Abstract
In this chapter, we concentrate on fuzzy geometry. Fuzzy geometry has been studied from different perspectives. The theory presented in this chapter is applicable to pattern recognition, computer graphics and image processing and follows closely the theory as developed by Rosenfeld, [37,47–51]. Buckley and Eslami, [7,8], are developing a fuzzy plane geometry which is quite different, but has the potential for applications in various fields of computer science.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ahuja, N., Davis, L.S., Milgram, D.L., and Rosenfeld, A., Piece-wise approximation of pictures using maximal neighborhoods, IEEE Trans. Comput. 27: 375–379, 1978.
Ahuja, N. and Hoff, W., Augmented medial axis transformation, Proc. 7th Intl. Conf. on Pattern Recognition, 336–338, 1984.
Bezdek, J.C., Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum, New York, 1981.
Bogomolny, A., On the perimeter and area of fuzzy sets, Fuzzy Sets and Systems 23: 241–246, 1987.
Boxer, L., On Hausdorff-like metrics for fuzzy sets, Pattern Recognition Letters 18: 15–118, 1997.
Buckley, J.J., Solving fuzzy equations, Fuzzy Sets and Systems 50: 114, 1992.
Buckley, J.J. and Eslami, E., Fuzzy plane geometry I: Points and lines, Fuzzy Sets and Systems 86: 179–187, 1997.
Buckley, J.J. and Eslami, E., Fuzzy plane geometry II: Circles and polygons, Fuzzy Sets and Systems 87: 79–85, 1997.
Buckley, J. J. and Qu, Y., Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38: 43–59, 1990.
Buckley, J.J. and Qu, Y., Solving fuzzy equations, a new solution concept, Fuzzy Sets and Systems 39: 291–301, 1991.
Buckley, J.J. and Qu, Y., Solving systems of fuzzy linear equations, Fuzzy Sets and Systems 43: 33–43, 1991.
Chaudhuri, B.B. and Rosenfeld, A., On a metric distance between fuzzy sets, Pattern Recognition Letters 17: 1157–1160, 1996.
Chaudhuri, B.B. and Rosenfeld, A., A modified Hausdorff distance between fuzzy sets, Information Sciences 118: 159–171, 1999.
De Luca, A. and Terminiu, S., A definition of non-probabilistic entropy in the setting of fuzzy set theory, Inform. and Control 20: 301–312, 1972.
Dubois, D. and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 38–40, 1980.
Dubois, D. and Prade, H., On distances between fuzzy points and their use for plausible reasoning, Proc. Internat. Conf. Systems Man Cybernet., 300–303, 1983.
Dubuisson, M. P. and Jain, A. K., A modified Hausdorff’ distance for object matching, in: Proc. 12th Int. Conf. on Pattern Recognition, 1: 566–568, 1994.
Dyer, C.R. and Rosenfeld, A., Thinning algorithms for gray-scale pictures, IEEE Trans. PAMI, 1: 88–89, 1979.
Fan, J., Note on Hausdorff’ like metrics for fuzzy sets, Pattern Recognition Letters 19: 793–796, 1998.
Gelfand, I.M. and Shilov, G.E., Generalized Functions, Academic Press, New York ( 1964, Vol. 1; 1968, Vol. 2 ).
Goetschel, R. and Voxman, W., Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10: 87–99, 1983.
Goetschel, R. and Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems 18: 31–43, 1986.
Gupta, K.C. and Ray, S., Fuzzy plane projective geometry, Fuzzy Sets and Systems 54: 191–206, 1993.
Heyting, A., Axiomatic Projective Geometry, North Holland, Amsterdam 1963.
Huttenlocher, D. P., Klanderman, G. A., and William, J. R., Comparing images using Hausdorff distances, IEEE Trans. PAMI, 15: 850863, 1993.
Kaufmann, A., Introduction to the Theory of Fuzzy Subsets - Fundamental Elements, vol. 1 Academic Press, New York, 1975.
Levi, G. and Montanari, U., A gray-weighted skeleton, Information and Control 17: 62–91, 1970.
Nakagawa, Y. and Rosenfeld, A., A note on the use of local min and max operations on digital picture processing, IEEE Trans. Systems Man Cybernet. 8: 632–635, 1978.
Pal, S.K., Fuzzy skeletonization of an image, Pattern Recognition Letters 10: 17–23, 1989.
Pal, S. K., A note on the quantitative measure of image enhancement through fuzziness, IEEE Trans. PAMI 4: 204–208, 1982.
Pal, S. K., A measure of edge ambiguity using fuzzy sets, Pattern Recognition Letters 4: 51–56, 1986.
Pal, S. K. and Chakraborty, B., Fuzzy set theoretic measure for automatic evaluation, IEEE Trans. SMC 16: 754–760, 1986.
Pal, S. K., King, R. A., and Hashim, A. A., Automatic grey level thresholding through index of fuzziness and entropy, Pattern Recognition Letters 1: 141–146, 1983.
Pal, S. K. and Majumder, D.D., Fuzzy Mathematical Approach to Pattern Recognition, Wiley ( Halsted Press ), New York, 1986.
Pal, S. K. and Pramanik, P. K., Fuzzy measures in determining seed points in clustering, Pattern Recognition Letters, 4: 159–164, 1986.
Pal, S.K. and Rosenfeld, A., Image enhancement and thresholding by optimization of fuzzy compactness, Pattern Recognition Letters 7: 77–86, 1988.
Pal, S.K. and Rosenfeld, A., A fuzzy medial axis transformation based on fuzzy disks, Pattern Recognition Letters 12: 585–590, 1991.
Paumard, J., Robust comparison of binary images, Pattern Recognition Letters 18: 1057–1063, 1997.
Peleg, S. and Rosenfeld, A., A min-max medial axis transformation, IEEE Trans. PAMI 3: 208–210, 1981.
Pickert, G., Projective Ebenen, Springer-Verlag, Berlin-GottingenHeidelberg, 1955.
Prewitt, J.M.S., Object enhancement and extraction. In: B. S. Lipkin and A. Rosenfeld, Eds., Picture Processing and Psychopictorics, Academic Press, New York, p. 121, 1970.
Puri, M. L. and Ralescu, D. A., Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications, 91: 552–558, 1983.
Rosenfeld, A., A note on perimeter and diameter in digital pictures, Inform. and Control 24: 384–388, 1974.
Rosenfeld, A., Fuzzy digital topology, Information and Control 40: 7687, 1979.
Rosenfeld, A., On connectivity properties of grayscale pictures, Pattern Recognition 16: 47–50, 1983.
Rosenfeld, A., The fuzzy geometry of image subsets, Pattern Recognition Letters 2: 311–317, 1984.
Rosenfeld, A., The diameter of a fuzzy set, Fuzzy Sets and Systems 13: 241–246, 1984.
Rosenfeld, A., Distances between fuzzy sets, Pattern Recognition Letters 3: 229–233, 1985.
Rosenfeld, A., Fuzzy rectangles, Pattern Recognition Letters 11: 677679, 1990.
Rosenfeld, A., Fuzzy plane geometry: triangles, Pattern Recognition Letters 15: 1261–1264, 1994.
Rosenfeld, A. and Haber, S., The perimeter of a fuzzy set, Pattern Recognition Letters 18: 125–130, 1985.
Rosenfeld, A. and Kak, A.C., Digital Picture Processing, Academic Press, New York, Vol. 2, 1980.
Rosenfeld, A. and Kak, A.C., Digital Picture Processing, Academic Press, New York, Sections 11.1.2, 11.2.1., 1982.
Rosenfeld, A. and Klette, R., Degree of adjancency or surroundedness, Pattern Recognition Letters 18: 169–177, 1985.
Royden, H.L., Real Analysis, Macmillan, New York, 2nd ed., 1968.
Serra, J., Image Analysis and Mathematical Morphology, Academic Press, London, 1982, Ch XII.
Shilov, G.E., Generalized Functions and Partial Differential Equations, Gordon and Breach, Science Publishers, Inc., New York 1968.
Smart, J.R., Modern Geometries, Wadsworth, Belmont, CA., 1978.
Thielman, H.P., Theory of Functions of Real Variables, Prentice Hall, New York, 1953.
Titchmarsh, E.C., Theory of Functions, Oxford University Press, 2nd ed., 1935.
Wang, S., Wu, A., and Rosenfeld, A., Image approximation from grayscale medial axis, IEEE Trans. PAMI 3: 687–696, 1981.
Zadeh, L.A., Fuzzy sets, Inform. Contr. 8: 338–353, 1965.
Zadeh, L. A., Calculus of fuzzy restrictions, IN: L. A. Zadeh et al., Eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press, London 1–39, 1975.
Zucker, S. W., Region growing: childhood and adolescence, Comput. Graphics Image Process. 5: 382–399, 1976.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mordeson, J.N., Nair, P.S. (2001). Fuzzy Geometry. In: Fuzzy Mathematics. Studies in Fuzziness and Soft Computing, vol 20. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1808-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-7908-1808-6_5
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-2494-0
Online ISBN: 978-3-7908-1808-6
eBook Packages: Springer Book Archive