Skip to main content
SpringerLink
Log in

Rough Mereology and Analytical Morphology

  • Chapter
Incomplete Information: Rough Set Analysis

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 13))

Abstract

We present two theories that emerge in connection with rough set-based methods for classifying dynamic populations of objects. The first theory, referred to as rough mereology aims at the analysis of complex objects in terms of properties of their parts. The second theory — analytical morphology of rough sets is a generalization of mathematical morphology obtained by imposing a geometrical structure on the attributes in information systems.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. Bleau A., deGuise J., LeBlanc R.: A new set of fast algorithms for mathematical morphology I; Idempotent geodesic transfornms, Computer Vision Graphics and Image Processing: Image Understanding 56 (2) (1992), 178 - 209

    MATH  Google Scholar 

  2. Bleau A., deGuise J., LeBlanc R.:A new set of fast algorithms for mathematical morphology II; Identification of topographic features on grayscale images, Computer Vision Graphics and Image Processing: Image Understanding 56 (2) (1992), 210 - 229

    MATH  Google Scholar 

  3. Bazan J., Skowron A., Synak P.:Dynamic reducts as a tool for extracting laws from decision tables,Proc. of the Symp. on Methodologies for Intelligent Systems}, Charlotte, NC, October 16-19, 1994, Lecture Notes in Artificial Intelligence Vol. 869, Springer-Verlag 1994, 346 - 355.

    Google Scholar 

  4. Bazan J., Skowron A., Synak P.: Market data analysis: A rough set approach, ICS Research Report 6/94 Warsaw University of Technology 1994

    Google Scholar 

  5. Bazan J., Son H.N., Trung T.N., Skowron A., Synak P.: Some logic and rough set applications for classifying objects, ICS Research Report 38/94, Warsaw University of Technology 1994.

    Google Scholar 

  6. Choquet G.: Convergences, Ann. Univ. Grenoble, 23 (1948), 55 - 112

    MathSciNet  Google Scholar 

  7. Clarke B.L.: A calculus of individuals based on “Connection”, Notre Dame Journal of Formal Logic, 22 (1981), 204 - 218

    Article  MathSciNet  MATH  Google Scholar 

  8. Clarke B.L.: Individuals and points, Notre Dame Journal of Formal Logic, 26 (1985), 61 - 75

    Article  MathSciNet  MATH  Google Scholar 

  9. Giardina C.R., Dougherty E.R.: Morphological Methods in Image and Signal Processing, Prentice Hall, Englewood Cliffs, 1988

    Google Scholar 

  10. Haralick R.M., Sternberg S., Zhuong X.: Image analysis using mathematical morphology, IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-9(1987), 532-550

    Google Scholar 

  11. Hausdorff F.: Grundzuge der Mengenlehre, Leipzig (Wien), 1914

    Google Scholar 

  12. Heijmans H.J.A.M., Ronse C.: The algebraic basis of mathematical morphology I. Dilations and erosions, Computer Vision Graphics and Image Processing 50 (1990), 245 - 295

    Article  MATH  Google Scholar 

  13. Kendall D.G.: Foundations of the theory of random sets, in: Stochastic Geometry (E.F. Harding and D.G. Kendall, eds. ), Wiley, 1973, pp. 322 - 376

    Google Scholar 

  14. Kodratoff Y., Michalski R.: Machine learning: An Artificial Intelligence approach, Vol. 3. San Mateo: Morgan Kaufmann 1990

    Google Scholar 

  15. Lesniewski, S.: Foundations of the general theory of sets (in Polish), Moscow, 1916; engl. trans!. in:Surma, Srzednicki, Barnett, Rickey (eds.), Stanislaw Lesniewski, Collected Works, Kluwer, Dordrecht 1992, 128 - 173

    Google Scholar 

  16. Lesniewski S.: On the foundations of Mathematics. Ch.I-III (in Polish), Przeglad Filozoficzny, 30 (1927), 164-206; Eng. transl. in [SSBR; 175-226]

    Google Scholar 

  17. Lesniewski S.: On the foundations of Mathematics. Ch.IV (in Polish), Przeglad Filozoficzny 31 (1928), 261-291; Eng. trans} in [SSBR; 227-263]

    Google Scholar 

  18. Lesniewski S.: On the foundations of Mathematics. Ch.V (in Polish), Przeglad Filozoficzny,32(1929), 60-101; Eng. transl. in [SSBR; 264-314]

    Google Scholar 

  19. Lesniewski S.: On the foundations of Mathematics. Ch. VI-Ch.IX (in Polish), Przeglad Filozoficzny, 33 (1930), 77-105; Eng. transl. in [SSBR; 315-349]

