Skip to main content
SpringerLink
Log in

Relations in Mathematical Morphology with Applications to Graphs and Rough Sets

  • Conference paper
Spatial Information Theory (COSIT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 4736))

Included in the following conference series:

Abstract

Rough sets have been applied in spatial information theory to construct theories of granularity – presenting information at different levels of detail. Mathematical morphology can also be seen as a framework for granularity, and the question of how rough sets relate to mathematical morphology has been raised by Bloch. This paper shows how by developing mathematical morphology in terms of relations we obtain a framework which includes the basic constructions of rough set theory as a special case. The extension of the relational framework to mathematical morphology on graphs rather than sets is explored and new operations of dilations and erosions on graphs are obtained.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bloch, I.: On links between mathematical morphology and rough sets. Pattern Recognition 33, 1487–1496 (2000)

    Article  Google Scholar 

  2. Bloch, I.: Spatial reasoning under imprecision using fuzzy set theory, formal logics and mathematical morphology. International Journal of Approximate Reasoning 41, 77–95 (2006)

    Article  MATH  Google Scholar 

  3. Brown, R., Morris, I., Shrimpton, J., Wensley, C.D.: Graphs of Morphisms of Graphs. Bangor Mathematics Preprint 06.04, Mathematics Department, University of Wales, Bangor (2006)

    Google Scholar 

  4. Galton, A.: The mereotopology of discrete space. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 251–266. Springer, Heidelberg (1999)

    Google Scholar 

  5. Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Prentice-Hall, Englewood Cliffs (2001)

    Google Scholar 

  6. Heijmans, H.: Mathematical morphology: A modern approach in image processing based on algebra and geometry. SIAM Review 37, 1–36 (1995)

    Article  MATH  Google Scholar 

  7. Hollings, C.: Partial actions of monoids. Semigroup Forum (to appear)

    Google Scholar 

  8. Hsueh, Y.C.: Relation-based variations of the discrete Radon transform. Computers and Mathematics with Applications 31, 119–131 (1996)

    Article  MATH  Google Scholar 

  9. Heijmans, H., Vincent, L.: Graph Morphology in Image Analysis. In: Dougherty, E.R. (ed.) Mathematical Morphology in Image Processing, ch. 6. Marcel Dekker, pp. 205–254 (1993)

    Google Scholar 

  10. Müller, J.C., Lagrange, J.P., Weibel, R. (eds.): GIS and Generalisation: Methodology and Practice. Taylor and Francis, London (1995)

    Google Scholar 

  11. Orłowska, E. (ed.): Incomplete Information – Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol. 13. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  12. Roerdink, J.B.T.M.: Mathematical Morphology with Non-Commutative Symmetry Groups. In: Dougherty, E.R. ed.: Mathematical Morphology in Image Processing, ch. 7. Marcel Dekker, pp. 171–203 (1993)

    Google Scholar 

  13. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1983)

    Google Scholar 

  14. Stell, J.G.: Granulation for graphs. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 417–432. Springer, Heidelberg (1999)

    Google Scholar 

  15. Stell, J.G., Worboys, M.F.: The algebraic structure of sets of regions. In: Frank, A.U. (ed.) COSIT 1997. LNCS, vol. 1329, pp. 163–174. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  16. Stell, J.G., Worboys, M.F.: Generalizing graphs using amalgamation and selection. In: Güting, R.H., Papadias, D., Lochovsky, F.H. (eds.) SSD 1999. LNCS, vol. 1651, pp. 19–32. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  17. Yao, Y.Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning 15, 291–317 (1996)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, University of Leeds, Leeds, LS2 9JT, UK

    John G. Stell

Authors
  1. John G. Stell
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stephan Winter Matt Duckham Lars Kulik Ben Kuipers

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Stell, J.G. (2007). Relations in Mathematical Morphology with Applications to Graphs and Rough Sets. In: Winter, S., Duckham, M., Kulik, L., Kuipers, B. (eds) Spatial Information Theory. COSIT 2007. Lecture Notes in Computer Science, vol 4736. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74788-8_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-74788-8_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74786-4

  • Online ISBN: 978-3-540-74788-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation