seen as an important goal for the visualization community by leading researchers in visualization. Yet, engaging in a scientific collaboration can be challenging. Physical sciences, with its array of research directions, provide many exciting chall ..."
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Cited by 8 (2 self)
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Close collaboration with other scientific fields is seen as an important goal for the visualization community by leading researchers in visualization. Yet, engaging in a scientific collaboration can be challenging. Physical sciences, with its array of research directions, provide many exciting challenges for a visualization scientist which in turn create ample possibilities for collaboration. We present the first survey of its kind that provides a comprehensive view on existing work on visualization for the physical sciences. We introduce a novel classification scheme based on application area, data dimensionality and main challenge addressed and apply this classification scheme to each contribution from the literature. Our classification highlights mature areas in visualization for the physical sciences and suggests directions for future work. Our survey serves as a useful starting point for those interested in visualization for the physical sciences, namely astronomy, chemistry, earth sciences and physics.
A Review of Uncertainty in Data Visualization
, 2012
"... Most visualization techniques have been designed on the assumption that the data to be represented are free from uncertainty. Yet this is rarely the case. Recently the visualization community has risen to the challenge of incorporating an indication of uncertainty into visual representations, and i ..."
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Most visualization techniques have been designed on the assumption that the data to be represented are free from uncertainty. Yet this is rarely the case. Recently the visualization community has risen to the challenge of incorporating an indication of uncertainty into visual representations, and in this article we review their work. We place the work in the context of a reference model for data visualization, that sees data pass through a pipeline of processes. This allows us to distinguish the visualization of uncertainty which considers how we depict uncertainty specified with the data and the uncertainty of visualization which considers how much inaccuracy occurs as we process data through the pipeline. It has taken some time for uncertain visualization methods to be developed, and we explore why uncertainty visualization is hard one explanation is that we typically need to find another display dimension and we may have used these up already! To organise the material we return to a typology developed by one of us in the early days of visualization, and make use of this to present a catalogue of visualization techniques describing the research that has been done to extend each method to handle uncertainty. Finally we note the responsibility on us all to incorporate any known uncertainty into a visualization, so that integrity of the discipline is maintained.
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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Cited by 4 (0 self)
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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
M.: Visiting the Gödel Universe
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract — Visualization of general relativity illustrates aspects of Einstein’s insights into the curved nature of space and time to the expert as well as the layperson. One of the most interesting models which came up with Einstein’s theory was developed by Kurt Gödel in 1949. The Gödel universe i ..."
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Abstract — Visualization of general relativity illustrates aspects of Einstein’s insights into the curved nature of space and time to the expert as well as the layperson. One of the most interesting models which came up with Einstein’s theory was developed by Kurt Gödel in 1949. The Gödel universe is a valid solution of Einstein’s field equations, making it a possible physical description of our universe. It offers remarkable features like the existence of an optical horizon beyond which time travel is possible. Although we know that our universe is not a Gödel universe, it is interesting to visualize physical aspects of a world model resulting from a theory which is highly confirmed in scientific history. Standard techniques to adopt an egocentric point of view in a relativistic world model have shortcomings with respect to the time needed to render an image as well as difficulties in applying a direct illumination model. In this paper we want to face both issues to reduce the gap between common visualization standards and relativistic visualization. We will introduce two techniques to speed up recalculation of images by means of preprocessing and lookup tables and to increase image quality through a special optimization applicable to the Gödel universe. The first technique allows the physicist to understand the different effects of general relativity faster and better by generating images from existing datasets interactively. By using the intrinsic symmetries of Gödel’s spacetime which are expressed by the Killing vector field, we are able to reduce the necessary calculations to simple cases using the second technique. This even makes it feasible to account for a direct illumination model during the rendering process. Although the presented methods are applied to Gödel’s universe, they can also be extended to other manifolds, for example light propagation in moving dielectric media. Therefore, other areas of research can benefit from these generic improvements. Index Terms—General relativity, Gödel universe, nonlinear ray tracing, time travel. 1