Skip to main content
SpringerLink
Log in
Book cover

Computational Proximity pp 1–62Cite as

Computational Proximity

  • Chapter
  • First Online:

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 102))

Abstract

This chapter introduces computational proximity. Basically, computational proximity (CP) is an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. The methods used by CP to find either near sets or remote sets result from the study of structures called proximity spaces.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

Notes

  1. 1.

    For more about this, see C. Stover and E.W. Weisstein, Point, http://mathworld.wolfram.com/Point.html.

  2. 2.

    Many thanks to Anna Di Concilio for pointing this out.

  3. 3.

    Many thanks to A. Di Concilio for pointing this out and suggesting overlap or strong contact as a more appropriate name for . For an in-depth view of overlap, see [21], especially [22, Sect. 3].

References

  1. Di Maio, G., Naimpally, S.A., Meccariello, E.: Theory and applications of proximity, nearness and uniformity. Seconda Università di Napoli, Napoli (2009), 264 pp., MR1269778

    Google Scholar 

  2. Naimpally, S., Peters, J., Wolski, M.: Foreword [near set theory and applications]. Math. Comput. Sci. 7(1), 1–2 (2013)

    Article  MathSciNet  Google Scholar 

  3. Peters, J.: Proximal relator spaces. Filomat (2016). Accepted

    Google Scholar 

  4. Peters, J., Guadagni, C.: Strongly proximal continuity and strong connectedness, pp. 1–11 (2015). arXiv:1504.02740

  5. Peters, J.: Topology of digital images - visual pattern discovery in proximity spaces, intelligent systems reference library, vol. 63. Springer, Berlin (2014). Xv + 411 pp. Zentralblatt MATH Zbl 1295 68010

    Google Scholar 

  6. Peters, J., Guadagni, C.: Strong proximities on smooth manifolds and Voronoï diagrams. Adv. Math.: Sci. J. 4(2), 91–107 (2015)

    MATH  Google Scholar 

  7. Naimpally, S., Peters, J.: Topology with applications. Topological spaces via near and far. World Scientific, Singapore (2013). Xv + 277 pp. Am. Math. Soc. MR3075111

    Google Scholar 

  8. Dochviri, I., Peters, J.: Topological sorting of finitely many near sets. Math. Comput. Sci. 1–6 (2015). Communicated

    Google Scholar 

  9. Naimpally, S.: Proximity Spaces. Cambridge University Press, Cambridge (1970). X + 128 pp., ISBN 978-0-521-09183-1

    Google Scholar 

  10. Lodato, M.: On topologically induced generalized proximity relations. Ph.D. thesis. Rutgers University (1962). Supervisor: S. Leader

    Google Scholar 

  11. Guadagni, C.: Bornological convergences on local proximity spaces and \(\omega _{\mu }\)-metric spaces. Ph.D. thesis, Università degli Studi di Salerno, Salerno (2015). Supervisor: A. Di Concilio, 79 pp

    Google Scholar 

  12. Di Concilio, A.: Proximity: a powerful tool in extension theory, functions spaces, hyperspaces, boolean algebras and point-free geometry. In: Mynard, F., Pearl, E. (eds.) Beyond Topology. AMS Contemporary Mathematics, vol. 486, pp. 89–114. American Mathematical Society, Providence (2009)

    Chapter  Google Scholar 

  13. Peters, J.: Local near sets: pattern discovery in proximity spaces. Math. Comput. Sci. 7(1), 87–106 (2013). doi:10.1007/s11786-013-0143-z, MR3043920

    Google Scholar 

  14. Hettiarachchi, R., Peters, J.: Multi-manifold LLE learning in pattern recognition. Pattern Recognit. Elsevier 48(9), 2947–2960 (2015)

    Article  Google Scholar 

  15. Hettiarachchi, R., Peters, J.: Multi-manifold skin classifier for feature space voronoï region-based skin segmentation. Image Vis. Comput. (2015). Communicated

    Google Scholar 

  16. İnan, E., Öztürk, M.: Near groups on nearness approximation spaces. Hacet. J. Math. Stat. 41(4), 545–558 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Peters, J., İnan, M.A., Öztürk, E.: Spatial and descriptive isometries in proximity spaces. Gen. Math. Notes 21(2), 125–134 (2014)

    Google Scholar 

  18. Peters, J., Öztürk, M., Uçkun, M.: Klee-Phelps convex groupoids, pp. 1–5 (2014). arXiv:1411.0934, Mathematica Slovaca 2015. Accepted

  19. Edelsbrunner, H., Harer, J.: Computational Topology. An Introduction. American Mathematical Society, Providence (2010). Xii + 110 pp., MR2572029

    Google Scholar 

  20. Peters, J.: Visibility in proximal Delaunay meshes and strongly near Wallman proximity. Adv. Math.: Sci. J. 4(1), 41–47 (2015)

    MATH  Google Scholar 

  21. Di Concilio, A., Gerla, G.: Quasi-metric spaces and point-free geometry. Math. Struct. Comput. Sci. 16(1), 115–137 (2006). MR2220893

    Google Scholar 

  22. Di Concilio, A.: Point-free geometries: proximities and quasi-metrics. Math. Comput. Sci. 7(1), 31–42 (2013). MR3043916

    Google Scholar 

  23. Beer, G., Lucchetti, R.: Weak topologies for the closed subsets of a metrizable space. Trans. Am. Math. Soc. 335(2), 805–822 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Aurenhammer, F.: Voronoi diagrams–a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)

    Article  Google Scholar 

  25. Peters, J., Ramanna, S.: Proximal three-way decisions: theory and applications in social networks. Knowl.-Based Syst. 1–12 (2014). http://dx.doi.org/10.1016/j.knosys.2015.07.021 (in press)

  26. Pták, P., Kropatsch, W.: Nearness in digital images and proximity spaces. LNCS. In: Proceedings of the 9th International Conference on Discrete Geometry, vol. 1953, pp. 69–77 (2000)

    Google Scholar 

  27. Beer, G., Di Concilio, A., Di Maio, G., Naimpally, S., Pareek, C., Peters, J.: Somashekhar Naimpally, 1931–2014. Topol. Appl. 188, 97–109 (2015). doi:10.1016/j.topol.2015.03.010, MR3339114

    Google Scholar 

  28. Di Maio, G., Naimpally, S.: Preface, theory and applications of proximity, nearness and uniformity, Quaderni di Matematica [Mathematics Series], vol. 22. Department of Mathematics, Seconda Universit di Napoli, Caserta (2008). viii + 364 pp., MR2760945

    Google Scholar 

  29. Plato: the allegory of the cave, in Republic, VII, 514a,2–517a, 7. Stanford University, Stanford (c300BC, 2015). Translated by T. Sheehan

    Google Scholar 

  30. Frege, G.: Foundations of arithmetic, Translated by J.L. Austin. Blackwell, Oxford (1953)

    Google Scholar 

  31. Whitehead, A.: An Inquiry into the Principles of Natural Knowledge. Cambridge University Press, Cambridge (1920)

    Google Scholar 

  32. Whitehead, A.: Process and Reality. Macmillan, London (1929)

    MATH  Google Scholar 

  33. Hooke, R.: Micrographia: or, some physiological descriptions of minute bodies made by magnifying glasses. With observations and inquiries thereupon. Martyn and J. Allestry, London (1665). 270 pp

    Google Scholar 

  34. Prince, S.: Computer Vision. Models, Learning, and Inference. Cambridge University Press, Cambridge (2012). Xvii + 580 pp

    Google Scholar 

  35. Favorskaya, M., Jain, L.C.: Computer Vision in Control Systems-1. Mathematical Theory. Springer, Berlin (2015). Xvii + 371 pp

    Google Scholar 

  36. Arkowitz, M.: Introduction to Homotopy Theory. Springer, New York (2011). Xiv + 344 pp. ISBN: 978-1-4419-7328-3, MR2814476

    Google Scholar 

  37. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). Xii + 544 pp. ISBN: 0-521-79160-X; 0-521-79540-0, MR1867354

    Google Scholar 

  38. Whitehead, J.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 45, 243–327 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  39. Krantz, S.: Essentials of topology with applications. CRC Press, Boca Raton (2010). Xvi + 404 pp. ISBN: 978-1-4200-8974-5, MR2554895

    Google Scholar 

  40. Euclid: The Thirteen Books of Euclid’s Elements, 2nd ed. Dover Publications, New York (c300BC, 1956). Translated by T.L. Heath, from the text by Heiberg, xi + 432pp; i + 436pp; i + 546 pp, MR0075873

    Google Scholar 

  41. Ronse, C.: Regular open or closed sets. Philips Research Laboratory Series, Brussels WD59, pp. 1–8 (1990)

    Google Scholar 

  42. Peters, J., Naimpally, S.: Applications of near sets. Notices Am. Math. Soc. 59(4), 536–542 (2012). http://dx.doi.org/10.1090/noti817, MR2951956

    Google Scholar 

  43. Edelsbrunner, H.: Geometry and Topology of Mesh Generation. Cambridge University Press, Cambridge (2001). 209 pp

    Google Scholar 

  44. Deritei, D., Lázár, Z., Papp, I., Járai-Szabó, F., Sumi, R., Varga, L., Regan, E., Ercsey-Ravasz, M.: Community detection by graph voronoi diagrams. New J. Phys. 16, 1–17 (2014)

