Skip to main content
SpringerLink
Log in
Book cover

Computational Geometry, Topology and Physics of Digital Images with Applications pp 223–269Cite as

Surface Shapes and Their Proximities

  • Chapter
  • First Online:

  • 711 Accesses

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 162))

Abstract

This chapter introduces of two basic types of proximities in the study of relationships between sub-complexes in cell complexes, namely, spatial and descriptive proximities. These proximities are useful in clustering and separating subcomplexes in triangulated finite, bounded surface regions such as those found in visual scenes. This chapter introduces a number of connectedness proximities useful in probing, analyzing, comparing and classifying cell complexes on triangulated surface regions.

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   149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
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.

    Many thanks to Braden Cross for the webcam image in Fig. 5.2, captured using the Matlab Computer Vision System toolbox and Matlab implementation of the Canny edge detection algorithm. 

  2. 2.

    Many thanks to M.Z. Ahmad for the LaTeX script used to display this bar graph, which does not depend on an external file.

  3. 3.

    Many thanks to Ron Enns for this picture of an Ontario, Canada maple tree, captured with a cell phone during the Thanksgiving holiday in October, 2018.

References

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

  2. Naimpally, S., Warrack, B.: Proximity Spaces. Cambridge Tract in Mathematics, vol. 59. Cambridge University Press, Cambridge (1970). X+128 pp. Paperback (2008), MR0278261

    Google Scholar 

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

    Google Scholar 

  4. Naimpally, S.: Near and far. A centennial tribute to Frigyes Riesz. Sib. Electron. Math. Rep. 2, 144–153 (2009)

    Google Scholar 

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

    Google Scholar 

  6. Concilio, A.D.: Proximal set-open topologies on partial maps. Acta Math. Hungar. 88(3), 227–237 (2000). MR1767801

    Google Scholar 

  7. Concilio, A.D.: Topologizing homeomorphism groups of rim-compact spaces. Topol. Its Appl. 153(11), 1867–1885 (2006)

    Article  MathSciNet  Google Scholar 

  8. Concilio, A.D.: 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 486, pp. 89–114. American Mathematical Society (2009)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  12. Peters, J., Ramanna, S.: Pattern discovery with local near sets. In: R. Alarcón, P. Barceló (eds.) Proceedings of Jornadas Chilenas de Computación 2012 workshop on pattern recognition, pp. 1–4. The Chilean Computing Society, Valparaiso (2012)

    Google Scholar 

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

  14. 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 

  15. Di Maio, G., S.A. Naimpally, E.: Theory and Applications of Proximity, Nearness and Uniformity. Seconda Università di Napoli, Napoli, Italy (2009). 264 pp. MR1269778

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Smirnov, J.M.: On proximity spaces. Math. Sb. (N.S.) 31(73), 543–574 (1952). English translation: Am. Math. Soc. Trans. Ser. 2, 38, 1964, 5-35

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  20. Peters, J.: Computational Proximity. Excursions in the Topology of Digital Images. Intelligent Systems Reference Library, vol. 102 (2016). Xxviii+433 pp. https://doi.org/10.1007/978-3-319-30262-1, MR3727129 and Zbl 1382.68008

  21. Murphy, J., MacManus, D.: Ground vortex aerodynamics under crosswind conditions. Exper. Fluids 50(1), 109–124 (2011)

    Article  Google Scholar 

  22. Barata, J., N. Bernardo, P.S., Silva, A.: Experimental study of a ground vortex: the effect of the crossflow velocity. In: 49th AIAA Aerospace Sciences Meeting, pp. 1–9. AIAA (2011)

    Google Scholar 

  23. Silva, A., ao, D.D., Barata, J., Santos, P., Ribeiro, S.: Laser-doppler analysis of the separation zone of a ground vortex flow. In: 14th Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, pp. 7–10. Universidade Beira Interior (2008)

    Google Scholar 

  24. P.R. Spalart M. Kh. Strelets, A.T., Slur, M.: Modeling the interaction of a vortex pair with the ground. Fluid Dyn. 36(6), 899–908 (1999)

    Google Scholar 

  25. Dzedolik, I.: Vortex properties of a photon flux in a dielectic waveguide. Tech. Phys. 75(1), 137–140 (2005)

    Google Scholar 

  26. van Leunen, H.: The hilbert book model project. Technical report, Deparment of Applied Physics, Technische Universiteit Eindhoven (2018). https://www.researchgate.net/project/The-Hilbert-Book-Model-Project

  27. Litchinitser, N.: Structured light meets structured matter. Sci. New Ser. 337(6098), 1054–1055 (2012)

    Google Scholar 

  28. Hance, M.: Algebraic structures on nearness approximation spaces. Ph.D. thesis, University of Pennsylvania, Department of Physics and Astronomy (2015). Supervisor: H.H. Williams, vii+113 pp

    Google Scholar 

  29. Ahmad, M., Peters, J.: Descriptive unions. A fibre bundle characterization of the union of descriptively near sets. 1–19 (2018). arXiv:1811.11129v1

  30. Cottet, G.H., Koumoutsakos, P.: Vortex methods. Theory and Practice. Cambridge University Press, Cambridge (2000). xiv+313 pp. ISBN:0-521-62186-0, MR1755095

    Google Scholar 

  31. Tian, S., Gao, Y., Dong, X., Liu, C.: A definition of vortex vector and vortex. 1–26 (2017). arXiv:1712.03887

  32. Nye, J.: Events in fields of optical vortices: rings and reconnection. J. Opt. 18, 1–11 (2016). https://doi.org/10.1088/2040-8978/18/10/105602

    Article  Google Scholar 

  33. Dudley, J., Dias, F., Erkintalo, M., Gentry, G.: Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014). www.nature.com/naturephotonics

  34. Concilio, A.D., Guadagni, C., Peters, J., Ramanna, S.: Descriptive proximities. properties and interplay between classical proximities and overlap. Math. Comput. Sci. 12(1), 91–106 (2018). MR3767897, Zbl 06972895

    Google Scholar 

  35. Peters, J.: Local near sets: pattern discovery in proximity spaces. Math. Comput. Sci. 7(1), 87–106 (2013). https://doi.org/10.1007/s11786-013-0143-z, MR3043920, ZBL06156991

  36. Lu, J., Li, J., Yan, Z., Zhang, C.: Zero-shot learning by generating pseudo feature representations. 1–18 (2017). arXiv:1703.06389v1

  37. Molina, M., Sánchez, J.: Zero-shot learning with partial attributes. In: Brito-Loeza, C., Espinosa-Romero, A. (eds.) Intelligent Computing Systems. ISICS 2018. Communications in Computer and Information Science, vol. 820, pp. 147–158. Springer Nature, Switzerland AG (2018). https://doi-org.uml.idm.oclc.org/10.1007/978-3-319-76261-6_12

  38. Hariharan, B., Girshick, R.: Low-shot visual recognition by shrinking and hallucinating features. 1–10 (2016). arXiv:1606.02819v4

  39. Peters, J.: Proximal relator spaces. Filomat 30(2), 469–472 (2016). https://doi.org/10.2298/FIL1602469P

    Article  MathSciNet  MATH  Google Scholar 

  40. Száz, A.: Basic tools and mild continuities in relator spaces. Acta Math. Hungar. 50(3–4), 177–201 (1987). MR0918156

    Google Scholar 

  41. Leader, S.: Local proximity spaces. Math. Ann. 169, 275–281 (1967)

    Article  MathSciNet  Google Scholar 

  42. Peters, J.: Proximal relator spaces. Filomat 30(2), 469–472 (2016). MR3497927

    Google Scholar 

  43. Peters, J.: Computational Proximity. Excursions in the Topology of Digital Images. Intelligent Systems Reference Library, vol. 102. Springer, Berlin (2016). viii+445 pp. https://doi.org/10.1007/978-3-319-30262-1

  44. Peters, J.: Proximal vortex cycles and vortex nerve structures. non-concentric, nesting, possibly overlapping homology cell complexes. J. Math. Sci. Model. 1(2), 56–72 (2018). ISSN 2636-8692, www.dergipark.gov.tr/jmsm, See, also, https://arxiv.org/abs/1805.03998

  45. Worsley, A., Peters, J.: Enhanced derivation of the electron magnetic moment anomaly from the electron charge using geometric principles. Appl. Phys. Res. 10(6), 1–14 (2018). http://apr.ccsenet.org

  46. Dennis, M., King, R., Jack, B., Holleran, K., Padgett, M.: Isolated optical vortex knots. Nat. Phys. 6(2), 118–121 (2010). https://doi.org/10.1103/PhysRevD.81.066004

    Article  Google Scholar 

  47. Pike, O., Mackenroth, F., Hill, E., Rose, S.: A photon-photon collider in a vacuum hohlraum. Nat. Photon. 8(6), 434–436 (2014). https://doi.org/10.1038/nphoton.2014.95

    Article  Google Scholar 

  48. Worsley, A.: The formulation of harmonic quintessence and a fundamental energy equivalence equation. Phys. Essays 23(2), 311–319 (2010). https://doi.org/10.4006/1.3392799

    Article  MathSciNet  Google Scholar 

  49. Worsley, A.: Harmonic quintessence and the derivation of the charge and mass of the electron and the proton and quark masses. Phys. Essays 24(2), 240–253 (2011). https://doi.org/10.4006/1.3567418

    Article  Google Scholar 

  50. Worsley, A.: The formulation of harmonic quintessence and a fundamental energy equivalence equation. Phys. Essays 23(2), 311–319 (2010). https://doi.org/10.4006/1.3392799, ISSN 0836-1398

  51. Nye, J.: Natural Focusing and Fine Structure of Light. Caustics and Dislocations. Institute of Physics Publishing, Bristol (1999). xii+328 pp. MR1684422

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Engineering and Information Technology Complex, 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

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Peters, J.F. (2020). Surface Shapes and Their Proximities. In: Computational Geometry, Topology and Physics of Digital Images with Applications. Intelligent Systems Reference Library, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-030-22192-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation