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Year 2019, Volume: 2 Issue: 3, 213 - 218, 30.09.2019
https://doi.org/10.33434/cams.541413
https://izlik.org/JA59SY36YW

Abstract

References

  • [1] I. Dochviri, J.F. Peters, Topological sorting of finitely near sets, Math. Comput. Sci., 10(2) (2016), 273–277. 1
  • [2] E. Andres, T. Roussillon, Analytical description of digital circles, Lecture Notes Comput. Sci., 6607 (2011), 901–917. 1
  • [3] M.D. McIlroy, Best approximate circles on integer grids, ACM Transactions Graph., 2(4) (1983), 237–263. 1
  • [4] J.-L. Toutant, E. Andres, T. Roussillon, Digital circles, spheres and hyperspheres: from morphological models to analytical characterizations and topological properties, Discrete Appl. Math., 161(16-17) (2011), 2662–2677. 1
  • [5] O. Fujita, Metrics based on average distance between sets, Jpn. J. Ind. Appl. Math., 30(1) (2013), 1–19. 2
  • [6] A. Gardner, J. Kanno, C.A. Duncan, R. Selmic, Measuring distance between unordered sets of different sizes, Proceeding 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014), 137–143. 2
  • [7] S. Kosub, A note on the triangle inequality for the Jaccard distance, Pattern Recognition Letters, 120 (2019), 36–38. 2
  • [8] M. Sharir, Intersection and closest-pair problems for a set of planar discs, SIAM J. Comput., 14(2) (1985), 448–468. 2
  • [9] E. Bishop, R.R. Phelps, The support functionals of a convex set, 1963 Proc. Sympos. Pure Math., Amer. Math. Soc., VII (1985), 27–35. 2, 2.5
  • [10] J. F. Peters, Proximal planar shape signatures. Homology nerves and descriptive proximity, Adv. Math.: Sci. J., 6(2) (2017), 71–85. 2
  • [11] P. Alexandroff, Elementary Concepts of Topology, Dover Publications, New York, 1965. 3
  • [12] P. Alexandroff, Simpliziale approximationen in der allgemeinen topologie, Math. Ann., 101(1) (1926), 452–456. 3
  • [13] P.S. Alexandrov, Combinatorial Topology, Graylock Press, Baltimore, Md, USA, 1956. 3
  • [14] P. Alexandroff, H. Hopf, Topologie. Band I, Springer, Berlin, 1935. 3
  • [15] J.F. Peters, Proximal planar shapes. Correspondence between triangulated shapes and nerve complexes, Bull. Allahabad Math. Soc., 33(1) (2018), 113–137. 3

Computable Proximity of $\ell_1$-Discs on the Digital Plane

Year 2019, Volume: 2 Issue: 3, 213 - 218, 30.09.2019
https://doi.org/10.33434/cams.541413
https://izlik.org/JA59SY36YW

Abstract

This paper investigates problems in the characterization of the proximity of digital discs.  Based on the $l_1$-metric structure for the 2D digital plane and using a Jaccard-like metric, we determine numerical characters for intersecting digital discs.

