Abstract
The main problem we persue in this paper is the question of when a given path-connectedness in Z 2 and Z 3 coincides with a topological connectedness. We answer this question provided the path-connectedness is induced by a homogeneous and symmetric neighbourhood structure. On the way we make a study of topological structures, arguing that the point-neighbourhood formalism can be well applied in the digital picture investigations.
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References
J. Adámek, H. Herrlich and G. Strecker: Abstract and concrete categories, J. WileyInterscience Publ., New York, 1990
E. ech: Topological Spaces, J. Wiley-Interscience Publ., New York, 1969
U. Eckhardt, L. Latecki: Digital topology. Trends in Pattern Recognition, Council of Scientific Information, Vilayil Gardens, Trilandrun, India 1994
J. M. Chassery: Connectivity and consecutivity in digital pictures, Computer Graphics and Image Processing 9, 294–300, 1979
E. R. Khalinsky, E. R. Kopperman and P. R. Meyer: Computer graphics and connected topologies on finite ordered sets, Topology Appl. 36, 117, 1980
T. Y. Kong, A. W. Roscoe and A. Rosenfeld: Concept of digital topology, Topology Appl. 46 (1992), 219–262
V. A. Kovalevski: Finite topology as applied to image analysis, Computer Vision, Graphics and Image Processing 45, (1989), 141–161
Mac Lane: Categories for the working mathematician, Springer-Verlag, 1971
L. Latecki: Digitale und Allgemeine Topologie in der bildhaven Wissensrepräsentation, Thesis, Hamburg, 1992
L. Latecki: Topological connectedness and 8-connectedness in digital pictures, Computer Vision, Graphics and Image Processing: Image Understanding 57, (1993), 261–262
P Pták, H. Koffer, W. Kropatsch: Digital topologies revisited (the approach based on the topological point-neighbourhood), In: Proc. 7th Discrete Geometry for Computer Imagery, Montpellier 1997, 151–159
A. Rosenfeld: Digital topology, Am. Math. Monthly 86, (1979), 621–630
F. Wyse and D. Marcus et al.: Solution to Problem 5712, Am. Math. Monthly 77, 1119, 1970.
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Kropatsch, W., Pták, P. (1998). The path-connectedness in Z 2 and Z 3 and classical topologies. In: Amin, A., Dori, D., Pudil, P., Freeman, H. (eds) Advances in Pattern Recognition. SSPR /SPR 1998. Lecture Notes in Computer Science, vol 1451. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0033236
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DOI: https://doi.org/10.1007/BFb0033236
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