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