Skip to main content
SpringerLink
Log in

Boundary of a fuzzy set and its application in GIS: a review

  • Published:
Artificial Intelligence Review Aims and scope Submit manuscript

Abstract

The boundary of a fuzzy set is an important topic that is defined using the concepts of fuzzy topology. The inherent vagueness in the fuzzy sets allows researchers to construct various definitions of the fuzzy boundary. The fuzzy boundary is not only crucial for the theoretical purpose, it is also used to express uncertain geographical objects in geographical information systems (GIS). Specifically, the fuzzy boundary is used for constructing algebraic models such as fuzzy 9 and 16-intersection matrices, which evaluate the topological relation between two uncertain geographical objects. Considering that, this paper analyzes the concept of the fuzzy boundary of a set in the fuzzy topological space. Next, several algebraic models where the fuzzy boundary has been used for analyzing the spatial objects are discussed.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Bai L, Yan L, Ma ZM (2012) Modeling topological relations between fuzzy spatiotemporal regions over time. In: 2012 IEEE international conference on fuzzy systems. IEEE, pp 1–8

  • Bai L, Zhu L, Zhao Y (2018) Determining topological relations of fuzzy spatiotemporal regions over time in xml. IEEE Access 6:65620–65634

    Google Scholar 

  • Beaubouef T, Ladner R, Petry F (2004) Rough set spatial data modeling for data mining. Int J Intell Syst 19(7):567–584

    MATH  Google Scholar 

  • Beaubouef T, Petry FE, Ladner R (2007) Spatial data methods and vague regions: a rough set approach. Appl Soft Comput 7(1):425–440

    Google Scholar 

  • Bejaoui L, Pinet F, Bédard Y, Schneider M (2009) Qualified topological relations between spatial objects with possible vague shape. Int J Geogr Inf Sci 23(7):877–921

    Google Scholar 

  • Benz UC, Hofmann P, Willhauck G, Lingenfelder I, Heynen M (2004) Multi-resolution, object-oriented fuzzy analysis of remote sensing data for GIS-ready information. ISPRS J Photogramm Remote Sens 58(3–4):239–258

    Google Scholar 

  • Bjørke JT (2004) Topological relations between fuzzy regions: derivation of verbal terms. Fuzzy Sets Syst 141(3):449–467

    MathSciNet  MATH  Google Scholar 

  • Blakemore M (1984) Generalisation and error in spatial data bases. Cartographica 21(2–3):131–139

    Google Scholar 

  • Blaschke T (2010) Object based image analysis for remote sensing. ISPRS J Photogramm Remote Sens 65(1):2–16

    Google Scholar 

  • Burrough PA (1996) Opportunities and limitations of GIS-based modeling of solute transport at the regional scale. Appl GIS Model Non-point Source Pollut Vadose Zone 48:19–38

    Google Scholar 

  • Burrough PA, Frank AU (1995) Concepts and paradigms in spatial information: are current geographical information systems truly generic? Int J Geogr Inf Syst 9(2):101–116

    Google Scholar 

  • Caniani D, Lioi DS, Mancini IM, Masi S (2011) Application of fuzzy logic and sensitivity analysis for soil contamination hazard classification. Waste Manag 31(3):583–594

    Google Scholar 

  • Carniel AC, Schneider M (2016) A conceptual model of fuzzy topological relationships for fuzzy regions. In: 2016 IEEE international conference on fuzzy systems (FUZZ-IEEE). IEEE, pp 2271–2278

  • Chamuah H, Chetia B (2015) Application of fuzzy topological relation in flood prediction. Int J Comput Appl 122(7):8–13

    Google Scholar 

  • Chang C-L (1968) Fuzzy topological spaces. J Math Anal Appl 24(1):182–190

    MathSciNet  MATH  Google Scholar 

  • Chang N-B, Parvathinathan G, Breeden JB (2008) Combining GIS with fuzzy multicriteria decision-making for landfill siting in a fast-growing urban region. J Environ Manag 87(1):139–153

    Google Scholar 

  • Chen J, Li Z, Li C, Gold C (1998) Describing topological relations with Voronoi-based 9-intersection model. Int Arch Photogramm Remote Sens 32:99–104

    Google Scholar 

  • Chen J, Li C, Li Z, Gold C (2001) A Voronoi-based 9-intersection model for spatial relations. Int J Geogr Inf Syst 15(3):201–220

    Google Scholar 

  • Cheng H, Wang R, Li P, Xu H (2019) Representing and reasoning fuzzy spatio-temporal knowledge with description logics: a survey. Intell Data Anal 23(S1):113–132

    Google Scholar 

  • Cheng H, Ma Z, Li P (2020) A fuzzy spatial description logic for the semantic web. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-020-01864-9

    Article  Google Scholar 

  • Clementini E, Di Felice P (1996) An algebraic model for spatial objects with indeterminate boundaries. Geogr Objects Indeterm Bound 2:155–169