    Google Scholar 

  20. Lesniewski S.: On the foundations of Mathematics. Ch.X, Ch.XI (in Polish), Przeglad Filozoficzny 34 (1931), 142-170; Eng. transl. in [SSBR; 350-382]

    Google Scholar 

  21. Marcus S.: Structures linguistiques et structures topologiques, Rev. Math. Pures Appl. 6 (30), (1961), 501 - 506

    MathSciNet  MATH  Google Scholar 

  22. Mamdani E.H., Assilian S.: An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies, 7, 1975, 1 - 13

    Article  MATH  Google Scholar 

  23. Matheron G.: Random Sets and Integral Geometry, Wiley, 1975

    Google Scholar 

  24. Marcus S.: Modeles mathematiques pour la categorie grammaticale du cas, Rev. Math. Pures Appl. 8 (4) (1963), 585 - 610

    MATH  Google Scholar 

  25. Michael E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152 - 183

    Article  MathSciNet  MATH  Google Scholar 

  26. Marek W., Rasiowa H.: Approximating sets with equivalence relations, Theoretical Computer Sc. 48 (1986), 145 - 152

    Article  MathSciNet  MATH  Google Scholar 

  27. Pawlak, Z.: Rough sets, International Journal of Information and Computer Science 11 (1982), 344 - 356

    MathSciNet  Google Scholar 

  28. Pawlak Z.: Rough sets: Theoretical aspects of reasoning about data. Dordrecht: Kluwer 1991

    MATH  Google Scholar 

  29. Pawlak Z.: Vagueness and Uncertainty a Rough Set Perspective, ICS ICS Research Report 19/94, Warsaw University of Technology 1994

    Google Scholar 

  30. Polkowski L.T.: Mathematical morphology of rough sets, Bull. Acad. Polon. Sci., Ser. Sci. Math. 41 (3) (1993), 241 - 273

    MathSciNet  MATH  Google Scholar 

  31. Pompeiu D.: Ann. de Toulouse (2), 7, (1905)

    Google Scholar 

  32. Polkowski L., Skowron A.: Rough mereology, Proc. of the Symp. On Methodologies for Intelligent Systems, Charlotte, NC, October 16-19, 1994, Lecture Notes in Artificial Intelligence Vol. 869, Springer Verlag, Berlin 1994, 85-94.

    Google Scholar 

  33. Polkowski L., Skowron A.: Decision Support Systems, Rough Set Approach, Warszawa, 1994.

    Google Scholar 

  34. Pawlak Z., Skowron A: Rough membership functions: A tool for reasoning with uncertainty, Algebraic Methods in Logic and their Computer Science, Banach Center Publications Vol.28., Polish Academy of Sciences, Warszawa 1993, 135 - 150

    Google Scholar 

  35. Pawlak Z., Skowron A: A rough set approach for decision rules generation. ICS Research Report 23/93, Warsaw University of Technology 1993, Proc. of the IJCAI’93 Workshop: The Management of Uncertainty in AI, France 1993

    Google Scholar 

  36. Pawlak Z., Skowron A: Rough membership functions, In: M.Fedrizzi, J. Kacprzyk and R.R. Yager (eds.): Advances in the Dempster-Shafer Theory of Evidence. New York: John Wiley and Sons 1994, 251 - 271

    Google Scholar 

  37. Ronse C., Heijmans H.J.A.M.: The algebraic basis of mathematical morphology II. Openings and closings, Computer Vision Graphics and Image Processing: Image Understanding 54 (1) (1991), 74 - 97

    MATH  Google Scholar 

  38. Serra J.: Image Analysis and Mathematical Morphology, Academic Press, New York 1982

    MATH  Google Scholar 

  39. Skowron A.: On topology in information systems, Bull. Acad. Polon. Sci. Math., 36 (1988)

    Google Scholar 

  40. Skowron A: The rough set theory as a basis for the evidence theory, ICS Research Report 2/91, Warsaw University of Technology 1991, 1 - 53

    Google Scholar 

  41. Skowron A: Boolean reasoning for decision rules generation. Proceedings of the 7-th International Symposium ISMIS’93, Trondheim, Norway 1993, In: J.Komorowski and Z.Ras (eds.): Lecture Notes in Artificial Intelligence, Vol.689. Springer-Verlag 1993, 295 - 305

    Google Scholar 

  42. Skowron A: A synthesis of decision rules: Applications of discernibility matrices, Proc. of the Conf. Intelligent Information Systems, Augustow, June 7-11, 1993, 30 - 46

    Google Scholar 

  43. Skowron A: Extracting laws from decision tables - a rough set approach, Proc. of the International Workshop on Rough Sets and Knowledge Discovery RSKD’93, Banff, Canada 1993, 101 - 104

    Google Scholar 

  44. Sk93c] Skowron, A: Management of uncertainty: A rough set approach, ICS Research Report 46/93, Warsaw University of Technology 1994, also in: Proc. Int. Workshop Incompleteness and ncertainty in Information Systems, Montreal, October 8-9, 1993, 36-61, Springer Verlag, in print.

