In this session we will explain and demonstrate to the attendants the relevance and use of correlation in uncertainty visualization. Our focus is on elucidating the potential of correlation to indicate the possible variations of structures in uncertain 2D and 3D scalar fields. We will address uncertain 2D height fields and isosurfaces in uncertain 3D scalar fields. We will first introduce structural uncertainty as the possible variations of the data values at different points relative to each other, and we will demonstrate how to use correlation to predict the structural uncertainty of particular features in the data. Our discussion is based on the stochastic modeling of the uncertainty via multivariate Gaussian distributions so that mean values and standard deviations exist, and the statistical dependency is given by the correlation. To demonstrate the use of correlation as an indicator for structural uncertainty, we will present a number of well-designed examples which clearly show the interdependencies between correlation structures and the variability of specific features in the data. We will use these examples to strengthen the awareness of the relevance of correlation analysis for estimating possible structural changes of relevant features due to uncertainty.

We will proceed by discussing methods for analyzing the structural variability via correlation visualization. Visualizing correlations is challenging since it is not clear how the visualization of complicated long-range dependencies can be integrated into standard visualizations of spatial data. Therefore, we will first discuss methods for visualizing local correlations in the vicinity of isosurfaces based on anisotropic correlation models. By visually distinguishing between the local correlations between points on the surface and along the surface’s normal directions, attendants obtain an improved understanding of the geometric and topological variability of uncertain isosurfaces. Since correlation is typically anisotropic, we will discuss general techniques for visualizing the directional structure of correlation. Texture- and glyph-based techniques will be demonstrated and compared to each other with respect to how well they can communicate these structures. The intention of this tutorial part is to equip the attendants with a notion of how complicated even the visualization of local correlation structures is, and how a visual depiction of structural uncertainty can be achieved in general.

We will then discuss approaches for visualizing positive and inverse global correlations, with a focus on correlation structures in uncertain 2D scalar fields. We will first review techniques which visualize the covariance matrix as a separate 2D structure, and we will then discuss alternatives which allow putting the correlation information into a spatial context. In particular, we will elaborate on the simultaneous visualization of correlation and standard deviation to obtain a comprehensive depiction of positional and geometric variations. We will show that by using such approaches, a local and global analysis of the stability of structures in a 2D scalar field is possible.

To demonstrate the practical relevance of correlation visualization we will use a number of synthetic and real-world data sets, where the uncertainty is either modeled via standard random distributions or given implicitly by a set of ensemble simulations. By using synthetic data sets in which certain correlation distributions have been enforced, we will demonstrate that correlation visualization is a mandatory ingredient of uncertainty visualization to reveal the otherwise hidden structural variability. As a result it should become clear that correlation visualization enables a more comprehensive uncertainty analysis and opens up new directions for further research. We further demonstrate the use of correlation visualization for analyzing the structural uncertainty in ensembles of 3D scalar fields from geology and atmosphere physics. By means of these examples we will show that correlation visualization techniques allow concluding on the stochastic stability of mean structures in simulated scalar fields. In this way, the attendants will learn that correlation visualization can help revealing regions which are strongly and weakly affected by structural uncertainty.

• Introduction
• Stochastic uncertainty modeling
• Definition of structural uncertainty
• Examples
• Correlation visualization
• Local correlation modeling
• Global correlation structures
• Applications

In this session we will explain and demonstrate the relevance and use of correlation in uncertainty visualization. Attendants will learn that correlations have to be considered in uncertainty visualization to obtain a more reliable estimation of the possible variations of structures due to uncertainty. A number of visualization techniques for correlation information will be described, so that the attendants obtain a good overview of the current state of the art in this field.

• T. Pfaffelmoser, M.Reitinger, R. Westermann, Visualization of Global Correlation Structures in Uncertain 2D Scalar Fields, Computer Graphics Forum, volume 31, Issue 3pt2, pages 1025-1034, June 2012
• Christoph Petz, Kai Pöthkow, Hans-Christian Hege, Probabilistic Local Features in Uncertain Vector Fields with Spatial Correlation, Computer Graphics Forum, volume 31, Issue 3pt2, pages 1045-1054, June 2012
• T. Pfaffelmoser, M. Reitinger, R. Westermann, Visualizing the Positional and Geometrical Variability of Isosurfaces in Uncertain Scalar Fields, Computer Graphics Forum Volume 30, Issue 3, pages 951–960, June 2011
• Kai Pöthkow, Britta Weber, Hans-Christian Hege, Probabilistic Marching Cubes, Computer Graphics Forum Volume 30, Issue 3, pages 931–940, June 2011

Technical University Munich

Rüdiger Westermann, born in Mai 1966, is Professor of Computer Graphics and Visualization in the Computer Science Department of the Technische Universität München. He serves as Coordinator of the Center for Computational and Visual Data Exploration, and he is member of the board of the TUM International Graduate School for Science and Engineering and the TUM Institute for Advanced Studies. His research interests include visual data exploration, visual simulation and real-time simulation and computer graphics. He was recently awarded with an Advanced Grant (2.3 million Euro) for pursuing research on uncertainty visualization by the European Research Council. He received his Diploma in Computer Science from the Technische Universität Darmstadt in 1991 and his Doctoral degree "with highest honours" from the University of Dortmund in 1996. From 1992 to 1997 he was a member of the research staff at the German National Institute for Mathematics and Computer Science in St. Augustin, Bonn, where he worked together with Wolfgang Krüger on parallel graphics algorithms. In 1998, he joined the Computer Graphics Group at the University of Erlangen-Nuremberg as a Research Scientist. Before he became an Assistant Professor in the Visualization Group at the University of Stuttgart in 1999 he was a Research Assistant in the Mulitres Group at Caltech and a Visiting Professor with the Scientific Computing Laboratory at the University of Utah. In 2001 he was appointed by the RWTH-Aachen as an Associate Professor for Scientific Visualization in the Department of Computer Science. Since 2003, Rüdiger Westermann is Chair of the Computer Graphics and Visualization group at the Technische Universität München.

Technical University Munich

Tobias Pfaffelmoser studied Technomathematics with minor subjects in computer science and medical engineering at Technische Universität München, where he received his Diploma degree in 2009. His Diploma Thesis focused on numerical algorithms for artifact correction in medical imaging procedures. Since 2009, he is with the Computer Graphics and Visualization Group at Technische Universität München, where he is currently working on his Ph.D. Furthermore, he is a member of the Munich Center of Advanced Computing (MAC) and involved in an interdisciplinary project on "Efficient Inversion Methods for Parameter Identification in the Earth Sciences". His research includes uncertainty visualization, visualization in geophysics and volume rendering.