    Article  Google Scholar 

  45. Riesz, F.: Stetigkeitsbegriff und abstrakte mengenlehre. Atti del IV Congresso Internazionale dei Matematici, vol. II, pp. 182–109 (1908)

    Google Scholar 

  46. C̆ech, E.: Topological Spaces. Wiley, London (1966). Fr seminar, Brno, 1936–1939; rev. ed. Z. Frolik, M. Katĕtov

    Google Scholar 

  47. Efremovič, V.: The geometry of proximity I (in Russian). Mat. Sb. (N.S.) 31(73)(1), 189–200 (1952)

    Google Scholar 

  48. Peters, J.: Near sets. Special theory about nearness of objects. Fundamenta Informaticae 75, 407–433 (2007). MR2293708

    Google Scholar 

  49. Peters, J., Guadagni, C.: Strongly near proximity and hyperspace topology, pp. 1–6 (2015). arXiv:1502.05913

  50. Wallman, H.: Lattices and topological spaces. Ann. Math. 39(1), 112–126 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  51. Euclid: Elements. Alexandria (300 B.C.). English translation by R. Fitzpatrick, from Euclidis Elementa Latin text by B.G. Teubneri, 1883–1885 and the Greek text by J.L. Heiberg, 1883–1885

    Google Scholar 

  52. Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis. Morgan-Kaufmann Publishers, Amsterdam (2004)

    MATH  Google Scholar 

  53. Du, Q., Faber, V., Gunzburger, M.: Centroidal voronoi tessellations: applications and algorithms. SIAM Rev. 41(4), 637–676 (1999). MR

    Google Scholar 

  54. Boissonnat, J.D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.D., Teillaud, E. (eds.) Effective Computational Geometry for Curves and Surfaces, pp. 67–116. Springer, New York (2006)

    Chapter  Google Scholar 

  55. Kovalevsky, V.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46, 141–161 (1989)

    Article  Google Scholar 

  56. Grünbaum, B., Shephard, G.: Tilings and Patterns. W.H. Freeman and Co., New York (1987). Xii + 700 pp., MR0857454

    Google Scholar 

  57. Romesburg, H.: Cluster Analysis for Researchers. Lulu Press, North Carolina (2004). Xii + 334 pp., MR3155265

    Google Scholar 

  58. Peters, J.: Near sets. General theory about nearness of sets. Appl. Math. Sci. 1(53), 2609–2629 (2007)

    MathSciNet  MATH  Google Scholar 

  59. Peters, J.: Near sets: an introduction. Math. Comput. Sci. 7(1), 3–9 (2013). doi:10.1007/s11786-013-0149-6, MR3043914

    Google Scholar 

  60. Kuratowski, C.: Topologie I. Panstwowe Wydawnictwo Naukowe, Warsaw (1958). XIII + 494 pp

    Google Scholar 

  61. Kuratowski, K.: Introduction to Set Theory and Topology, 2nd edn. Pergamon Press, Oxford (1962, 1972). 349 pp

    Google Scholar 

  62. Naimpally, S., Warrack, B.: Proximity Spaces. Cambridge Tract in Mathematics, vol. 59. Cambridge University Press, Cambridge (1970). X + 128 pp., Paperback (2008)

    Google Scholar 

  63. Peters, J., Guadagni, C.: Strongly far proximity and hyperspace topology, pp. 1–6 (2015). arXiv:1502.02771

  64. Peters, J., Guadagni, C.: Strongly hit and far miss hypertopology and hit and strongly far miss hypertopology, pp. 1–8 (2015). arXiv:1503.02587

  65. Solan, V.: Introduction to the axiomatic theory of convexity [Russian with English and French summaries]. Shtiintsa, Kishinev (1984). 224 pp., MR0779643

    Google Scholar 

  66. Zelins’kyi, Y.: Generalized convex envelopes of sets and the problem of shadow. J. Math. Sci. 211(5), 710–717 (2015)

    Article  MathSciNet  Google Scholar 

  67. Kay, D., Womble, E.: Automatic convexity theory and relationships between the carathèodory, helly and radon numbers. Pac. J. Math. 38(2), 471–485 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  68. Tuz, V.: Axiomatic convexity theory [Russian]. Rossiïskaya Akademiya Nauk. Matematicheskie Zametki [Math. Notes Math. Notes] 20(5), 761–770 (1976)

    Google Scholar 

  69. Rocchi, N.: Parliamo Di Insiemi. Instituto Didattico Editoriale Felsineo. Bologna (1969). 316 pp

    Google Scholar 

  70. Bourbaki, N.: Elements of Mathematics. General Topology, Part 1. Hermann and Addison-Wesley, Paris and Reading (1966). I-vii, 437 pp

    Google Scholar 

  71. Edelsbrunner, H.: A Short Course in Computational Geometry and Topology. Springer, Berlin (2014). 110 pp

    Google Scholar 

  72. Frank, N., Hart, S.: A dynamical system using the Voronoi tessellation. Am. Math. Monthly 117(2), 92–112 (2010)

    MathSciNet  MATH  Google Scholar 

  73. Weisstein, E.: Voronoi diagram. Wolfram MathWorld (2015). http://mathworld.wolfram.com/VoronoiDiagram.html

  74. Liebling, T., Pourin, L.: Voronoi diagrams and Delaunay triangulations: Ubiquitous siamese twins. Documenta Mathematica Extra volume: Optimization stories, 419–431 (2012). MR2991503

    Google Scholar 

  75. Okabe, A., Boots, B., Sugihara, K., Chiu, S.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester (2000). Xvi + 671 pp. ISBN: 0-471-98635-6, MR1770006

    Google Scholar 

  76. Stover, C.: Point Process. Wolfram MathWorld (2015). http://mathworld.wolfram.com/PointProcess.html

  77. Weisstein, E.: Poissonprocess. Wolfram MathWorld (2015). http://mathworld.wolfram.com/PoissonProcess.html

  78. Floriani, L.D., Spagnuolo, M.: Shape Analysis and Structuring. Springer, Berlin (2008). Xiv + 296 pp. ISBN 978-3-540-33264-0

    Google Scholar 

  79. Hilhorst, H.: Statistical properties of planar Voronoi tessellations. Eur. Phys. J. B 64, 437–441 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  80. Anton, F., Mioc, D., Gold, C.: The Voronoi diagram of circles and its application to the visualization of the growth of particles. In: Gavrilova, M., Tan, C.K. (eds.) Transactions on Computational Science III, pp. 20–54. Springer, Berlin (2009). MR2912541

    Google Scholar 

  81. Surendran, S., Chitraprasad, D., Kaimal, M.: Voronoi diagrams-based geometric approach to social network analysis. In: Krishnan, G.S.S., et al. (eds.) Computational Intelligence, Cyber Security and Computational Models, Advances in Intelligent Systems and Computing, vol. 246 pp. 359–369. Springer, India (2014)

    Google Scholar 

  82. Liu, H.: Dynamic concept cartography for social networks. Master’s thesis, School of Information Technologies (2007)

    Google Scholar 

  83. Peters, J.: Proximal Voronoï regions, convex polygons, and leader uniform topology. Adv. Math.: Sci. J. 4(1), 1–5 (2015)

    Google Scholar 

  84. Munkres, J.: Topology, 2nd edn. Prentice-Hall, Englewood Cliffs (2000). Xvi + 537 pp., 1st edn. in 1975, MR0464128

    Google Scholar 

  85. Krantz, S.: A Guide to Topology. The Mathematical Association of America, Washington (2009). Ix + 107 pp

    Google Scholar 

  86. Leader, S.: On clusters in proximity spaces. Fundamenta Mathematicae 47, 205–213 (1959)

    MathSciNet  MATH  Google Scholar 

  87. Naimpally, S.: Proximity Approach to Problems in Topology and Analysis. Oldenbourg Verlag, Munich (2009). 73 pp., ISBN 978-3-486-58917-7, MR2526304

    Google Scholar 

  88. Mozzochi, C., Gagrat, M., Naimpally, S.: Symmetric Generalized Topological Structures. Exposition Press, Hicksville (1976). I + 73 pp

    Google Scholar 

  89. Di Concilio, A., Naimpally, S.: Proximal convergence. Monatsh. Math. 103, 93–102 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  90. Willard, S.: General Topology. Dover Publications Inc, Mineola (1970). Xii + 369 pp, ISBN: 0-486-43479-6 54-02, MR0264581

    Google Scholar 

  91. Install, M., Weisstein, E.: Connected set. Mathworld. A Wolfram Web Resource (2015). http://mathworld.wolfram.com/ConnectedSet.html

  92. Leader, S.: Extensions based on proximity and boundedness. Math. Z. 108, 137–144 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  93. Leader, S.: Local proximity spaces. Mathematische Annalen 169, 275–281 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  94. Sierpiński, W.: Sur une courbe dont tout point est un point de ramification. C.R.A.S. 160, 302–305 (1915)

    Google Scholar 

  95. Wolfram, S.: A New Kind of Science. Wolfram Media Inc, Champaign (2002). Xiv + 1197 pp., MR1920418

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

Authors
  1. James F. Peters
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James F. Peters .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Peters, J.F. (2016). Computational Proximity. In: Computational Proximity. Intelligent Systems Reference Library, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-30262-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30262-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30260-7

  • Online ISBN: 978-3-319-30262-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation