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

Year 2019, Volume: 2 Issue: 3, 213 - 218, 30.09.2019

James F. Peters , K. Kordzaya İ. Dochviri

https://doi.org/10.33434/cams.541413

Cited By: 1

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.

Keywords

Digital discs, $l_1$-metric, Jaccard like metric

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