Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T00:59:38.158Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity properties of attractors for iterated fuzzy set systems

Published online by Cambridge University Press:  17 February 2009

B. Forte ,
M. Lo Schiavo  and
E. R. Vrscay
B. Forte
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1.
M. Lo Schiavo
Affiliation:
Metodi e Mod. Mat. Sc. Appl., Universita' di Roma “La Sapienza”, 00161 Rome, Italy.
E. R. Vrscay
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An N-map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, Φ), is a system of N contraction maps wi: X → X over a compact metric space (X, d), with associated “grey level” maps øi: [0, 1] → [0, 1]. Associated with an IFZS (w, Φ) is a fixed point u ∈ f*(X), the class of normalized fuzzy sets on X, u: X → [0, 1]. We are concerned with the continuity properties of u with respect to changes in the wi, and the φi. Establishing continuity for the fixed points of IFZS is more complicated than for traditional Iterated Function Systems (IFS) with probabilities since a composition of functions is involved. Continuity at each specific attractor u may be established over a suitably restricted domain of φi maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 175 - 193
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Barnsley, M. F., Fractals everywhere (Academic Press, New York, 1988).Google Scholar
[2]Barnsley, M. F., Fractal image compression (A. K. Peters, Wellesley, MA, 1993).Google Scholar
[3]Barnsley, M. F., Hardin, D., Ervin, V. and Lancaster, J., “Solution of an inverse problem for fractals and other sets”, Proc. Nat. Acad. Sci. USA 83 (1986) 1975–1977.CrossRefGoogle ScholarPubMed
[4]Cabrelli, C. A., Forte, B., Molter, U. M. and Vrscay, E. R., “Iterated fuzzy set systems: a new approach to the inverse problem for fractals and other sets”, J. Math. Anal. Appl. 171 (1992) 79–100.CrossRefGoogle Scholar
[5]Cabrelli, C. A. and Molter, U. M., “Density of fuzzy set attractors: a step towards the solution of the inverse problem for fractals and other sets”, Proc. NATO Advanced Study Institute on Probabilistic and Stochastic Methods in Analysis with Applications (07 1991).Google Scholar
[6]Centore, P. and Vrscay, E. R., “Continuity of attractors and invariant measures for iterated function systems”, Can. Math. Bull. (1993), to appear.Google Scholar
[7]Diamond, P. and Kloeden, P., “Metric spaces of fuzzy sets”, Fuzzy Sets and Systems 35 (1990) 241–249.CrossRefGoogle Scholar
[8]Forte, B. and Vrscay, E. R., “Approximation of functions and images in L1(X) using iterated function systems”, in preparation.Google Scholar
[9]Hutchinson, J., “Fractals and self-similarity”, Indiana Univ. J. Math. 30 (1981) 713–747.CrossRefGoogle Scholar
[10]Jacquin, A., “A fractal theory of iterated Markov operators with applications to digital image coding”, Ph. D. Thesis, Georgia Institute of Technology, 1989.Google Scholar
[11]Vrscay, E. R., “Iterated function systems: theory, application, and the inverse problem”, in Proc. NATO Advanced Study Institute on Fractal Geometry and Analysis (eds. Béelair, J. and Dubuc, S.), NATO ASI Series C 346, (Kluwer, Dordrecht, The Netherlands, 1991), 405–468.CrossRefGoogle Scholar