Advertisement

The 9 + -Intersection: A Universal Framework for Modeling Topological Relations

  • Yohei Kurata
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5266)

Abstract

The 9 + -intersection is an extension of the 9-intersection, which distinguishes the topological relations between various spatial objects by the pattern of a nested matrix. This paper develops a small set of constraints on this matrix, which is applicable to arbitrary pairs of spatial objects in various spaces. Based on this set of universal constraints, the sets of matrix patterns, each representing a candidate for topological relations, are derived for every possible pair of basic objects (points, directed/non-directed line segments, regions, and bodies) embedded in R 1, R 2, R 3, S 1, and S 2. The derived sets of candidates are consistent with the sets of topological relations ever identified, as well as yield the identification of some missing sets of topological relations. Finally, the topological relations between a region and a region with a hole in R 2 and S 2 are identified to demonstrate the applicability of our approach to deriving topological relations between more complicated objects.

Keywords

Intersection Matrix Basic Object Spatial Object Topological Relation Matrix Pattern 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Egenhofer, M., Franzosa, R.: Point-Set Topological Spatial Relations. International Journal of Geographical Information Systems 5, 161–174 (1991)CrossRefGoogle Scholar
  2. 2.
    Egenhofer, M., Herring, J.: Categorizing Binary Topological Relationships between Regions, Lines and Points in Geographic Databases. In: Egenhofer, M., Herring, J., Smith, T., Park, K. (eds.): NCGIA Technical Reports 91-7. National Center for Geographic Information and Analysis, Santa Barbara, CA, USA (1991)Google Scholar
  3. 3.
    Randell, D., Cui, Z., Cohn, A.: A Spatial Logic Based on Regions and Connection. In: Nebel, B., Rich, C., Swarout, W. (eds.) 3rd International Conference on Knowledge Representation and Reasoning, pp. 165–176. Morgan Kaufmann, San Francisco (1992)Google Scholar
  4. 4.
    Schneider, M., Behr, T.: Topological Relationships between Complex Spatial Objects. ACM Transactions on Database Systems 31, 39–81 (2006)CrossRefGoogle Scholar
  5. 5.
    Kurata, Y., Egenhofer, M.: The Head-Body-Tail Intersection for Spatial Relations between Directed Line Segments. In: Raubal, M., Miller, H.J., Frank, A.U., Goodchild, M.F. (eds.) GIScience 2006. LNCS, vol. 4197, pp. 269–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Kurata, Y., Egenhofer, M.: The 9 + -Intersection for Topological Relations between a Directed Line Segment and a Region. In: Gottfried, B. (ed.) 1st International Symposium for Behavioral Monitoring and Interpretation, pp. 62–76 (2007)Google Scholar
  7. 7.
    Egenhofer, M., Vasardani, M.: Spatial Reasoning with a Hole. In: Winter, S., Duckham, M., Kulik, L., Kuipers, B. (eds.) COSIT 2007. LNCS, vol. 4736, pp. 303–320. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Allen, J.: An Interval-Based Representation of Temporal Knowledge. In: Hayes, P. (ed.): 7th International Joint Conference on Artificial Intelligence, pp. 221-226 (1981) Google Scholar
  9. 9.
    Zlatanova, S.: On 3D Topological Relationships. In: 11th International Workshop on Database and Expert Systems Applications, pp. 913–924. IEEE Computer Society, Los Alamitos (2000)CrossRefGoogle Scholar
  10. 10.
    Hornsby, K., Egenhofer, M., Hayes, P.: Modeling Cyclic Change. In: Chen, P., Embley, D., Kouloumdjian, J., Liddle, S., Roddick, J. (eds.) TABLEAUX 1997. LNCS, vol. 1227, pp. 98–109. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Egenhofer, M.: Spherical Topological Relations. Journal on Data Semantics III, 25–49 (2005)Google Scholar
  12. 12.
    Egenhofer, M., Sharma, J.: Topological Relations between Regions in R2 and Z2. In: Abel, D.J., Ooi, B.-C. (eds.) SSD 1993. LNCS, vol. 692, pp. 316–336. Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Mark, D.: Calibrating the Meanings of Spatial Predicates from Natural Language: Line-Region Relations. In: Waugh, T., Healey, R. (eds.) 6th International Symposium on Spatial Data Handling, pp. 538–553. Taylor Francis (1994)Google Scholar
  14. 14.
    Nedas, K., Egenhofer, M., Wilmsen, D.: Metric Details of Topological Line-Line Relations. International Journal of Geographical Information Science 21, 21–48 (2007)CrossRefGoogle Scholar
  15. 15.
    Clementini, E., Di Felice, P.: A Model for Representing Topological Relationships between Complex Geometric Features in Spatial Databases. Information Science 90, 121–136 (1996)zbMATHCrossRefGoogle Scholar
  16. 16.
    Alexandroff, P.: Elementary Concepts of Topology. Dover Publications, Mineola (1961)zbMATHGoogle Scholar
  17. 17.
    Egenhofer, M., Franzosa, R.: On the Equivalence of Topological Relations. International Journal of Geographical Information Systems 9, 133–152 (1995)CrossRefGoogle Scholar
  18. 18.
    Billen, R., Zlatanova, S., Mathonet, P., Boniver, F.: The Dimensional Model: A Framework to Distinguish Spatial Relationships. In: Richardson, D., van Oosterom, P. (eds.) 10th International Symposium on Spatial Data Handling, pp. 285–298. Springer, Heidelberg (2002)Google Scholar
  19. 19.
    Mark, D., Egenhofer, M.: Modeling Spatial Relations between Lines and Regions: Combining Formal Mathematical Models and Human Subjects Testing. Cartography and Geographical Information Systems 21, 195–212 (1994)Google Scholar
  20. 20.
    Renz, J.: A Spatial Odyssey of the Interval Algebra: 1. Directed Intervals. In: Nebel, B. (ed.) 7th International Joint Conference on Artificial Intelligence, pp. 51–56. Morgan Kaufmann, San Francisco (2001)Google Scholar
  21. 21.
    Billen, R.: Nouvelle Perception De La Spatialité Des Objets Et De Leurs Relations. Développment D’une Modélisation Tridimensionnelle De L’information Spatiale. Department of Geography, Ph.D. Thesis. University of Liège, Liège, Belgium (2002)Google Scholar
  22. 22.
    Pullar, D., Egenhofer, M.: Towards Formal Definitions of Topological Relations among Spatial Objects. In: Marble, D. (ed.) 3rd International Symposium on Spatial Data Handling, pp. 225–241 (1988)Google Scholar
  23. 23.
    Egenhofer, M., Clementini, E., Di Felice, P.: Topological Relations between Regions with Holes. International Journal of Geographical Information Science 8, 129–142 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yohei Kurata
    • 1
  1. 1.SFB/TR8 Spatial CognitionUniversität BremenBremenGermany

Personalised recommendations