    Google Scholar 

  45. Skowron,A., Polkowski,L.: Introducing rough mereological controllers: rough quality control, In: T.Y. Lin, and A.M. Wildberger (eds.), Soft Computing, Simulation Councils, Inc., San Diego 1995, 240 - 243

    Google Scholar 

  46. Skowron A., Polkowski L.: Analytical Morphology: Mathematical Morphology of Decision Tables, Fundaments Informaticae 27 (2-3) (1996), 255 - 272.

    MathSciNet  MATH  Google Scholar 

  47. Skowron, A. and Rauszer C.: The Discernibility Matrices and Functions in Information Systems. In: R. Slowinski (ed.): Intelligent Decision Support. Handbook of Applications and Advances of the Rough Sets Theory. Dordrecht: Kluwer 1992, 331 - 362

    Google Scholar 

  48. Song J., Delp E. J.: The analysis of morphological filters with multiple structuring elements, Computer Vision Graphics and Image Processing 50 (1990), 308 - 328

    Article  Google Scholar 

  49. Sriram D., Logcher R., Fukuda S.: Computer - Aided Cooperative Product Development, LNCS 492, Springer-Verlag, Berlin Heidelberg, 1991

    Google Scholar 

  50. Sunna S.J., Srzednicki J.T., Barnett D.I., Rickey V.F. (eds.): Stanislaw Lesniewski, Collected Works, Kluwer - Polish Scientific Publishers, Dordrecht-Warsaw, 1992

    Google Scholar 

  51. Sternberg R. S.: Grayscale morphology, Computer Vision Graphics and Image Processing 35 (3) (1986), 333 - 355

    Article  MathSciNet  Google Scholar 

  52. Ta29] Tarski A.: Les fondements de la geometrie des corps, Annales de la Societe Polonaise de Mathematique (Supplement), Krakow, 1929, 29-33; Eng. transi. in: [Wo; 24-29]

    Google Scholar 

  53. Ta35] Tarski A: Zur Grundlegung der Booleschen Algebra.I, Fundamenta Mathematicae, 24 (1935), 177-198; Eng. transi. in [Wo; 320-341]

    Google Scholar 

  54. Tarski A.: Appendix E in: Woodger, J.H., The Axiomatic Method in Biology, Cambridge University Press, Cambridge 1937

    Google Scholar 

  55. Tondeur: Ein Beispiel zur Allgemeinen Topologie: die Topologie einer Aequivalenz-relation, Annales Academiae Scientiarum Fennicae, ser. A, I. Math., No.344, Helsinki, 1964

    Google Scholar 

  56. Vietoris L.: Monat. Math. Ph., 31 (1921), 173 - 204

    Article  MathSciNet  MATH  Google Scholar 

  57. Whitehead A.N.: An Enquiry Concerning the Principles of Natural Knowledge,Cambridge U.Press, Cambridge, 1919

    Google Scholar 

  58. Wiweger A.: On topological rough sets, Bull. Acad. Polon. Sci. Math., 37 (1988), 89 - 93

    MathSciNet  Google Scholar 

  59. Wo56] Woodger J.H.,transl.:Logic, Semantics, Metamathematics. Papers from 1923 to 1938 by Alfred Tarski, Clarendon, Oxford, 1956

    Google Scholar 

  60. Zadeh L.A.: Fuzzy sets, Information and Control 8, (1965) 338 - 353

    MathSciNet  MATH  Google Scholar 

  61. Ziarko W.: Variable precision rough set model, Journal of Computer and System Sciences, 46 (1993), 39 - 59

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warszawa, Poland

    Andrzej Skowron

  2. Institute of Mathematics, Warsaw University of Technology, Pl.Politechniki 1, 00-650, Warszawa, Poland

    Lech Polkowski

Authors
  1. Andrzej Skowron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lech Polkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Telecommunications, ul. Szachowa 1, PL-04-894, Warsaw, Poland

    Ewa Orłowska

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Skowron, A., Polkowski, L. (1998). Rough Mereology and Analytical Morphology. In: Orłowska, E. (eds) Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol 13. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1888-8_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-7908-1888-8_13

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-2457-5

  • Online ISBN: 978-3-7908-1888-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation