SpringerLink
Log in

Geoinformatics and Topological Data Analysis

  • Chapter
  • First Online:
Geo-Topology

Part of the book series: GeoJournal Library ((GEJL,volume 133))

  • 114 Accesses

Abstract

This chapter presents concepts and methods of topology that are applicable in geoinformatics and spatial data analysis. After introducing the topology of boundaries and topological models for GIS and simplicial complexes, methods of TDA (Topological Data Analysis) are presented that are suitable for handling spatial data using Hasse diagrams and Vietoris-Rips complexes.

As for the mathematical magnitudes, length is generated first, then width and last is depth

“τὰ μαθηματικὰ μεγέθη … πρῶτον μὲν γὰρ ἐπὶ μῆκος γίγνεται,

εἶτα ἐπὶ πλάτος, τελευταῖον δ᾿ εἰς βάθος”

(Aristotle, 384–322 b.C., “Metaphysics” 12, 1077α)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Asao, Y., Nagase, J., Sakamoto, R., & Takagi, S. (2021). Image recognition via Vietoris-Rips complex. ar**v preprint ar**v:2109.02231.

    Google Scholar 

  • Atienza, N., Gonzalez-Diaz, R., & Rucco, M. (2019). Persistent entropy for separating topological features from noise in vietoris-rips complexes. Journal of Intelligent Information Systems, 52, 637–655.

    Google Scholar 

  • Beguin, H., & Thisse, J. F. (1979). An axiomatic approach to geographic space. Geographical Analysis, 11(4), 325–341.

    Google Scholar 

  • Billen, R., & Kurata, Y. (2008). Refining topological relations between regions considering their shapes. In M. Raunbal, J. Miller, A. U. Frank, et al. (Eds.), Geographic information science (Lecture notes in computer science) (pp. 18–32). Springer.

    Google Scholar 

  • Billen, R., & Zlatanova, S. (2003). 3D spatial relationship model: A useful concept for 3D cadastre? Computers, Environment and Urban Systems, 27, 411–425.

    Google Scholar 

  • Breunig, M., & Zlatanova, S. (2011). 3D geo-database research: Retrospective and future directions. Computers and Geosciences, 37(7), 791–803.

    Google Scholar 

  • Brisson, E. (1989). Representing geometric structures in d dimensions: Topology and order. In Proceedings of the fifth annual symposium on computational geometry SCG’89 (pp. 218–227). ACM.

    Google Scholar 

  • Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1), 77–102.

    Google Scholar 

  • Cardelli, L. (1984). A semantics of multiple inheritance. In International symposium on semantics of data types (pp. 51–67). Springer.

    Google Scholar 

  • Cardoze, D., Miller, G., & Phillips, T. (2006, July 26–28). Representing topological structures using cell-chains. In Proceedings of the geometric modeling and processing—GMP 2006. Kim, M.-S., Shimada, K., Eds). (pp. 248–266). : Springer.

    Google Scholar 

  • Carlson, E. (1987). Three dimensional conceptual modeling of subsurface structures. Technical Papers of ASPRS/ACSM Annual Convention, 4, 188–200.

    Google Scholar 

  • Carlsson, G., & Vejdemo-Johansson, M. (2021). Topological data analysis with applications. Cambridge University Press.

    Google Scholar 

  • Chazal, F., & Michel, B. (2021). An introduction to topological data analysis: Fundamental and practical aspects for data scientists. Frontiers in Artificial Intelligence, 4, 108.

    Google Scholar 

  • Chen, J., Li, C., Li, Z., & Gold, C. (2001). A Voronoi-based 9-intersection model for spatial relations. International Journal of Geographical Information Science, 15(3), 201–220.

    Google Scholar 

  • Clementini, E., & di Felice, P. (1997). Approximate topological relations. International Journal of Approximate Reasoning, 16, 173–204.

    Google Scholar 

  • Clementini, E., di Felice, P., & van Oosterom, P. J. M. (1993). A small set of formal topological relations suitable for end-user interaction. In Proceedings of the 3th international symposium on large spatial databases (pp. 277–295). Springer.

    Google Scholar 

  • Clementini, E., Sharma, J., & Egenhofer, M. J. (1994). Modelling topological spatial relations: Strategies for query processing. Computer Graphics, 18(6), 815–822.

    Google Scholar 

  • de Almeida, J. P., Morley, J. G., & Dowman, I. J. (2007). Graph theory in higher order topological analysis of urban scenes. Computers, Environment and Urban Systems, 31, 426–440.

    Google Scholar 

  • Deng, M., Cheng, T., Chen, X., & Li, Z. (2007). Multi-level topological relations between spatial regions based upon topological invariants. GeoInformatica, 11, 239–267.

    Google Scholar 

  • Dube, M.P., Barrett, J.V., & Egenhofer, M.J. (2015, October 12–16). From metric to topology: Determining relations in discrete space. In Spatial information theory: 12th international conference, COSIT 2015, Proceedings 12 (pp. 151–171). : Springer.

    Google Scholar 

  • Egenhofer, M. J., & Dube, M. P. (2009). Topological relations from metric refinements. In Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems (pp. 158–167).

    Google Scholar 

  • Egenhofer, M. J., & Franzosa, R. D. (1991). Point-set topological spatial relations. International Journal of Geographical Information Systems, 5, 161–174.

    Google Scholar 

  • Egenhofer, M. J., & Shariff, A. R. B. (1998). Metric details for natural-language spatial relations. ACM Transactions on Information Systems (TOIS), 16(4), 295–321.

    Google Scholar 

  • Egenhofer, M. J., Frank, A. U., & Jackson, J. P. (1989). A topological data model for spatial databases. Springer Lecture Notes in Computer Science, 7649, 271–286.

    Google Scholar 

  • Egenhofer, M. J., Clementini, E., & di Felice, P. (1994). Topological relations between regions with holes. International Journal of Geographical Information Systems, 8(2), 129–144.

    Google Scholar 

  • Ellul, C., & Haklay, M. (2006). Requirements for topology in 3D GIS. Transactions in GIS, 10, 157–175.

    Google Scholar 

  • Ellul, C., & Haklay, M. (2007). The research agenda for topological and spatial databases. Computers, Environment and Urban Systems, 31, 373–378.

    Google Scholar 

  • Epstein, C., Carlsson, G., & Edelsbrunner, H. (2011). Topological data analysis. Inverse Problems, 27(12), 120201.

    Google Scholar 

  • Escobar-Molano, M. L., Barret, D. A., Carson, E., et al. (2007). A representation for databases of 3D objects. Computers, Environment and Urban Systems, 31, 409–425.

    Google Scholar 

  • Godoy, F., & Rodríguez, A. (2002). A quantitative description of spatial configurations. In Advances in spatial data handling: 10th international symposium on spatial data handling (pp. 299–311). Springer.

    Google Scholar 

  • Kurata, Y. (2008, September 23–26). The 9+−intersection: A universal framework for modeling topological relations. In Geographic information science: 5th international conference, GIScience 2008, Park City, UT, USA. Proceedings 5 (pp. 181–198). : Springer.

    Google Scholar 

  • Kyzirakos, K., Karpathiotakis, M., & Koubarakis, M. (2012). Strabon: A semantic geospatial DBMS. In P. Cudré-Mauroux, J. Heflin, E. Sirin, T. Tudorache, J. Euzenat, M. Hauswirth, J. X. Parreira, J. Hendler, G. Schreiber, A. Bernstein, & E. Blomqvist (Eds.), ISWC 2012, Boston, USA, Springer lecture notes in computer science, 7649 (pp. 295–311).

    Google Scholar 

  • Lienhardt, P. (1991). Topological models for boundary representation: A comparison with n-dimensional generalized maps. Computer-Aided Design, 23, 59–82.

    Google Scholar 

  • Liu, K., & Shi, W. (2007). Extended model of topological relations between spatial objects in geographic information systems. International Journal of Applied Earth Observation and Geoinformation, 9(3), 264–275.

    Google Scholar 

  • Mazroob Semnani, N., Kuper, P. V., Breunig, M., & Al-Doori, M. (2018). Towards an intelligent platform for big 3d geospatial data management. ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 4, 133–140.

    Google Scholar 

  • Molenaar, M. (1990). A formal data structure for three dimensional GIS in geographic information systems. In Proceedings of 4th international symposium on spatial data handling, 2 (pp. 830–843).

    Google Scholar 

  • Munch, E. (2017). A user’s guide to topological data analysis. Journal of Learning Analytics, 4(2), 47–61.

    Google Scholar 

  • Nedas, K. A., Egenhofer, M. J., & Wilmsen, D. (2007). Metric details of topological line–line relations. International Journal of Geographical Information Science, 21(1), 21–48.

    Google Scholar 

  • O’Rourke, J. (1985). Finding minimal enclosing boxes. International Journal of Computer & Information Sciences, 14(3), 183–199.

    Google Scholar 

  • OGC (2010). GeoSPARQL – A geographic query language for RDF data.

    Google Scholar 

  • Papadimitriou, F. (2012). Modelling landscape complexity for land use management in Rio de Janeiro, Brazil. Land Use Policy, 29(4), 855–861.

    Google Scholar 

  • Pigot, S. (1991). Topological models for 3D spatial information systems. In Proceedings of the AutoCarto conference, Baltimore, MD, vol.6 (pp. 368–392). ASPRS (American Society for Photogrammetry and Remote Sensing).

    Google Scholar 

  • Pilouk, M., Tempfli, K., & Molenaar, M. (1994). A tetrahedron-based 3D vector data model for geo-information. In M. Molenaar & S. de Hoop (Eds.), AGDM’94 spatial data modelling and query languages for 2D and 3D applications (pp. 129–140). Netherlands Geodetic Comm.

    Google Scholar 

  • Scholl, M., & Voisard, A. (1989). Thematic map modelling: Design and implementation of large spatial databases. Springer Lecture Notes in Computer Science, 409, 167–190.

    Google Scholar 

  • Sen, A. (1976). On a class of map transformations. Geographical Analysis, 8, 23–37.

    Google Scholar 

  • Shariff, A. R. B., Egenhofer, M. J., & Mark, D. M. (1998). Natural-language spatial relations between linear and areal objects: The topology and metric of English-language terms. International Journal of Geographical Information Science, 12(3), 215–245.

    Google Scholar 

  • Shen, J., Zhou, T., & Chen, M. (2017). A 27-intersection model for representing detailed topological relations between spatial objects in two-dimensional space. ISPRS International Journal of Geo-Information, 6(2), 37.

    Google Scholar 

  • Sridhar, M., Cohn, A. G., & Hogg, D. C. (2011). From video to RCC8: exploiting a distance based semantics to stabilise the interpretation of mereotopological relations. In Spatial Information Theory: 10th International Conference, COSIT 2011, Belfast, ME, USA, September 12–16, 2011. Proceedings 10 (pp. 110–125). Springer.

    Google Scholar 

  • Taylor, D., Klimm, F., Harrington, H. A., Kramár, M., Mischaikow, K., Porter, M. A., & Mucha, P. J. (2015). Topological data analysis of contagion maps for examining spreading processes on networks. Nature Communications, 6(1), 7723.

    Google Scholar 

  • Tierny, J. (2017). Topological data analysis for scientific visualization (Vol. 3). Springer.

    Google Scholar 

  • Wasserman, L. (2018). Topological data analysis. Annual Review of Statistics and Its Application, 5, 501–532.

    Google Scholar 

  • Yu, Z., Luo, W., Yuan, L., Hu, Y., Zhu, A. X., & Lü, G. (2016). Geometric algebra model for geometry-oriented topological relation computation. Transactions in GIS, 20(2), 259–279.

    Google Scholar 

  • Yuan, L., Yu, Z., Luo, W., Yi, L., & Lü, G. (2014). Multidimensional-unified topological relations computation: A hierarchical geometric algebra-based approach. International Journal of Geographical Information Science, 28(12), 2435–2455.

    Google Scholar 

  • Zomorodian, A. (2010). Fast construction of the Vietoris-rips complex. Computers & Graphics, 34(3), 263–271.

    Google Scholar 

  • Zomorodian, A. (2012). Topological data analysis. Advances in Applied and Computational topology, 70, 1–39.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch-Naturwissenschaftliche Fakultät, Forschungsbereich Geographie, Universität Tübingen, Tübingen, Germany

    Fivos Papadimitriou

Authors
  1. Fivos Papadimitriou
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Papadimitriou, F. (2023). Geoinformatics and Topological Data Analysis. In: Geo-Topology. GeoJournal Library, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-031-48185-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-48185-7_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-48184-0

  • Online ISBN: 978-3-031-48185-7

  • eBook Packages: Social SciencesSocial Sciences (R0)

Publish with us

Policies and ethics

Search

Navigation