Summary
We describe how to determine the number of cases that arise in visualization al- gorithms such as Marching Cubes by applying the deBruijn extension of Pólya counting. This technique constructs a polynomial, using the cycle index, encoding the case counts that arise when a discrete function (or “color”) is evaluated at each vertex of a polytope. The technique can serve as a valuable aid in debugging visualization algorithms that extend Marching Cubes, Separating Surfaces, Interval Volumes, Sweeping Simplices, etc., to larger dimensions and to more colors.
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References
David C. Banks and Stephen A. Linton. Counting cases in marching cubes: Toward a generic algorithm for producing substitopes. In Proceedings of Visualization 2003, pages 51–58. IEEE, 2003.
David C. Banks, Stephen A. Linton, and Paul K. Stockmeyer. Counting cases in substitope algorithms. IEEE Transactions on Visualization and Computer Graphics, 10(4):371–384, 2004.
Praveen Bhaniramka, Rephael Wenger, and Roger Crawfis. Isosurfacing in higher dimensions. In Proceedings of IEEE Visualization 2000, pages 267–273. IEEE, 2000.
Jules Bloomenthal. Polygonization of implicit surfaces. In Computer Aided Geometric Design, volume 5, pages 341–355, 1988.
N. G. deBruijn. Generalization of pólya's fundamental theorem in enumerative combinatorial analysis. Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21, pages 59–69, 1959.
N. G. deBruijn. Pólya's theory of counting. In Edwin F. Bechenbach, editor, Applied Combinatorial Mathematics, pages 144–184. Wiley, New York, 1964.
Frank Harary and Ed Palmer. The power group enumeration theorem. Journal of Combinatorial Theory, 1:157–173, 1966.
Frank Harary and Edgar Palmer. Graphical Enumeration. Academic, New York, 1973.
Hans-Christian Hege, Martin Seebass, Detlev Stalling, and Malte Zöckler. A Generalized Marching Cubes Algorithm Based on Non-Binary Classifications. Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1997. Technical Report SC 97–05.
Ming Jiang, Raghu Machiraju, and David Thompson. A novel approach to vortex core region detection. In Proceedings of the symposium on data visualization 2002, pages 217–225. Eurographics Association, 2002.
Stefan F. Kirchberg. Marching Hypercubes – ein Verfahren zur Konstruktion von Hyperflächen aus 4D-Rasterdaten. Universität Dortmund, 1993. Doplomarbeit am Lehrstuhl 7.
William E. Lorensen and Harvey E. Cline. Marching cubes: A high resolution 3D surface construction algorithm. In Proceedings of SIGGRAPH 1987, pages 163–169. ACM, 1987.
Gregory M. Nielson and Richard Franke. Computing the separating surface for segmented data. In Proceedings of IEEE Visualization 1997, pages 229–233. IEEE, 1997.
Gregory M. Nielson and Junwon Sung. Interval volume tetrahedrization. In Proceedings of IEEE Visualization 1997, pages 221–228. IEEE, 1997.
E. M. Palmer. Graphical Enumeration and the Power Group (Ph.D. dissertation). University of Michigan, 1965.
G. Pólya. Kombinatorische anzahlbestimmungen für gruppen, graphen und chemische verbindungen. Acta Mathematica, 68:145–254, 1937.
G. Pólya and R. C. Read. Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer, New York, 1987.
Jonathan C. Roberts and Steve Hill. Piecewise linear hypersurfaces using the marching cubes algorithm. In Robert Erbacher and Alex Pang, editors, Visual Data Exploration and Analysis VI, Proceedings of SPIE, pages 170–181. IS&T and SPIE, January 1999.
Han-Wei Shen and Christopher R. Johnson. Sweeping simplices: A fast iso-surface extraction algorithm for unstructured grids. In Proceedings of IEEE Visualization 1995, pages 143–151. IEEE, 1995.
Detlev Stalling, Malte Zöckler, O. Sander, and Hans-Christian Hege. Weighted Labels for 3D Image Segmentation. Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1998. Technical Report SC 98–39.
Chris Weigle and David C. Banks. Complex-valued contour meshing. In Proceedings of IEEE Visualization 1996, pages 173–180. IEEE, 1996.
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Banks, D.C., Stockmeyer, P.K. (2009). DeBruijn Counting for Visualization Algorithms. In: Möller, T., Hamann, B., Russell, R.D. (eds) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/b106657_4
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DOI: https://doi.org/10.1007/b106657_4
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