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Minimal Simplex for IFS Fractal Sets

  • Elena Babače
  • Ljubiša Kocić
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)

Abstract

Fractal sets manipulation and modeling is a difficult task due to their complexity and unpredictability. One of the basic problems is to determine bounds of a fractal set given by some recursive definitions, for example by an Iterated Function System (IFS). Here we propose a method of bounding an IFS-generated fractal set by a minimal simplex that is affinely identical to the standard simplex. First, it will be proved that for a given IFS attractor, such simplex exists and it is unique. Such simplex is then used for definition of an Affine invariant Iterated Function System (AIFS) that then can be used for affine transformation of a given fractal set and for its modeling.

Keywords

Convex Hull Iterate Function System Nonempty Compact Subset Standard Simplex Standard Orthonormal Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Barnsley, M.F.: Fractals everywhere. Academic Press, London (1988)zbMATHGoogle Scholar
  2. 2.
    Berger, M.A.: Random affine iterated function systems: curve generation and wavelets. SIAM Review 34, 361–385 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dubuc, S., Elqortobi, A.: Approximation of fractal sets. J. Comput. Appl. Math. 29, 79–89 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981)Google Scholar
  5. 5.
    Kocić, L.M.: Descrete methods for visualising fractal sets. FILOMAT (Niš) 9(3), 753–764 (1995)Google Scholar
  6. 6.
    Kocić, L.M., Simoncelli, A.C.: Towards free-form fractal modelling. In: Daehlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 287–294. Vanderbilt University Press, Nashville (1998)Google Scholar
  7. 7.
    Kocić, L.M., Simoncelli, A.C.: Stochastic approach to affine invariant IFS. Prague Stochastic 1998 theory, Statistical Decision Functions and Random Processes 2, 317–320 (1998)Google Scholar
  8. 8.
    Kocić, L.M., Simoncelli, A.C.: Cantor Dust by AIFS. FILOMAT (Niš) 15, 265–276 (2001); Math. Subj. Class. 28A80 (65D17) (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kocić, L.M., Simoncelli, A.C.: Shape predictable IFS representations. In: Novak, M.M. (ed.) Emergent Nature, pp. 435–436. World Scientific, Singapore (2002); Math. Subj. Class. 65D17, 28A80 (1991)Google Scholar
  10. 10.
    Kocić, L.M., Stefanovska, L., Babače, E.: Affine invariant iterated function system and minimal simplex problem. In: Proceedings of the International Conference DGDS-2007, pp. 119–128. Geometry Balkan Press, Bucharest (2008)Google Scholar
  11. 11.
    Lawlor, O.S., Hart, J.C.: Bounding Recursive Procedural Models using Convex Optimization. In: Proc. Pacific Graphics 2003 (October 2003)Google Scholar
  12. 12.
    Mandelbrot, B.: The Fractal Geometry of Nature. Freeman and Company, New York (1977)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Elena Babače
    • 1
  • Ljubiša Kocić
    • 2
  1. 1.Faculty of Electrical Engineering and Information TechnologiesSs Cyril and Methodius UniversitySkopjeMacedonia
  2. 2.Faculty of Electronic EngineeringUniversity of NišSerbia

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