Fuzzy Three-Dimensional Voronoi Diagram and its Application to Geographical Data Analysis
Mian Dai, Fangyan Dong, and Kaoru Hirota
Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan
A concept of fuzzy three-dimensional Voronoi Diagram is presented for spatial relations analysis of real world three-dimensional geographical data, where it is an extension of well known two-dimensional Voronoi Diagram to three-dimensional representation with uncertain spatial relation information in terms of fuzzy set. It makes possible to analyze quantitatively complex boundaries of geographically intricate areas, to give human friendly fuzzy explanation of determining three-dimensional directions, and to express uncertain spatial relations by precise unified fuzzy description. It is applied to decide spatial direction relations of artificial geographicalmountain data, which includes 8 spatial directions with at most 60 relative direction relations, and it leads to detect threedimensional directions whereas the expression of traditional 4 directions and 12 relative directions indicate two-dimensional directions only. The proposed concept aims to discriminate neighbors’ class relations and spatial-temporal changes of specially appointed objects, and also aims to be a tool to achieve the intellective extraction and analysis of geographical data of a mountainous area located in northeast China.
-  J. Wang and Y. F. Chen, “Cartography Theoretic,” Beijing: The people’s Liberation Army Press, pp.11-12, 2000.
-  G. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. für die Reine und Angewandte Mathematik, Vol.133, pp. 97-178, 1907.
-  J. O’Rourke, “Computational geometry in C (2nd Edition) [M],” London: Cambridge Univ. Press, 1998, pp. 181-226, 1998.
-  G. L. Dirichlet, “Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen,” J. für die Reine und Angewandte Mathematik, Vol.40, pp. 209-227, 1850.
-  F. Aurenhammer, “Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure,” ACM Computing Surveys, Vol.23, No.3, pp. 345-405, 1991.
-  Z. Nilforoushan, A. Mohades, M. M. Rezaii, and A. Laleh, “3D hyperbolic Voronoi diagrams,” Computer-Aided Design Vol.42, pp. 759-767, 2010.
-  L. Wu and W. Shi, “Geographic Information Systems Theory and algorithm,” Beijing: Science Press, pp. 127-128, 2003.
-  M. Jooyandeh, A. Mohades, andM.Mirzakhah, “Uncertain Voronoi diagram,” Information Processing Letters Vol.109, pp. 709-712, 2010.
-  L. A. Zadeh, “Fuzzy sets,” Inform Control, Vol.8, pp. 338-353, 1965.
-  M. Jooyandeh, “A Mohades Fuzzy Voronoi Diagram,” 13th Int. CSI Computer Conf., Advances in Computer Science and Engineering, Vol.6, pp. 82-89, 2009.
-  D. Rutovitz, G. C. Cheng, D. K. Pollock, and A. Rosenfeld, “Pictorial Pattern Recognition,” Thompson, Washington, pp. 105-133, 1968.
-  D. Randell, Z. Cuiz, and A. Cohn, “A spatial logic based on regions and connection,” Proc. of the Knoweledge Representation and Reasoning. San Mateo: MorganKarfmann, pp. 165-176, 1992.
-  J. O. Roy and J. G. Stell, “Spatial Relations between indeterminate regions,” Int. J. of Approximate reasoning, Vol.27, pp. 205-234, 2001.
-  Autodesk 3ds max 9.0 provide powerful, integrated 3D modeling, animation, redering, and compositing tools that enable designers to more quickly ramp up for production.