Abstract
Determining the convex hull, its lower convex hull, and Voronoi diagram of a point set is a basic operation for many applications of pattern recognition, image processing, and data mining. To date, the lower convex hull of a finite point set is determined from the entire convex hull of the set. There arises a question “How can we determine the lower convex hull of a finite point set without relying on the entire convex hull?” In this paper, we show that the lower convex hull is wrapped by lower facets starting from an extreme edge of the lower convex hull. Then a direct method for determining the lower convex hull of a finite point set in 3D without the entire convex hull is presented. The actual running times on the set of random points (in the uniform distribution) show that our corresponding algorithm runs significantly faster than the incremental convex hull algorithm and some versions of the gift-wrapping algorithm.
The original version of this chapter was revised: Acknowledgement section has been updated. The erratum to this chapter is available at 10.1007/978-3-319-17996-4_37
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-17996-4_37
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References
Akl, S.G., Toussaint, G.: Efficient convex hull algorithms for pattern recognition applications. In: 4th Int’l Joint Conf. on Pattern Recognition, Kyoto, Japan, pp. 483–487 (1978)
An, P.T., Trang, L.H.: An efficient convex hull algorithm for finite point sets in 3D based on the Method of Orienting Curves. Optimization 62(7), 975–988 (2013)
Day, A.M.: An implementation of an algorithm to find the convex hull of a set of three-dimensional points. ACM Transactions on Graphics 9(1), 105–132 (1990)
Luo, D.: Pattern Recognition and Image Processing. Woodhead Publishing (1998)
McMullen, P., Shephard, G.C.: Convex Polytopes and the Upper Bound Conjecture. Cambridge University Press, Cambridge (1971)
Meethongjan, K., Dzulkifli, M., Rehman, A., Saba, T.: Face recognition based on fusion of Voronoi diagram automatic facial and wavelet moment invariants. International Journal of Video & Image Processing and Network Security 10(4), 1–8 (2010)
Na, H.S., Lee, C.N., Cheong, O.: Voronoi diagrams on the sphere. Computational Geometry 23(2), 183–194 (2002)
Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley & Sons Ltd (2000)
O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press (1998)
Preparata, F.P., Shamos, M.I.: Computational Geometry - An Introduction, 2nd edn. Second Edition. Springer, New York (1988)
Sugihara, K.: Robust gift wrapping for the three-dimensional convex hull. Journal of Computer and System Sciences 49, 391–407 (1994)
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Phan, T.A., Dinh, T.G. (2015). A Direct Method for Determining the Lower Convex Hull of a Finite Point Set in 3D. In: Le Thi, H., Nguyen, N., Do, T. (eds) Advanced Computational Methods for Knowledge Engineering. Advances in Intelligent Systems and Computing, vol 358. Springer, Cham. https://doi.org/10.1007/978-3-319-17996-4_2
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DOI: https://doi.org/10.1007/978-3-319-17996-4_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17995-7
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