Abstract
The concept of “nearness”, which has been dealt with as soon as one started studying digital images, finds one of its rigorous forms in the notion of proximity space. It is this notion, together with “nearness preserving mappings”, that we investigate in this paper. We first review basic examples as they naturally occur in digital topologies, making also brief comparison studies with other concepts in digital geometry. After this we characterize proximally continuous mappings in metric spaces. Finally, we show by example that the “proximite complexity” of a finite covering in a digital picture may be too high to be adequately depicted in a finite topological space. This combinatorial result may indicate another conceptual advantage of proximities over topologies.
This work was supported by the Austrian Science Foundation under grant S 7002-MAT, by the project of Czech Ministry of Education No. VS96049, and by the Grant Agency of the Czech Republic under the grant GACR 102/00/1679.
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P. Alexandro.: Elementary Concepts of Topology, A translation of P. Alexandro., 1932, by A. E. Farley, Dover, New York, 1961
L. Boxer: Digitally continuous functions, Pattern Recognition Lett. 15 (1994), 833–839 69
E. Cech: Topological Spaces, J. Wiley-Interscience Publ., New York, (1966) 70, 71, 72
J.-M. Chassery: Connectivity and consecutivity in digital pictures, Computer Graphics and Image Processing 9 (1979), 294–300 69
J. Isbell: Uniform Spaces, Publ. AMS, Providence, (1968) 70, 72, 73
H. Herrlich: A concept of nearness, General Topology Appl. 5 (1974), 191–212 70
E. R. Khalinsky, E. R. Kopperman and P. R. Meyer: Computer graphics and connected topologies on finite ordered sets, Topology Appl. 36, 117, (1990)
T. Y. Kong and A. Rosenfeld: Digital topology: introduction and survey, Computer Vision, Graphics, and Image Processing 48 (1989), 367–393 69, 75
T. Y. Kong: On the problem of determining whether a parallel reduction operator of n-dimensional binary images always preserves topology, Proc. SPIE’s Conf. on Vision Geometry, (1993), 69–77 69
V. A. Kovalevski: Finite topology as applied to image analysis, Computer Vision, Graphics and Image Processing 46, (1989), 141–161 75
W. Kropatsch, P. Pták: The path-connectedness in Z2 and Z3 and classical topologies. Advances in Pattern Recognition, Joint IAPR International Workshops SSPR’98 and SPR’98, Sydney 1998, Springer Verlag-Lecture Notes in Computer Science Vol. 1451, 181–189 71, 75, 76
L. Latecki, F. Prokop: Semi-proximity continuous functions in digital images, Pattern Recognition Letters 16 (1995), 1175–1187 69, 70, 71
L. Latecki: Topological connectedness and 8-connectedness in digital pictures, Computer Vision, Graphics and Image Processing: Image Understanding 57, (1993), 261–262 69, 75, 76
S. A. Naimpally and B. D. Warrack: Proximity Spaces, Cambridge Univ. Press, Cambridge 1970 70
J. Pelant, P. Pták: The complexity of ó-discretely decomposable families in uniform spaces, Comm. Math. Univ. Carolinae 22 (1981), 317–326
F. Riesz: Stetigkeitsbegri. und abstrakte Mengenlehre, Atti IV Congresso Internazionale dei Matematici, Roma, 1908, Vol. II, 18–24 70
A. Rosenfeld: Digital topology, Am. Math. Monthly 86, (1979), 621–630 73
A. Rosenfeld: “Continuous” functions on digital pictures, Pattern Recognition Lett. 4, (1986), 177–184 69, 73, 75
A. Rosenfeld and J. L. Pfaltz: Sequential operations in digital picture processing, J. Assoc. Comput. Mach. 13, (1966), 471–494 73
F. Wyse and D. Marcus et al.: Solution to Problem 5712, Am. Math. Monthly 77, 1119, 1970 71, 75
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Pták, P., Kropatsch, W.G. (2000). Nearness in Digital Images and Proximity Spaces. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds) Discrete Geometry for Computer Imagery. DGCI 2000. Lecture Notes in Computer Science, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44438-6_7
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