    Google Scholar 

  • Cohn A, Gotts M (1996) The ‘egg-yolk’ representation of regions with indeterminate. Geogr Objects Indeterm Bound 2:171–188

    Google Scholar 

  • Cuchillo-Ibáñez E, Tarrés J (1997) On the boundary of fuzzy sets. Fuzzy Sets Syst 89(1):113–119

    MathSciNet  MATH  Google Scholar 

  • Dilo A (2006) Representation of and reasoning with vagueness in spatial information: A system for handling vague objects, International Institute for Geo-information Science & Earth Observation (ITC), Enschede, the Netherlands ITC Dissertation Number: 135.

  • Dixon B (2005) Applicability of neuro-fuzzy techniques in predicting ground-water vulnerability: a GIS-based sensitivity analysis. J Hydrol 309(1–4):17–38

    Google Scholar 

  • Du S, Qin Q, Wang Q, Li B (2005a) Fuzzy description of topological relations I: a unified fuzzy 9-intersection model. In: International conference on natural computation. Springer, pp 1261–1273

  • Du S, Wang Q, Qin Q, Yang Y (2005b) Fuzzy description of topological relations II: computation methods and examples. In: International conference on natural computation. Springer, pp 1274–1279

  • Du S, Qin Q, Wang Q, Ma H (2008) Reasoning about topological relations between regions with broad boundaries. Int J Approx Reason 47(2):219–232

    MathSciNet  MATH  Google Scholar 

  • Egenhofer MJ (1991) Reasoning about binary topological relations. In: Symposium on spatial databases. Springer, pp 141–160

  • Egenhofer MJ, Franzosa RD (1991) Point-set topological spatial relations. Int J Geogr Inf Syst 5(2):161–174

    Google Scholar 

  • Egenhofer MJ, Herring J (1990) Categorizing binary topological relations between regions, lines, and points in geographic databases. Technical report, Department of Surveying Engineering, University of Maine, vol 9. pp 1–28

  • Egenhofer MJ, Sharma J, Mark DM et al (1993) A critical comparison of the 4-intersection and 9-intersection models for spatial relations: formal analysis AUTOCARTO-CONFERENCE-11. ASPRS AMERICAN SOCIETY FOR PHOTOGRAMMETRY AND REMOTE SENSING, 1993. http://www.dpi.inpe.br/gilberto/references/max_comparison_4_9_intersection.pdf

  • Egenhofer MJ, Clementini E, Di Felice P (1994) Topological relations between regions with holes. Int J Geogr Inf Sci 8(2):129–142

    Google Scholar 

  • Ellul C, Haklay MM (2009) Using a B-Rep structure to query 9-intersection topological relationships in 3D GIS—reviewing the approach and improving performance. In: 3D geo-information sciences. Springer, pp 127–151

  • Fisher P (1996) Boolean and fuzzy regions. In: Geographic objects with indeterminate boundaries, ISBN- 9780748403875, United Kingdom: Taylor & Francis. pp. 87–94

  • Fisher P (2000) Sorites paradox and vague geographies. Fuzzy Sets Syst 113(1):7–18

    Google Scholar 

  • Freeman J (1975) The modelling of spatial relations. Comput Graph Image Process 4(2):156–171

    Google Scholar 

  • Giamalaki M, Tsoutsos T (2019) Sustainable siting of solar power installations in Mediterranean using a GIS/AHP approach. Renew Energy 141:64–75

    Google Scholar 

  • Gigović L, Pamučar D, Lukić D, Marković S (2016) GIS-fuzzy DEMATEL MCDA model for the evaluation of the sites for ecotourism development: a case study of “Dunavski Ključ” region. Land Use Policy 58:348–365

  • Goodchild MF, Dubuc O (1987) A model of error for choropleth maps, with applications to geographic information systems. In: Proceedings, AutoCarto 8. Citeseer, pp 165–174

  • Goodchild M, Zhang J, Kyriakidis P (2009) Discriminant models of uncertainty in nominal fields. Trans GIS 13(1):7–23

    Google Scholar 

  • Guang-qing W, Chong-you Z (1991) Fuzzy boundary and characteristic properties of order-homomorphisms. Fuzzy Sets Syst 39(3):329–337

    MathSciNet  Google Scholar 

  • Gupta L, Dixit J (2022) A GIS-based flood risk mapping of Assam, India, using the MCDA-AHP approach at the regional and administrative level. Geocarto Int. https://doi.org/10.1080/10106049.2022.2060329

    Article  Google Scholar 

  • Hemetsberger M, Klinger G, Niederer S, Benedikt J (2002) Risk assessment of avalanches-a fuzzy GIS application. In: Computational intelligent systems for applied research. World Scientific, pp 395–402

