The Community for Technology Leaders
RSS Icon
Issue No.02 - Feb. (2013 vol.19)
pp: 263-277
B. Duffy , Oxford Centre for Collaborative Appl. Math. (OCCAM), Univ. of Oxford, Oxford, UK
H. Carr , Visualization & Virtual Reality Group, Univ. of Leeds, Leeds, UK
T. Moller , Graphics, Usability, & Visualization (GrUVi) Lab., Simon Fraser Univ., Burnaby, BC, Canada
Many data sets are sampled on regular lattices in two, three or more dimensions, and recent work has shown that statistical properties of these data sets must take into account the continuity of the underlying physical phenomena. However, the effects of quantization on the statistics have not yet been accounted for. This paper therefore reconciles the previous papers to the underlying mathematical theory, develops a mathematical model of quantized statistics of continuous functions, and proves convergence of geometric approximations to continuous statistics for regular sampling lattices. In addition, the computational cost of various approaches is considered, and recommendations made about when to use each type of statistic.
statistics, data visualisation, geometry, geometric approximations, isosurface statistics, histograms, data sets, regular lattices, Histograms, Size measurement, Jacobian matrices, Lattices, Quantization, Extraterrestrial measurements, Approximation methods, geometric statistics, Histograms, frequency distribution, integration
B. Duffy, H. Carr, T. Moller, "Integrating Isosurface Statistics and Histograms", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 2, pp. 263-277, Feb. 2013, doi:10.1109/TVCG.2012.118
[1] S. Tenginakai, J. Lee, and R. Machiraju, "Salient Iso-surface Detection with Model-Independent Statistical Signatures," Proc. IEEE Visualization, pp. 231-238, 2001.
[2] S. Tenginakai and R. Machiraju, "Statistical Computation of Salient Iso-Values," Proc. Symp. Data Visualisation (VisSym), pp. 19-24, 2002.
[3] H. Pfister, B. Lorensen, C. Bajaj, G. Kindlmann, W. Schroeder, L.S. Avila, K. Martin, R. Machiraju, and J. Lee, "The Transfer Function Bake-Off," IEEE Computer Graphics and Applications, vol. 21, no. 3, pp. 16-22, May 2001.
[4] G. Kindlmann and J.W. Durkin, "Semi-Automatic Generation of Transfer Functions for Direct Volume Rendering," Proc. IEEE Symp. Vol. Visualization, pp. 79-86, 1998.
[5] G. Kindlmann, "Semi-Automatic Generation of Transfer Functions for Direct Volume Rendering," master's thesis, Cornell Univ., 1999.
[6] G. Kindlmann, R. Whitaker, T. Tasdizen, and T. Möller, "Curvature-Based Transfer Functions for Direct Volume Rendering: Methods and Applications," Proc. IEEE Visualization '03, pp. 513-520, Oct. 2003.
[7] J. Kniss, G. Kindlmann, and C.D. Hansen, "Interactive Volume Rendering Using Multi-Dimensional Transfer Functions and Direct Manipulation Widgets," Proc. IEEE Visualization, pp. 255-262, 2001.
[8] J. Kniss, G. Kindlmann, and C.D. Hansen, "Multidimensional Transfer Functions for Interactive Volume Rendering," IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 3, pp. 270-285, July 2002.
[9] C. Lundström, P. Ljung, and A. Ynnerman, "Local Histograms for Design of Transfer Functions in Direct Volume Rendering," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 6, pp. 1570-1579, Nov./Dec. 2006.
[10] C. Bajaj, V. Pascucci, and D. Schikore, "The Contour Spectrum," Proc. IEEE Visualization, pp. 167-173, 1997.
[11] C.L. Bajaj, V. Pascucci, and D. Schikore, "Accelerated Isocontouring of Scalar Fields," Data Visualization Techniques. pp. 31-47, Wiley, 1999.
[12] H.-W. Shen, C.D. Hansen, Y. Livnat, and C.R. Johnson, "Isosurfacing in Span Space with Utmost Efficiency (ISSUE)," Proc. Visualization '96, pp. 287-294, 1996.
[13] I. Fujishiro and Y. Takeshima, "Coherence-Sensitive Solid Fitting," Computers and Graphics, vol. 26, pp. 417-427, 2002.
[14] H. Carr, B. Duffy, and B. Denby, "On Histograms and Isosurface Statistics," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1259-1266, Sept./Oct. 2006.
[15] 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.
[16] S. Bachthaler and D. Weiskopf, "Continuous Scatterplots," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1428-1435, Nov./Dec. 2008.
[17] H. Federer, Geometric Measure Theory. Springer-Verlag, 1965.
[18] F. Morgan, Geometric Measure Theory: A Beginner's Guide. Academic Press, 1988.
[19] B. Guo, "Interval Set: A Volume Rendering Technique Generalizing Isosurface Extraction," Proc. IEEE Visualization, pp. 3-10, 1995.
[20] I. Fujishiro, Y. Maeda, and H. Sato, "Interval Volume: A Solid Fitting Technique for Volumetric Data Display and Analysis," Proc. IEEE Visualization, pp. 151-158, 1995.
[21] H. Carr, T. Theußl, and T. Möller, "Isosurfaces on Optimal Regular Samples," Proc. Symp. Data Visualisation (VISSYM '03), pp. 39-48, 2003.
[22] T. Itoh and K. Koyamada, "Isosurface Generation by Using Extrema Graphs," Proc. IEEE Conf. Visualization, pp. 77-83, 1994.
[23] M. Khoury and R. Wenger, "On the Fractal Dimension of Isosurfaces," IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 6, pp. 1198-1205, Nov./Dec. 2010.
[24] A. van Gelder and J. Wilhelms, "Topological Considerations in Isosurface Generation," ACM Trans. Graphics, vol. 13, pp. 337-375, 1994.
[25] W.E. Lorensen and H.E. Cline, "Marching Cubes: A High Resolution 3D Surface Construction Algorithm," Proc. SIGGRAPH Conf. Computer Graphics and Interactive Techniques, pp. 163-169, 1987.
[26] T.S. Newman and H. Yi, "A Survey of the Marching Cubes Algorithm," Computers and Graphics, vol. 30, pp. 854-879, 2006.
[27] S.R. Marschner and R.J. Lobb, "An Evaluation of Reconstruction Filters for Volume Rendering," Proc. IEEE Conf. Visualization, pp. 100-107, 1994.
506 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool