Treasure Trove at Banacha. Set Patterns in Descriptive Proximity Spaces

• Autorzy: Peters J.F. , Ramanna S.
• Języki publikacji: EN

Abstrakty

EN

This paper introduces descriptive set patterns that originated from our visits with Zdzisław Pawlak and Andrzej Skowron at Banacha and environs in Warsaw. This paper also celebrates the generosity and caring manner of Andrzej Skowron, who made our visits to Warsaw memorable events. The inspiration for the recent discovery of descriptive set patterns can be traced back to our meetings at Banacha. Descriptive set patterns are collections of near sets that arise rather naturally in the context of an extension of Solomon Leader's uniform topology, which serves as a base topology for compact Hausdorff spaces that are proximity spaces. The particular form of proximity space (called EF-proximity) reported here is an extension of the proximity space introduced by V. Efremovič during the first half of the 1930s. Proximally continuous functions introduced by Yu.V. Smirnov in 1952 lead to pattern generation of comparable set patterns. Set patterns themselves were first considered by T. Pavlidis in 1968 and led to U. Grenander's introduction of pattern generators during the 1990s. This article considers descriptive set patterns in EF-proximity spaces and their application in digital image classification. Images belong to the same class, provided each image in the class contains set patterns that resemble each other. Image classification then reduces to determining if a set pattern in a test image is near a set pattern in a query image.

Słowa kluczowe

EN

descriptive set pattern; EF-proximity; Grenander pattern generator; near sets; proximally continuous function; proximity space; uniform topology

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2013
• Tom: Vol. 127, nr 1-4
• Strony: 357--367
• Opis fizyczny: Bibliogr. 30 poz., fot., rys.

Twórcy

autor

Peters J.F.

• Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, E1-526 EITC, 75A Chancellor's Circle, Winnipeg, MB R3T 5V6, Canada

autor

Ramanna S.

• Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, E1-526 EITC, 75A Chancellor's Circle, Winnipeg, MB R3T 5V6, Canada

Bibliografia

• [1] Beckman, F., Quarles, D.: On isometries of Euclidean space, Proc. Amer. Math. Soc., 4, 1953, 810-815.
• [2] Efremovic, V: The geometry of proximity I, Mat. Sb., 31, 1951,189-200 (in Russian), MR 14, 1106.
• [3] Engelking, R.: General Topology, Revised & completed edition, Heldermann Verlag, Berlin, 1989.
• [4] Grenander, U.: General Pattern Theory. A Mathematical Study of Regular Structures, Oxford Univ. Press, Oxford, UK, 1993, xxi + 883 pp.
• [5] Grunbaum, B., Shepard, G.: Tilings and Patterns, W.H. Freeman and Co., New York, 1987, ix + 700 pp.
• [6] Hausdorff, F.: Set Theory, AMS Chelsea Publishing, Providence, RI, 1914, 352 pp.
• [7] Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis, Morgan- KaufmannPub., Amsterdam, The Netherlands, 2004.
• [8] Leader, S.: On clusters in proximity spaces, Fundamenta Mathematicae, 47, 1959, 205-213.
• [9] Naimpally, S., Peters, J.: Topology with Applications. Topological Spaces Via Near and Far, World Scientific, Singapore, 2013.
• [10] Naimpally, S., Warrack, B.: Proximity Spaces, Cambridge University Press, Cambridge,UK, 1970, X+128 pp., ISBN 978-0-521-09183-1.
• [11] Pavlidis, T.: Analysis of set patterns, Pattern Recog., 1, 1968,165-178.
• [12] Pawlak, Z., Peters, J.: Jakblisko (How near), Systemy Wspomagania Decyzji, I, 2002,2007, 57, 109, ISBN 83-920730-4-5.
• [13] Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning, Inf. Sci., 177(1), 2007, 41-73.
• [14] Pawlak, Z., Skowron, A.: Rough sets: Some extensions, Inf. Sci., 177(1), 2007, 28-40.
• [15] Pawlak, Z., Skowron, A.: Rudiments of rough sets, Inf. Sci., 177(1), 2007, 3-27.
• [16] Peters, J.: Local near sets. Pattern discovery in proximity spaces, Math. Comp. Sci., 7(1), 2013, 87-106, DOI 10.1007/s11786-013-0143-z.
• [17] Peters, J.: Near sets. An introduction, Math. in Comp. Sci., 7(1), 2013, 3-9, DOI 10.1007/s11786- 013-0149-6.
• [18] Peters, J.: Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces, Springer, Berlin, 2013.
• [19] Peters, J., Hettiarachichi, R.: Visual motif patterns in separation spaces, Theory and Applications of Mathematics in Computer Sci., 3(2), 2013, 36-58.
• [20] Peters, J., Naimpally, S.: Applications of near sets, Notices of the Amer Math. Soc., 59(4), 2012, 536-542.
• [21] Peters, J., Skowron, A., Stepaniuk, J.: Nearness in approximation spaces, Proc. Concurrency, Specification and Programming (CS&P 2006), Humboldt Universität, 2006.
• [22] Peters, J., Skowron, A., Stepaniuk, J.: Nearness of objects: Extension of approximation space model, Fundamenta Informaticae, 79(3-4), 2007, 497-512.
• [23] Skowron, A.: On topology in information systems, Bull. Polish Acad. of Sci. Math., 36(7-8), 1988, 477-479, Communicated by Z. Pawlak, 11 March 1988.
• [24] Skowron, A., Stepaniuk, J.: Approximations and Classifiers, Rough Sets and Current Trends in Computing - 7th International Conference, RSCTC 2010, Warsaw, Poland, June 28-30,2010. Proceedings, LNCS 6086, 2010, ISBN 978-3-642-13528-6.
• [25] Skowron, A., Stepaniuk, J., Peters, J. F., Swiniarski, R. W.: Calculi of Approximation Spaces, Fundam. Inform., 72(1-3), 2006, 363-378.
• [26] Skowron, A., Stepaniuk, J., Swiniarski, R. W.: Modeling rough granular computing based on approximation spaces, Inf Sci., 184(1), 2012, 20-43.
• [27] Skowron, A., Synak, P.: Complex Patterns, Fundam. Inform., 60(1-4), 2004, 351-366.
• [28] Skowron, A., Wasilewski, P.: Interactive information systems: Toward perception based computing, Theor. Comput. Sci., 454, 2012, 240-260.
• [29] Smirnov, Y. M.: On proximity spaces, Math. Sb. (N.S.), 31(73), 1952, 543-574, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 5-35.
• [30] Weihrauch, K.: Computable separation in topology, from T0 to T2, J. Univ. Comp. Sci., 16(18), 2010, 27332753.