References

  • [1] I. Dochviri, J.F. Peters, Topological sorting of finitely near sets, Math. Comput. Sci., 10(2) (2016), 273–277. 1
  • [2] E. Andres, T. Roussillon, Analytical description of digital circles, Lecture Notes Comput. Sci., 6607 (2011), 901–917. 1
  • [3] M.D. McIlroy, Best approximate circles on integer grids, ACM Transactions Graph., 2(4) (1983), 237–263. 1
  • [4] J.-L. Toutant, E. Andres, T. Roussillon, Digital circles, spheres and hyperspheres: from morphological models to analytical characterizations and topological properties, Discrete Appl. Math., 161(16-17) (2011), 2662–2677. 1
  • [5] O. Fujita, Metrics based on average distance between sets, Jpn. J. Ind. Appl. Math., 30(1) (2013), 1–19. 2
  • [6] A. Gardner, J. Kanno, C.A. Duncan, R. Selmic, Measuring distance between unordered sets of different sizes, Proceeding 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014), 137–143. 2
  • [7] S. Kosub, A note on the triangle inequality for the Jaccard distance, Pattern Recognition Letters, 120 (2019), 36–38. 2
  • [8] M. Sharir, Intersection and closest-pair problems for a set of planar discs, SIAM J. Comput., 14(2) (1985), 448–468. 2
  • [9] E. Bishop, R.R. Phelps, The support functionals of a convex set, 1963 Proc. Sympos. Pure Math., Amer. Math. Soc., VII (1985), 27–35. 2, 2.5
  • [10] J. F. Peters, Proximal planar shape signatures. Homology nerves and descriptive proximity, Adv. Math.: Sci. J., 6(2) (2017), 71–85. 2
  • [11] P. Alexandroff, Elementary Concepts of Topology, Dover Publications, New York, 1965. 3
  • [12] P. Alexandroff, Simpliziale approximationen in der allgemeinen topologie, Math. Ann., 101(1) (1926), 452–456. 3
  • [13] P.S. Alexandrov, Combinatorial Topology, Graylock Press, Baltimore, Md, USA, 1956. 3
  • [14] P. Alexandroff, H. Hopf, Topologie. Band I, Springer, Berlin, 1935. 3
  • [15] J.F. Peters, Proximal planar shapes. Correspondence between triangulated shapes and nerve complexes, Bull. Allahabad Math. Soc., 33(1) (2018), 113–137. 3
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

James F. Peters 0000-0002-1026-4638

K. Kordzaya This is me

İ. Dochviri This is me

Submission Date March 18, 2019
Acceptance Date August 1, 2019
Publication Date September 30, 2019
DOI https://doi.org/10.33434/cams.541413
IZ https://izlik.org/JA59SY36YW
Published in Issue Year 2019 Volume: 2 Issue: 3

Cite

APA Peters, J. F., Kordzaya, K., & Dochviri, İ. (2019). Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences, 2(3), 213-218. https://doi.org/10.33434/cams.541413
AMA 1.Peters JF, Kordzaya K, Dochviri İ. Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences. 2019;2(3):213-218. doi:10.33434/cams.541413
Chicago Peters, James F., K. Kordzaya, and İ. Dochviri. 2019. “Computable Proximity of $\ell_1$-Discs on the Digital Plane”. Communications in Advanced Mathematical Sciences 2 (3): 213-18. https://doi.org/10.33434/cams.541413.
EndNote Peters JF, Kordzaya K, Dochviri İ (September 1, 2019) Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences 2 3 213–218.
IEEE [1]J. F. Peters, K. Kordzaya, and İ. Dochviri, “Computable Proximity of $\ell_1$-Discs on the Digital Plane”, Communications in Advanced Mathematical Sciences, vol. 2, no. 3, pp. 213–218, Sept. 2019, doi: 10.33434/cams.541413.
ISNAD Peters, James F. - Kordzaya, K. - Dochviri, İ. “Computable Proximity of $\ell_1$-Discs on the Digital Plane”. Communications in Advanced Mathematical Sciences 2/3 (September 1, 2019): 213-218. https://doi.org/10.33434/cams.541413.
JAMA 1.Peters JF, Kordzaya K, Dochviri İ. Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences. 2019;2:213–218.
MLA Peters, James F., et al. “Computable Proximity of $\ell_1$-Discs on the Digital Plane”. Communications in Advanced Mathematical Sciences, vol. 2, no. 3, Sept. 2019, pp. 213-8, doi:10.33434/cams.541413.
Vancouver 1.Peters JF, Kordzaya K, Dochviri İ. Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences [Internet]. 2019 Sept. 1;2(3):213-8. Available from: https://izlik.org/JA59SY36YW

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