Positional Uncertainty of Isocontours: Condition Analysis and Probabilistic Measures

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.

[1]  Hans-Christian Hege,et al.  Uncertain 2D Vector Field Topology , 2010, Comput. Graph. Forum.

[2]  Alex T. Pang,et al.  Visualizing scalar volumetric data with uncertainty , 2002, Comput. Graph..

[3]  T. Palmer,et al.  Development of a European Multi-Model Ensemble System for Seasonal to Inter-Annual Prediction (DEMETER) , 2004 .

[4]  Hans-Christian Hege,et al.  Vortex and Strain Skeletons in Eulerian and Lagrangian Frames , 2007, IEEE Transactions on Visualization and Computer Graphics.

[5]  J. Azaïs,et al.  Level Sets and Extrema of Random Processes and Fields , 2009 .

[6]  Paolo Fornasini,et al.  The Uncertainty in Physical Measurements: An Introduction to Data Analysis in the Physics Laboratory , 2008 .

[7]  Kwan-Liu Ma,et al.  Importance-Driven Time-Varying Data Visualization , 2008, IEEE Transactions on Visualization and Computer Graphics.

[8]  Andrew P. Morse,et al.  DEVELOPMENT OF A EUROPEAN MULTIMODEL ENSEMBLE SYSTEM FOR SEASONAL-TO-INTERANNUAL PREDICTION (DEMETER) , 2004 .

[9]  Peter Deuflhard,et al.  Numerical Analysis in Modern Scientific Computing , 2003 .

[10]  Jong-Jin Baik,et al.  Flow and dispersion in an urban cubical cavity , 2009 .

[11]  R. Adler The Geometry of Random Fields , 2009 .

[12]  Björn Zehner,et al.  Visualization of gridded scalar data with uncertainty in geosciences , 2010, Comput. Geosci..

[13]  Weldon A Lodwick Fuzzy Surfaces in GIS and Geographical Analysis : Theory, Analytical Methods, Algorithms and Applications , 2007 .

[14]  Penny Rheingans,et al.  Point-based probabilistic surfaces to show surface uncertainty , 2004, IEEE Transactions on Visualization and Computer Graphics.

[15]  Chi-Wing Fu,et al.  Visualizing Large-Scale Uncertainty in Astrophysical Data , 2007, IEEE Transactions on Visualization and Computer Graphics.

[16]  Ignacio Lira Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance , 2002 .

[17]  H. J. Arnold Introduction to the Practice of Statistics , 1990 .

[18]  Mark Gahegan,et al.  Visualizing Geospatial Information Uncertainty: What We Know and What We Need to Know , 2005 .

[19]  Alex T. Pang,et al.  Visualizing Sparse Gridded Data Sets , 2000, IEEE Computer Graphics and Applications.

[20]  R M Harrison,et al.  A comparison of two methods for measuring the signal to noise ratio on MR images , 1999, Physics in medicine and biology.

[21]  J. Thong,et al.  Single-image signal-to-noise ratio estimation. , 2006, Scanning.

[22]  Rüdiger Westermann,et al.  Acceleration techniques for GPU-based volume rendering , 2003, IEEE Visualization, 2003. VIS 2003..

[23]  J. Hunt Vorticity and vortex dynamics in complex turbulent flows , 1987 .

[24]  Robert S. Laramee,et al.  Uncertainty Visualization Methods in Isosurface Rendering , 2003, Eurographics.

[25]  Daniel Weiskopf,et al.  Texture-based visualization of uncertainty in flow fields , 2005, VIS 05. IEEE Visualization, 2005..

[26]  Anders Ynnerman,et al.  Uncertainty Visualization in Medical Volume Rendering Using Probabilistic Animation , 2007, IEEE Transactions on Visualization and Computer Graphics.

[27]  Rafael Wiemker,et al.  Fast detection of meaningful isosurfaces for volume data visualization , 2001, Proceedings Visualization, 2001. VIS '01..

[28]  Ecmwf Newsletter,et al.  EUROPEAN CENTRE FOR MEDIUM-RANGE WEATHER FORECASTS , 2004 .

[29]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[30]  Alex T. Pang,et al.  Glyphs for Visualizing Uncertainty in Vector Fields , 1996, IEEE Trans. Vis. Comput. Graph..

[31]  R. Kanwal Generalized Functions: Theory and Technique , 1998 .

[32]  Alex T. Pang,et al.  Approaches to uncertainty visualization , 1996, The Visual Computer.

[33]  A. Hohmann,et al.  Numerical Analysis in Modern Scientific Computing: An Introduction , 2003 .

[34]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[35]  M. Sheelagh T. Carpendale,et al.  Exploration of uncertainty in bidirectional vector fields , 2008, Electronic Imaging.

[36]  Ignacio Lira,et al.  Evaluating the Measurement Uncertainty , 2002 .

[37]  Paolo Fornasini,et al.  The Uncertainty in Physical Measurements , 2008 .

[38]  Valerio Pascucci,et al.  The contour spectrum , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[39]  Leonidas J. Guibas,et al.  Uncertainty and Variability in Point Cloud Surface Data , 2004, PBG.

[40]  Joe Michael Kniss,et al.  Statistically quantitative volume visualization , 2005, VIS 05. IEEE Visualization, 2005..

[41]  Jennifer L. Dungan,et al.  Visualizing Spatial Distribution Data Sets , 2003, VisSym.

[42]  Gerik Scheuermann,et al.  Multifield visualization using local statistical complexity , 2007, IEEE Transactions on Visualization and Computer Graphics.

[43]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[44]  Ross T. Whitaker,et al.  Curvature-based transfer functions for direct volume rendering: methods and applications , 2003, IEEE Visualization, 2003. VIS 2003..

[45]  Heidrun Schumann,et al.  The Visualization of Uncertain Data: Methods and Problems , 2006, SimVis.

[46]  Cláudio T. Silva,et al.  Revisiting Histograms and Isosurface Statistics , 2008, IEEE Transactions on Visualization and Computer Graphics.

[47]  Chris R. Johnson,et al.  A Next Step: Visualizing Errors and Uncertainty , 2003, IEEE Computer Graphics and Applications.

[48]  T. O’Neil Geometric Measure Theory , 2002 .