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A Voronoi-Based Map Algebra

  • Hugo Ledoux
  • Christopher Gold

Abstract

Although the map algebra framework is very popular within the GIS community for modelling fields, the fact that it is solely based on raster structures has been severely criticised. Instead of representing fields with a regular tessellation, we propose in this paper using the Voronoi diagram (VD), and argue that it has many advantages over other tessellations. We also present a variant of map algebra where all the operations are performed directly on VDs. Our solution is valid in two and three dimensions, and permits us to circumvent the gridding and resampling processes that must be performed with map algebra.

Keywords

Geographical Information System Voronoi Diagram Voronoi Cell Natural Neighbour Triangulate Irregular Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Akima H (1978) A method of bivariate interpolation and smooth surface fitting for irregularly distributed data points. ACM Transactions on Mathematical Software, 4(2):148–159.CrossRefGoogle Scholar
  2. Berry JK (1993) Cartographic Modeling: The Analytical Capabilities of GIS. In M Goodchild, B Parks, and L Steyaert, editors, Environmental Modeling with GIS, chapter 7, pages 58–74. Oxford University Press, New York.Google Scholar
  3. Boudriault G (1987) Topology in the TIGER File. In Proceedings 8th International Symposium on Computer Assisted Cartography. Baltimore, USA.Google Scholar
  4. Bruns HT and Egenhofer M (1997) Use Interfaces for Map Algebra. Journal of the Urban and Regional Information Systems Association, 9(1):44–54.Google Scholar
  5. Caldwell DR (2000) Extending Map Algebra with Flag Operators. In Proceedings 5th International Conference on GeoComputation. University of Greenwich, UK.Google Scholar
  6. Couclelis H (1992) People Manipulate Objects (but Cultivate Fields): Beyond the Raster-Vector Debate in GIS. In A Frank, I Campari, and U Formentini, editors, Theories and Methods of Spatio-Temporal Reasoning in Geographic Space, volume 639 of LNCS, pages 65–77. Springer-Verlag.Google Scholar
  7. Devillers O (2002) On Deletion in Delaunay Triangulations. International Journal of Computational Geometry and Applications, 12(3):193–205.CrossRefGoogle Scholar
  8. Eastman J, Jin W, Kyem A, and Toledano J (1995) Raster procedures for multicriteria/multi-objective decisions. Photogrammetric Engineering & Remote Sensing, 61(5):539–547.Google Scholar
  9. Edelsbrunner H and Shah NR (1996) Incremental Topological Flipping Works for Regular Triangulations. Algorithmica, 15:223–241.CrossRefGoogle Scholar
  10. Fisher PF (1997) The Pixel: A Snare and a Delusion. International Journal of Remote Sensing, 18(3):679–685.CrossRefGoogle Scholar
  11. Fortune S (1987) A Sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153–174.CrossRefGoogle Scholar
  12. Gold CM (1989) Surface Interpolation, spatial adjacency and GIS. In J Raper, editor, Three Dimensional Applications in Geographic Information Systems, chapter 3, pages 21–35. Taylor & Francis.Google Scholar
  13. Gold CM and Edwards G (1992) The Voronoi spatial model: two-and threedimensional applications in image analysis. ITC Journal, 1:11–19.Google Scholar
  14. Gold CM, Nantel J, and Yang W (1996) Outside-in: An Alternative Approach to Forest Map Digitizing. International Journal of Geographical Information Science, 10(3):291–310.CrossRefGoogle Scholar
  15. Goodchild MF (1992) Geographical Data Modeling. Computers & Geosciences, 18(4):401–408.CrossRefGoogle Scholar
  16. Guibas LJ and Stolfi J (1985) Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams. ACM Transactions on Graphics, 4:74–123.CrossRefGoogle Scholar
  17. Haklay M (2004) Map Calculus in GIS: A Proposal and Demonstration. International Journal of Geographical Information Science, 18(2):107–125.CrossRefGoogle Scholar
