Advertisement

Associated Near Sets of Distance Functions in Pattern Analysis

  • James F. Peters
Conference paper
  • 717 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7080)

Abstract

This paper introduces description-based associated sets of distance functions, where members are topological structures helpful in pattern analysis and machine intelligence. An associated set of a function is a collection containing members with one or more common properties. This study has important implications in discerning patterns shared by members of an associated set. The focus in this paper is on defining and characterising distance functions relative to structures that are collections of sufficiently near (far) neighbourhoods, filters, grills and clusters. Naimpally-Peters-Tiwari distance functions themselves define approach spaces that generalise the traditional notion of a metric space. An important side-effect of this work is the discovery of various patterns that arise from the descriptions (perceptions) of associated set members. An application of the proposed approach is given in the context of camouflaged objects.

Keywords

Apartness approach space associated set C̆ech distance cluster collection near sets pattern analysis topological structure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Peters, J.: Near sets. Special theory about nearness of objects. Fund. Inform. 75(1-4), 407–433 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Peters, J., Wasilewski, P.: Foundations of near sets. Info. Sci. 179, 3091–3109 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Peters, J.: Metric spaces for near sets. Ap. Math. Sci. 5(2), 73–78 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Peters, J., Naimpally, S.: Approach spaces for near families. Gen. Math. Notes 2(1), 159–164 (2011)zbMATHGoogle Scholar
  5. 5.
    Peters, J., Tiwari, S.: Approach merotopies and near filters. Gen. Math. Notes 3(1), 1–15 (2011)zbMATHGoogle Scholar
  6. 6.
    Peters, J.: How near are Zdzisław Pawlak’s paintings? Merotopic distance between regions of interest. In: Skowron, A., Suraj, S. (eds.) Intelligent Systems Reference Library volume dedicated to Prof. Zdzisław Pawlak, pp. 1–19. Springer, Berlin (2011)Google Scholar
  7. 7.
    Naimpally, S., Peters, J.: Topology with Applications. World Scientific, Singapore (to appear, 2012)zbMATHGoogle Scholar
  8. 8.
    Coble, A.: Associated sets of points. Trans. Amer. Math. Soc. 24(1), 1–20 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bruckner, A.: On characterizing classes of functions in terms of associated sets. Canad. Math. Bull. 10(2), 227–231 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Peters, J., Tiwari, S.: Associated near sets. Theory and application. Demo. Math. (2011), communicatedGoogle Scholar
  11. 11.
    Ramanna, S., Peters, J.: Nearness of associated rough sets: Case study in image analysis. In: Peters, G., Lingras, P., Yao, Y., Slezak, D. (eds.) Selected Methods and Applications of Rough Sets in Management and Engineering, pp. 62–73. Springer, Heidelberg (2011)Google Scholar
  12. 12.
    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74 (1906)CrossRefzbMATHGoogle Scholar
  13. 13.
    Sutherland, W.: Introduction to Metric & Topological Spaces. Oxford University Press, Oxford (1974,2009); 2nd ed. (2008)zbMATHGoogle Scholar
  14. 14.
    Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, pp. viii + 253. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  15. 15.
    Čech, E.: Topological Spaces, revised Ed. by Z. Frolik and M. Katătov. John Wiley & Sons, NY (1966)Google Scholar
  16. 16.
    Katětov, M.: On continuity structures and spaces of mappings. Comment. Math. Univ. Carolinae 6, 257–278 (1965)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tiwari, S.: Some Aspects of General Topology and Applications. Approach Merotopic Structures and Applications, supervisor: M. Khare. PhD thesis, Department of Mathematics, Allahabad (U.P.), India (January 2010)Google Scholar
  18. 18.
    Lowen, R., Vaughan, D., Sioen, M.: Completing quasi metric spaces: an alternative approach. Houstan J. Math. 29(1), 113–136 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bourbaki, N.: Elements of Mathematics. In: General Topology, Part 1, pp. i-vii, 437. Hermann & Addison-Wesley, Paris & Reading, MA, U.S.A (1966)Google Scholar
  20. 20.
    Peters, J.: ε-Near Collections. In: Yao, J.-T., Ramanna, S., Wang, G., Suraj, Z. (eds.) RSKT 2011. LNCS, vol. 6954, pp. 533–542. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Khare, M., Tiwari, S.: Grill determined L-approach merotopological spaces. Fund. Inform. 48, 1–12 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Peters, J., Tiwari, S.: Completing extended metric spaces: An alternative approach (2011), communicatedGoogle Scholar
  23. 23.
    Khare, M., Tiwari, S.: L-approach merotopies and their categorical perspective. Demonstratio Math. 48 (to appear, 2012)Google Scholar
  24. 24.
    Ramanna, S., Peters, J.: Approach Space Framework for Image Database Classification. In: Hruschka Jr., E.R., Watada, J., do Carmo Nicoletti, M. (eds.) INTECH 2011. CCIS, vol. 165, pp. 75–89. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  25. 25.
    Beer, G., Lechnicki, A., Levi, S., Naimpally, S.: Distance functionals and suprema of hyperspace topologies. Annali di Matematica pura ed applicata CLXII(IV), 367–381 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hausdorff, F.: Grundzüge der Mengenlehre, pp. viii + 476. Veit and Company, Leipzig (1914)zbMATHGoogle Scholar
  27. 27.
    Leader, S.: On clusters in proximity spaces. Fundamenta Mathematicae 47, 205–213 (1959)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Naimpally, S.: Proximity Spaces, pp. x+128. Cambridge University Press, Cambridge (1970) ISBN 978-0-521-09183-1zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • James F. Peters
    • 1
  1. 1.Computational Intelligence Laboratory, Department of Electrical & Computer EngineeringUniv. of ManitobaWinnipegCanada

Personalised recommendations