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

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

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 Articles
Authors

James F. Peters 0000-0002-1026-4638

K. Kordzaya This is me

İ. Dochviri This is me

Publication Date September 30, 2019
Submission Date March 18, 2019
Acceptance Date August 1, 2019
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 Peters JF, Kordzaya K, Dochviri İ. Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences. September 2019;2(3):213-218. doi:10.33434/cams.541413
Chicago Peters, James F., K. Kordzaya, and İ. Dochviri. “Computable Proximity of $\ell_1$-Discs on the Digital Plane”. Communications in Advanced Mathematical Sciences 2, no. 3 (September 2019): 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 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, 2019, doi: 10.33434/cams.541413.
ISNAD Peters, James F. et al. “Computable Proximity of $\ell_1$-Discs on the Digital Plane”. Communications in Advanced Mathematical Sciences 2/3 (September 2019), 213-218. https://doi.org/10.33434/cams.541413.
JAMA 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, 2019, pp. 213-8, doi:10.33434/cams.541413.
Vancouver Peters JF, Kordzaya K, Dochviri İ. Computable Proximity of $\ell_1$-Discs on the Digital Plane. Communications in Advanced Mathematical Sciences. 2019;2(3):213-8.

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