  • Jiang Z, Jiang Y, Wang Y, Zhang H, Cao H, Tian G (2019) A hybrid approach of rough set and case-based reasoning to remanufacturing process planning. J Intell Manuf 30(1):19–32

    Google Scholar 

  • Jun X (2019a) An improved spatial topological relationship model and algorithm. In: 2019 3rd international conference on electronic information technology and computer engineering (EITCE). IEEE, pp 993–997

  • Jun X (2019b) A scalable fuzzy division model of topological space on GIS. IOP Conf Ser Mater Sci Eng 592:012089

  • Kaymaz ÇK, Çakır Ç, Birinci S, Kızılkan Y (2021) GIS-fuzzy DEMATEL MCDA model in the evaluation of the areas for ecotourism development: a case study of “uzundere’’, Erzurum-Turkey. Appl Geogr 136:102577

    Google Scholar 

  • Kazemi H, Akinci H (2018) A land use suitability model for rainfed farming by multi-criteria decision-making analysis (MCDA) and geographic information system (GIS). Ecol Eng 116:1–6

    Google Scholar 

  • Kumar S, Srivastava PK, Snehmani (2017) GIS-based MCDA-AHP modelling for avalanche susceptibility mapping of Nubra valley region, Indian Himalaya. Geocarto Int 32(11):1254–1267

  • Leng L, Wang F, Wang M, Yang G, Niu X, Zhang X (2022) A generalized 9-intersection model for topological relations between regions with holes. ISPRS Int J Geo-Inf 11(4):218

    Google Scholar 

  • Liu K, Shi W (2006) Computing the fuzzy topological relations of spatial objects based on induced fuzzy topology. Int J Geogr Inf Sci 20(8):857–883

    Google Scholar 

  • Liu K, Shi W (2007) Extended model of topological relations between spatial objects in geographic information systems. Int J Appl Earth Obs Geoinf 9(3):264–275

    Google Scholar 

  • Liu K, Shi W (2009) Quantitative fuzzy topological relations of spatial objects by induced fuzzy topology. Int J Appl Earth Obs Geoinf 11(1):38–45

    MathSciNet  Google Scholar 

  • Liu B, Li D, Ruan J, Zhang L, You L, Wu H (2015a) A model of fuzzy topological relations for simple spatial objects in GIS. Boletim de Ciências Geodésicas 21(2):389–408

    Google Scholar 

  • Liu B, Li D, Xia Y, Ruan J, Xu L, Wu H (2015b) Combinational reasoning of quantitative fuzzy topological relations for simple fuzzy regions. PLoS ONE 10(3):e0117379

    Google Scholar 

  • Lowen R (1976) Fuzzy topological spaces and fuzzy compactness. J Math Anal Appl 56(3):621–633

    MathSciNet  MATH  Google Scholar 

  • Mahanta J, Das PK (2014) On fuzzy boundary. J Fuzzy Math 22(1):197–201

    MathSciNet  MATH  Google Scholar 

  • Mazgareanu I, Biron PM, Buffin-Bélanger T (2020) A fuzzy GIS model to determine confluence morphological sensitivity to tributary inputs at the watershed scale. Geomorphology 357:107095

    Google Scholar 

  • Molenaar M (1996) A syntactic approach for handling the semantics of fuzzy spatial. Geogr Objects Indeterm Bound 2:207

    Google Scholar 

  • Mu W, Tong D (2019) Choropleth mapping with uncertainty: a maximum likelihood-based classification scheme. Ann Am Assoc Geogr 109(5):1493–1510

    Google Scholar 

  • Palanisamy M, Sentamilselvi M, Sitheeswari A (2019) A fuzzy topology for operator in spatial object quantitatively in flood prediction. Int J Transform Appl Math Stat 2(1):8–22

    Google Scholar 

  • Pao-Ming P, Ying-Ming L (1980) Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore–Smith convergence. J Math Anal Appl 76(2):571–599

    MathSciNet  MATH  Google Scholar 

  • Randell DA, Cui Z, Cohn AG (1992) A spatial logic based on regions and connection. KR 92:165–176

    Google Scholar 

  • Salamo M, Lopez-Sanchez M (2011) Rough set based approaches to feature selection for case-based reasoning classifiers. Pattern Recogn Lett 32(2):280–292

    Google Scholar 

  • Salleh S, Ujang U, Azri S (2022) Representing 3D topological adjacencies between volumes using a 36-intersection model. Geomat Environ Eng 16(2):127–155

    Google Scholar 

  • Sarkar S, Parihar SM, Dutta A (2016) Fuzzy risk assessment modelling of east Kolkata wetland area: a remote sensing and GIS based approach. Environ Model Softw 75:105–118

    Google Scholar 

  • Schneider M (1999) Uncertainty management for spatial data in databases: fuzzy spatial data types. In: International symposium on spatial databases. Springer, pp 330–351

  • Schneider M (2008) Fuzzy spatial data types for spatial uncertainty management in databases. In: Handbook of research on fuzzy information processing in databases. IGI Global, pp 490–515

  • Schockaert S, De Cock M, Cornelis C, Kerre EE (2008a) Fuzzy region connection calculus: an interpretation based on closeness. Int J Approx Reason 48(1):332–347

    MathSciNet  MATH  Google Scholar 

  • Schockaert S, De Cock M, Cornelis C, Kerre EE (2008b) Fuzzy region connection calculus: representing vague topological information. Int J Approx Reason 48(1):314–331

    MathSciNet  MATH  Google Scholar 

  • Schockaert S, De Cock M, Kerre EE (2009) Spatial reasoning in a fuzzy region connection calculus. Artif Intell 173(2):258–298

    MathSciNet  MATH  Google Scholar 

  • Shahi E, Alavipoor FS, Karimi S (2018) The development of nuclear power plants by means of modified model of fuzzy DEMATEL and GIS in Bushehr, Iran. Renew Sustain Energy Rev 83:33–49

    Google Scholar 

  • Shi W, Liu K (2004) Modeling fuzzy topological relations between uncertain objects in a GIS. Photogramm Eng Remote Sens 70(8):921–929

    Google Scholar 

  • Shi W, Liu K (2007) A fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitatively in GIS. Comput Geosci 33(7):898–915

    Google Scholar 

  • Shi W, Liu K, Huang C (2009) A fuzzy-topology-based area object extraction method. IEEE Trans Geosci Remote Sens 48(1):147–154

    Google Scholar 

  • Sulaiman BM, Ismail TH (2019) Computing the fuzzy topological relations of countries affected by congenital disease. J Al-Qadisiyah Comput Sci Math 11(4):94

    Google Scholar 

  • Tang X, Kainz W (2002) Analysis of topological relations between fuzzy regions in a general fuzzy topological space. In: Symposium on geospatial theory, processing and applications. Citeseer, pp 1–15

  • Tang X, Kainz W, Wang H (2010) Topological relations between fuzzy regions in a fuzzy topological space. Int J Appl Earth Obs Geoinf 12:151–165

    Google Scholar 

  • Tao C, Molenaar M, Bouloucos T (1997) Identification of fuzzy objects from field observation data. In: International conference on spatial information theory. Springer, pp 241–259

  • Thiruchelvi M, Ilango G (2020) Application of fuzzy topological relation in cyclone prediction. Malaya J Mat. https://doi.org/10.26637/MJM0S20/0016

  • Tong X, Sun T, Fan J, Goodchild MF, Shi W (2013) A statistical simulation model for positional error of line features in geographic information systems (GIS). Int J Appl Earth Obs Geoinf 21:136–148

    Google Scholar 

  • Tong X-Y, Xia G-S, Lu Q, Shen H, Li S, You S, Zhang L (2020) Land-cover classification with high-resolution remote sensing images using transferable deep models. Remote Sens Environ 237:111322

    Google Scholar 

  • Vahidnia MH, Alesheikh AA, Alimohammadi A (2009) Hospital site selection using fuzzy AHP and its derivatives. J Environ Manag 90(10):3048–3056

    Google Scholar 

  • Warren RH (1977) Boundary of a fuzzy set. Indiana Univ Math J 26(2):191–197

    MathSciNet  MATH  Google Scholar 

  • Winter S (2000) Uncertain topological relations between imprecise regions. Int J Geogr Inf Sci 14(5):411–430

    Google Scholar 

  • Xiao N, Calder CA, Armstrong MP (2007) Assessing the effect of attribute uncertainty on the robustness of choropleth map classification. Int J Geogr Inf Sci 21(2):121–144

    Google Scholar 

  • Yariyan P, Zabihi H, Wolf ID, Karami M, Amiriyan S (2020) Earthquake risk assessment using an integrated fuzzy analytic hierarchy process with artificial neural networks based on GIS: a case study of Sanandaj in Iran. Int J Disaster Risk Reduct 50:101705

    Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Google Scholar 

  • Zhang L, Zhang L, Du B (2016) Deep learning for remote sensing data: a technical tutorial on the state of the art. IEEE Geosci Remote Sens Mag 4(2):22–40

    Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewers for their insightful comments and suggestions.

Funding

The authors have no relevant financial or non-financial interests to disclose.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Silchar, Silchar, Assam, 788010, India

    Subhankar Jana & Juthika Mahanta

Authors
  1. Subhankar Jana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juthika Mahanta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subhankar Jana.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jana, S., Mahanta, J. Boundary of a fuzzy set and its application in GIS: a review. Artif Intell Rev 56, 6477–6507 (2023). https://doi.org/10.1007/s10462-022-10331-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10462-022-10331-0

Keywords

Search

Navigation