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Revisiting Histograms and Isosurface Statistics
November/December 2008 (vol. 14 no. 6)
pp. 1659-1666
Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant under isosurface-preserving transformations. We apply Federerâ™s Coarea Formula from geometric measure theory to explain these discrepancies. We describe an improved substitute for histograms based on weighting with the inverse gradient magnitude, develop a statistical model that is invariant under isosurface-preserving transformations, and argue that this provides a consistent method for algorithm evaluation across multiple datasets based on histogram equalization. We use our corrected formulation to reevaluate recent results on average isosurface complexity, and show evidence that noise is one cause of the discrepancy between the expected figure and the observed one.
[1] S. Bachthaler and D. Weiskopf, Continuous scatterplots. IEEE Transactions on Visualization and Computer Graphics (Proceedings of Visualization 2008), 2008.
[2] C. Bajaj, V. Pascucci, and D. Schikore, The contour spectrum. In Proceedings of IEEE Visualization 1997, pages 167–173, 1997.
[3] H. Carr, B. Duffy, and B. Denby, On histograms and isosurface statistics. IEEE Transactions on Visualization and Computer Graphics (Proceedings of IEEE Visualization 2006), 12 (5): 1259–1265, 2006.
[4] H. Edelsbrunner and E. P. Mucke, Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. In Symposium on Computational Geometry, pages 118–133, 1988.
[5] R. C. Gonzalez and R. E. Woods, Digital Image Processing. Prentice-Hall Inc., Englewood-Cliffs, NJ, 2nd ed. edition, 2002.
[6] Y. Ioannidis, The history of histograms. In Proceedings of Very Large Databases (VLDB) 2003, 2003.
[7] T. Itoh and K. Koyamada, Automatic isosurface propagation using an extrema graph and sorted boundary lists. IEEE Transactions on Visualization and Computer Graphics, 1 (4): 319–327, 1995.
[8] G. Kindlmann and J. W. Durkin, Semi-automatic generation of transfer functions for direct volume rendering. In IEEE Symposium on Volume Visualization, pages 79–86, 1998.
[9] J. Kniss, G. Kindlmann, and C. Hansen, Interactive volume rendering using multi-dimensional transfer functions and direct manipulation widgets. In Proceedings of IEEE Visualization 2001, pages 255–262, 2001.
[10] W. E. Lorenson and H. E. Cline, Marching Cubes: A High Resolution 3D Surface Construction Algorithm. In Proceedings of ACM SIGGRAPH 1987, pages 163–169, 1987.
[11] C. Lundstrom, P. Ljung, and A. Ynnerman, Local histograms for design of transfer functions in direct volume rendering. IEEE Transactions on Visualization and Computer Graphics, 12 (6): 1570–1579, 2006.
[12] F. Morgan, Geometric Measure Theory — A Beginner's Guide. Elsevier, 2000.
[13] P. Mullen, A. McKenzie, Y. Tong, and M. Desbrun, A variational approach to eulerian surface processing. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2007), 26 (3): 66, 2007.
[14] T. S. Newman and H. Yi, A survey of the marching cubes algorithm. Computers And Graphics, pages 854–879, 2006.
[15] V. Pekar, rafael Wiemker, and D. Hempel, Fast detection of meaningful isosurfaces for volume data visualization. In Proceedings of IEEE Visualization 2001, pages 223–230, 2001.
[16] S. Tenginakai, J. Lee, and R. Machiraju, Salient iso-surface detection with model-independent statistical signatures. In Proceedings of IEEE Visualization 2001, pages 231–238, 2001.
[17] G. Wyvill, C. McPheeters, and B. Wyvill, Data Structure for Soft Objects. Visual Computer, 2: 227–234, 1986.
Index Terms:
Index Terms—Isosurfaces, Histograms, Coarea Formula
Citation:
Carlos E. Scheidegger, John M. Schreiner, Brian Duffy, Hamish Carr, Cláudio T. Silva, "Revisiting Histograms and Isosurface Statistics," IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 6, pp. 1659-1666, Nov./Dec. 2008, doi:10.1109/TVCG.2008.160
