On Generalized Digital Topology and Root Images of Median Filters

Year 2018, Issue: 20, 13 - 26, 03.01.2018

Osama Tantawy Sobhy El Shiekh Mohamed Yakout Sawsan El Sayed

Abstract

In this paper, we extend the concepts of semi-open sets and λ-open sets in the digital topology. In addition, we introduce the concepts of regular semi-open and regular λ-open sets. A relationship between digital topology and image processing is established.

Keywords

Digital topology, median filter, root image, regular open

References

• [1] M. E. Abd El Monsef, A. M. Kozae, M. J. Iqelan, Near approximations in topological spaces, Int. Journal of Math. Analysis, Vol. 4, no. 6, p.p. 279-290, 2010.
• [2] P. Alexandroff, Diskrete Räume, Mathematicheskii Sbornik (Rwceul Mathématique), Vol.2, no.3, p.p 502-519, 1937.
• [3] A. Alpers, Digital topology: Regular sets and root image of cross-median filter, Journal of Mathematical imaging vision, Vol. 17, p.p. 7-14, 2002.
• [4] Arenas, F. G. Dontchev, J. and Ganster. M, on λ–sets and dual of generalized continuity. Questions Answers. Gen. Topology. No 15, p.p 3-13, 1997.
• [5] M. Caidas, S. Jafri, G. Navalagi, More on λ-closed sets in topological spaces, Revesita Colombinnade Mathemάtices, Vol. 41. No. 2, p.p. 355-369, 2007.
• [6] H. U. Döhler, Generation of root signals of two-dimensional median filters, Signal Processing, Vol. 18, p.p. 269-276, 1989. In: Ulrich Eckhardt, Root images of median filters, Journal of Mathematical Imaging and Vision, No. 19, p.p 63-70, 2003.
• [7] U. Eckhardt, L. J. Latecki, Topologies for the digital spaces and, Computer vision and image understanding, Vol. 90, p.p. 295-312,2003.
• [8] U. Eckhardt, Root image of median filters: Semi-Topological Approach, T. Asano et al. (Eds): Geometry, Morphology, LNCS 2616, p.p. 176-195, Springer Verlag Berlin Heidelberg, 2003.
• [9] U. Eckhardt, Root Images of Median Filters, Journal of Mathematical Imaging and Vision, Vol. 19, p.p. 63-70, 2003.
• [10] E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl. Vol. 36, p.p. 1-17, 1990.
• [11] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, Vol. 70, p.p. 36-41, 1963. In: M. E. Abd El-Monsef, A M. Kozea, M. J. Iqelan, Near approximations in topological spaces, Int. Journal of Math. Analysis, Vol. 4, No 6, p.p 279-290, 2010.
• [12] Maki, H. Generalized -sets and the associated closure operator. The Special Issue in commemoration of Prof. Kazusada Ikeda's Retirement, Vol. 1, p.p 139-146, 1986. In: Miguel Caladas, Saeid Jafari, Govindappa Navalagi, More on λ-closed sets in topological spaces, Revisita Colombiana de Mathemáticas, Vol. 41, p.p. 355-369, no. 2, 2007.
• [13] D. Marcus. F. Wyse et al., A special topology for the integers (Problem 5712). Amer. Math. Monthly, Vol. 77, p.p. 85-1119, 1970. In: Ulrich Eckhardt, Longin J. Latecki, Topologies for the digital spaces and, Computer Vision and Image Understanding, Vol. 90, p.p. 295-312, 2003.
• [14] S. I. Nada, Semi-open and semi-closed sets in digital spaces, Commun. Fac.Sci. Univ. Ank. Series, Vol 53, no.1, p.p.1-6, 2004.
• [15] A. Rosenfeld, Digital topology, American Mathematical Monthly, vol. 86, p.p. 621-630, 1979.
• [16] Tuckey, J. W.: Exploratory Data Analysis. Addision-Wesley, Reading. Mass. Vol. 177, 1977. In: Ulrich Eckhardt, Root images of Median filters-Semi-topological approach, Geomtry Morphology, LNCS 2616, P.P. 176-195, 2003, Springer-Verlag. Berlin Heidelberg, 2003.