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Issue No.10 - October (2011 vol.17)
pp: 1393-1406
Kai Pöthkow , Konrad-Zuse-Zentrum für Informationstechnik, Berlin
Hans-Christian Hege , Konrad-Zuse-Zentrum für Informationstechnik, Berlin
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.
Uncertainty, probability, isolines, isosurfaces, numerical condition, error analysis, volume visualization.
Kai Pöthkow, Hans-Christian Hege, "Positional Uncertainty of Isocontours: Condition Analysis and Probabilistic Measures", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 10, pp. 1393-1406, October 2011, doi:10.1109/TVCG.2010.247
[1] B. Taylor and C. Kuyatt, Guidelines for Expressing and Evaluating the Uncertainty of NIST Experimental Results, NIST Technical Note 1297, 1994.
[2] ISO/IEC Guide 98-3:2008 Uncertainty of Measurement—Part 3: Guide to the Expression of Uncertainty in Measurement (GUM), Int'l Organization for Standardization, 2008.
[3] W. Feller, Introduction to Probability Theory and Its Applications. John Wiley and Sons, 1968 and 1971.
[4] P. Fornasini, The Uncertainty in Physical Measurements. Springer, 2008.
[5] I. Lira, Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance. Inst. of Physics Publishing, 2002.
[6] C.R. Johnson and A.R. Sanderson, “A Next Step: Visualizing Errors and Uncertainty,” IEEE Computer Graphics and Applications, vol. 23, no. 5, pp. 6-10, Sept./Oct. 2003.
[7] G. Grigoryan and P. Rheingans, “Point-Based Probabilistic Surfaces to Show Surface Uncertainty,” IEEE Trans. Visualization and Computer Graphics, vol. 10, no. 5, pp. 564-573, Sept./Oct. 2004.
[8] G. Kindlmann, R. Whitaker, T. Tasdizen, and T. Möller, “Curvature-Based Transfer Functions for Direct Volume Rendering: Methods and Applications,” Proc. IEEE Visualization 2003 Conf., pp. 513-520, Oct. 2003.
[9] P.J. Rhodes, R.S. Laramee, R.D. Bergeron, and T.M. Sparr, “Uncertainty Visualization Methods in Isosurface Rendering,” Proc. Eurographics 2003, Short Papers, pp. 83-88, 2003.
[10] A.T. Pang, C.M. Wittenbrink, and S.K. Lodha, “Approaches to Uncertainty Visualization,” The Visual Computer, vol. 13, no. 8, pp. 370-390, 1997.
[11] H. Griethe and H. Schumann, “The Visualization of Uncertain Data: Methods and Problems,” SimVis, T. Schulze, G. Horton, B. Preim, and S. Schlechtweg, eds., pp. 143-156, SCS Publishing House e.V., Mar. 2006.
[12] P. Fornasini, The Uncertainty in Physical Measurements: An Introduction to Data Analysis in the Physics Laboratory. Springer, 2008.
[13] A. Luo, D. Kao, and A. Pang, “Visualizing Spatial Distribution Data Sets,” Proc. Symp. Data Visualization 2003 (VISSYM '03), pp. 29-38, 2003.
[14] L. Zadeh, “Fuzzy Sets,” Information Control, vol. 8, pp. 338-353, 1965.
[15] W.A. Lodwick, Fuzzy Surfaces in GIS and Geographical Analysis: Theory, Analytical Methods, Algorithms and Applications. CRC Press, 2008.
[16] S. Djurcilov, K. Kim, P. Lermusiaux, and A. Pang, “Visualizing Scalar Volumetric Data with Uncertainty,” Computers and Graphics, vol. 26, no. 2, pp. 239-248, 2002.
[17] C.M. Wittenbrink, A.T. Pang, and S.K. Lodha, “Glyphs for Visualizing Uncertainty in Vector Fields,” IEEE Trans. Visualization and Computer Graphics, vol. 2, no. 3, pp. 266-279, Sept. 1996.
[18] R.P. Botchen, D. Weiskopf, and T. Ertl, “Texture-Based Visualization of Uncertainty in Flow Fields,” Proc. IEEE Visualization 2005 Conf., pp. 647-654, 2005.
[19] T. Zuk, J. Downton, D. Gray, S. Carpendale, and J. Liang, “Exploration of Uncertainty in Bidirectional Vector Fields,” Proc. SPIE and IS&T Conf. Electronic Imaging, Visualization and Data Analysis 2008, p. 68090B, 2008.
[20] M. Otto, T. Germer, H.-C. Hege, and H. Theisel, “Uncertain 2D Vector Field Topology,” Computer Graphics Forum, vol. 29, pp. 347-356, 2010.
[21] H. Li, C.-W. Fu, Y. Li, and A. Hanson, “Visualizing Large-Scale Uncertainty in Astrophysical Data,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1640-1647, Nov./Dec. 2007.
[22] J.M. Kniss, R.V. Uitert, A. Stephens, G.-S. Li, T. Tasdizen, and C. Hansen, “Statistically Quantitative Volume Visualization,” Proc. IEEE Visualization 2005 Conf., pp. 287-294, Oct. 2005.
[23] C. Lundstrom, P. Ljung, A. Persson, and A. Ynnerman, “Uncertainty Visualization in Medical Volume Rendering Using Probabilistic Animation,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1648-1655, Nov./Dec. 2007.
[24] M. Pauly, N. Mitra, and L. Guibas, “Uncertainty and Variability in Point Cloud Surface Data,” Proc. Eurographics Symp. Point-Based Graphics, pp. 77-84, 2004.
[25] S. Djurcilov and A. Pang, “Visualizing Sparse Gridded Data Sets,” IEEE Computer Graphics and Applications, vol. 20, no. 5, pp. 52-57, Sept./Oct. 2000.
[26] B. Zehner, N. Watanabe, and O. Kolditz, “Visualization of Gridded Scalar Data with Uncertainty in Geosciences,” to be published in Computers and Geoscience, 2010.
[27] A. MacEachren, A. Robinson, S. Hopper, S. Gardner, R. Murray, M. Gahegan, and E. Hetzler, “Visualizing Geospatial Information Uncertainty: What We Know and What We Need to Know,” Cartography and Geographic Information Science, vol. 32, no. 3, pp. 139-161, 2005.
[28] T. Zuk, “Visualizing Uncertainty,” PhD thesis, Univ. of Calgary, Apr. 2008.
[29] H. Jänicke, A. Wiebel, G. Scheuermann, and W. Kollmann, “Multifield Visualization Using Local Statistical Complexity,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1384-1391, Nov./Dec. 2007.
[30] C. Wang, H. Yu, and K.-L. Ma, “Importance-Driven Time-Varying Data Visualization,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1547-1554, Nov./Dec. 2008.
[31] R.J. Adler, J.E. Taylor, and K.J. Worsley, Applications of Random Fields and Geometry, Preprint, Technion-Israel Inst. of Tech nology, 2009.
[32] D.S. Moore, G.P. McCabe, and M.J. Evans, Introduction to the Practice of Statistics. W. H. Freeman & Co., 2005.
[33] R.J. Adler and J. Taylor, Random Fields and Geometry. Springer, 2007.
[34] J.-M. Azaïs and M. Wschebor, Level Sets and Extrema of Random Processes and Fields, ch. 6. John Wiley & Sons, 2009.
[35] R.J. Adler, The Geometry of Random Fields. John Wiley and Sons, 1981.
[36] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” ACM SIGGRAPH Computer Graphics, vol. 21, no. 4, pp. 163-169, 1987.
[37] P. Deuflhard and A. Hohmann, Numerical Analysis in Modern Scientific Computing: An Introduction. Springer, 2003.
[38] H. Federer, Geometric Measure Theory. Springer, 1969.
[39] C.E. Scheidegger, J.M. Schreiner, B. Duffy, H. Carr, and C.T. Silva, “Revisiting Histograms and Isosurface Statistics,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1659-1666, Nov./Dec. 2008.
[40] J.C.R. Hunt, “Vorticity and Vortex Dynamics in Complex Turbulent Flows,” Canadian Soc. for Mechanical Eng. Trans., vol. 11, no. 1, pp. 21-35, 1987.
[41] J. Sahner, T. Weinkauf, N. Teuber, and H.-C. Hege, “Vortex and Strain Skeletons in Eulerian and Lagrangian Frames,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 5, pp. 980-990, Sept./Oct. 2007.
[42] Y.-H. Ryu and J.-J. Baik, “Flow and Dispersion in an Urban Cubical Cavity,” Atmospheric Environment, vol. 43, no. 10, pp. 1721-1729, 2009.
[43] C.L. Bajaj, V. Pascucci, and D.R. Schikore, “The Contour Spectrum,” Proc. IEEE Visualization 1997 Conf., pp. 167-174, 1997.
[44] V. Pekar, R. Wiemker, and D. Hempel, “Fast Detection of Meaningful Isosurfaces for Volume Data Visualization,” Proc. IEEE Visualization 2001 Conf., pp. 223-230, 2001.
[45] R.P. Kanwal, Generalized Functions: Theory and Technique. Birkhäuser, 1998.
[46] S. Winitzki, “A Handy Approximation for the Error Function and Its Inverse,” Winitzkierf-approx.pdf, 2008.
[47] J. Krüger and R. Westermann, “Acceleration Techniques for GPU-Based Volume Rendering,” Proc. IEEE Visualization 2003 Conf., pp. 287-292, 2003.
[48] M. Firbank, A. Coulthard, R. Harrison, and E. Williams, “A Comparison of Two Methods for Measuring the Signal to Noise Ratio on MR Images,” Physics in Medicine and Biology, vol. 44, no. 12, pp. N261-N264, 1999.
[49] J. Thong, K. Sim, and J. Phang, “Single-Image Signal-to-Noise Ratio Estimation,” Scanning, vol. 23, no. 5, pp. 328-336, 2001.
[50] T.N. Palmer et al., “Development of a European Multi-Model Ensemble System for Seasonal to Inter-Annual Prediction (DEMETER),” Technical Memorandum, European Centre for Medium-Range Weather Forecasts, 2004.
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