  18. Kemp KK (1993) Environmental Modeling with GIS: A Strategy for Dealing with Spatial Continuity. Technical Report 93-3, National Center for Geographic Information and Analysis, University of California, Santa Barbara, USA.Google Scholar
  19. Kemp KK and Vckovski A (1998) Towards an ontology of fields. In Proceedings 3rd International Conference on GeoComputation. Bristol, UK.Google Scholar
  20. Ledoux H and Gold CM (2004) An Efficient Natural Neighbour Interpolation Algorithm for Geoscientific Modelling. In P Fisher, editor, Developments in Spatial Data Handling — 11th International Symposium on Spatial Data Handling, pages 97–108. Springer.Google Scholar
  21. Ledoux H, Gold CM, and Baciu G (2005) Flipping to Robustly Delete a Vertex in a Delaunay Tetrahedralization. In Proceedings International Conference on Computational Science and its Applications — ICCSA 2005, LNCS 3480, pages 737–747. Springer-Verlag, Singapore.Google Scholar
  22. Mennis J, Viger R, and Tomlin CD (2005) Cubic Map Algebra Functions for Spatio-Temporal Analysis. Cartography and Geographic Information Science, 32(1):17–32.CrossRefGoogle Scholar
  23. Morehouse S (1985) ARC/INFO: A Geo-Relational Model for Spatial Information. In Proceedings 7th International Symposium on Computer Assisted Cartography. Washington DC, USA.Google Scholar
  24. Mücke EP, Saias I, and Zhu B (1999) Fast randomized point location without preprocessing in two-and three-dimensional Delaunay triangulations. Computational Geometry-Theory and Applications, 12:63–83.Google Scholar
  25. Peuquet DJ (1984) A Conceptual Framework and Comparison of Spatial Data Models. Cartographica, 21(4):66–113.Google Scholar
  26. Pullar D (2001) MapScript: A Map Algebra Programming Language Incorporating Neighborhood Analysis. GeoInformatica, 5(2):145–163.CrossRefGoogle Scholar
  27. Ritter G, Wilson J, and Davidson J (1990) Image Algebra: An Overview. Computer Vision, Graphics, and Image Processing, 49(3):297–331.CrossRefGoogle Scholar
  28. Sambridge M, Braun J, and McQueen H (1995) Geophysical parameterization and interpolation of irregular data using natural neighbours. Geophysical Journal International, 122:837–857.Google Scholar
  29. Sibson R (1981) A brief description of natural neighbour interpolation. In V Barnett, editor, Interpreting Multivariate Data, pages 21–36. Wiley, New York, USA.Google Scholar
  30. Stevens S (1946) On the Theory of Scales and Measurement. Science, 103:677–680.Google Scholar
  31. Suzuki A and Iri M (1986) Approximation of a tesselation of the plane by a Voronoi diagram. Journal of the Operations Research Society of Japan, 29:69–96.Google Scholar
  32. Takeyama M (1996) Geo-Algebra: A Mathematical Approach to Integrating Spatial Modeling and GIS. Ph.D. thesis, Department of Geography, University of California at Santa Barbara, USA.Google Scholar
  33. Theobald DM (2001) Topology revisited: Representing spatial relations. International Journal of Geographical Information Science, 15(8):689–705.CrossRefGoogle Scholar
  34. Tomlin CD (1983) A Map Algebra. In Proceedings of the 1983 Harvard Computer Graphics Conference, pages 127–150. Cambridge, MA, USA.Google Scholar
  35. Watson DF (1981) Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Computer Journal, 24(2):167–172.CrossRefGoogle Scholar
  36. Watson DF (1992) Contouring: A Guide to the Analysis and Display of Spatial Data. Pergamon Press, Oxford, UK.Google Scholar
  37. Worboys MF and Duckham M (2004) GIS: A Computing Perspective. CRC Press, second edition.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hugo Ledoux
    • 1
  • Christopher Gold
    • 1
  1. 1.GIS Research Centre, School of ComputingUniversity of Glamorgan PontypriddWalesUK

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