Advertisement

From Fractal Image Compression to Fractal-Based Methods in Mathematics

  • Edward R. Vrscay
Part of the The IMA Volumes in Mathematics and its Application book series (IMA, volume 132)

Abstract

In keeping with the philosophy of this workshop, the aim of this presentation is to provide an overview of the research done over the years at Waterloo on fractal-based methods of approximation and associated inverse problems. Near the end, some new and encouraging results regarding “fractal enhancement” will be presented. The paper concludes with thoughts and challenges on how the mathematical methods that underlie fractal image compression could be used in other areas of mathematics.

Keywords

Inverse Problem Invariant Measure Grey Level Wavelet Coefficient Iterate Function System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Bacry, J.F. Muzy, and A. Arnéodo, Singularity spectrum of fractal signals from wavelet analysis: exact results, J. Stat. Phys. 70, 635–674 (1993).CrossRefMATHGoogle Scholar
  2. [2]
    M. Bajraktarevic, Sur une équation fonctionelle, Glasnik Mat.-Fiz. I Astr. 12, 201–205 (1957).MathSciNetGoogle Scholar
  3. [3]
    S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund. Math. 3, 133–181 (1922).MATHGoogle Scholar
  4. [4]
    M.F. Barnsley, Fractal interpolation functions, Constr. Approx. 2, 303–329 (1986).MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    M.F. Barnsley, Fractals Everywhere, Academic Press, New York (1988).MATHGoogle Scholar
  6. [6]
    M.F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London A 399, 243–275 (1985).MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M.F. Barnsley, S.G. Demko, J. Elton, and J.S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. H. Poincaré 24, 367–394 (1988).MathSciNetMATHGoogle Scholar
  8. [8]
    M.F. Barnsley, V. Ervin, D. Hardin, and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Nat. Acad. Sci. USA 83, 1975–1977 (1985).MathSciNetCrossRefGoogle Scholar
  9. [9]
    M.F. Barnsley and L.P. Hurd, Fractal Image Compression, A.K. Peters, Wellesley, Mass. (1993).MATHGoogle Scholar
  10. [10]
    K.U. Barthel, S. Brandau, W. Hermesmeier, and G. Heising, Zerotree wavelet coding using fractal prediction, Proc. IEEE Conf. Data Compression 1997, pp. 314–317.Google Scholar
  11. [11]
    C.A. Cabrelli, B. Forte, U.M. Molter, and E.R. Vrscay, Iterated Fuzzy Set Systems: a new approach to the inverse problem for fractals and other sets, J. Math. Anal. Appl. 171, 79–100 (1992).MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    C.A. Cabrelli and U.M. Molter, Generalized self-similarity, J. Math. Anal. Appl. 230, 251–260 (1999).MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    P. Centore and E.R. Vrscay, Continuity properties for attractors and invariant measures for iterated function systems, Canadian Math. Bull. 37 315–329 (1994).MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    K. Daoudi, J. Levy Vehel, and Y. Meyer, Construction of continuous functions with prescribed local regularity, Constr. Approx. 14, 349–385 (1998).CrossRefMATHGoogle Scholar
  15. [15]
    I. Daubechies, Ten Lectures on Wavelets, SIAM Press, Philadelphia (1992).CrossRefMATHGoogle Scholar
  16. [16]
    G. Davis, A wavelet-based analysis fractal image compression, IEEE Trans. Image Proc. 7, 141–154 (1998).CrossRefMATHGoogle Scholar
  17. [17]
    P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35, 241–249 (1990).MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    S. Dubuc, Interpolation fractale, in Fractal Geometry and Analysis, J. Béelair and S. Dubuc, Eds., NATO ASI Series C, Vol. 346, Kluwer, Dordrecht (1991).Google Scholar
  19. [19]
    C.J.G. Evertesz and B.B. Mandelbrot, Multifractal measures, in Chaos and Fractals: New Frontiers of Science, H.-O. Peitgen, H. Jürgens and D. Saupe, Springer Verlag, New York (1994).Google Scholar
  20. [20]
    K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge (1985).CrossRefMATHGoogle Scholar
  21. [21]
    K. Falconer, Techniques in Fractal Geometry, Wiley, Chichester (1997).MATHGoogle Scholar
  22. [22]
    Y. Fisher, A discussion of fractal image compression, in Chaos and Fractals, New Frontiers of Science, H.-O. Peitgen, H. Jürgens, and D. Saupe, Springer-Verlag, Heidelberg (1994).Google Scholar
  23. [23]
    Y. Fisher, Fractal Image Compression, Theory and Application, Springer-Verlag, New York (1995).CrossRefGoogle Scholar
  24. [24]
    B. Forte, M. LoSchiavo, and E.R. Vrscay, Continuity properties of attractors for iterated fuzzy set systems, J. Aust. Math. Soc. B 36, 175–193 (1994).MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    B. Forte, F. Mendivil, and E.R. Vrscay, “IFS-Type Operators on Integral Transforms,” in Fractals: Theory and Applications in Engineering, ed. M. Dekking, J. Levy-Vehel, E. Lutton, and C. Tricot, Springer Verlag, London (1999).Google Scholar
  26. [26]
    B. Forte and E.R. Vrscay, Solving the inverse problem for measures using iterated function systems: A new approach, Adv. Appl. Prob. 27, 800–820 (1995).MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    B. Forte and E.R. Vrscay, Solving the inverse problem for functions and image approximation using iterated function systems, Dyn. Cont. Impul. Sys. 1 177–231 (1995).MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    B. Forte and E.R. Vrscay, Theory of generalized fractal transforms, in Fractal Image Encoding and Analysis, Y. Fisher, Ed., NATO ASI Series F 159, Springer Verlag, New York (1998).Google Scholar
  29. [29]
    B. Forte and E.R. Vrscay, Inverse Problem Methods for Generalized Fractal Transforms, in Fractal Image Encoding and Analysis, ibid.Google Scholar
  30. [30]
    M. Ghazel and E.R. Vrscay, An effective hybrid fractal-wavelet image coder using quadtree partitioning and pruning, Proc. Can. Conf. Elect. Comp. Eng., CCECE 2000, Halifax, Nova Scotia (2000).Google Scholar
  31. [31]
    W. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl. 83, 264–274 (1981)MathSciNetCrossRefMATHGoogle Scholar
  32. [31a]
    W. Gilbert, Fractal geometry derived from complex bases, Math. Intelligencer, 4, 78–86 (1981)CrossRefGoogle Scholar
  33. [31b]
    W. Gilbert, Geometry of radix expansions, in The Geometric Vein, The Coxeter Festschrift, C. Davis, B. Grünbaum and F.A. Sherk, Eds., Springer Verlag, New York (1982).Google Scholar
  34. [32]
    W. Gilbert, The division algorithm in complex bases, Can. Math. Bull. 39, 47–54 (1996).CrossRefMATHGoogle Scholar
  35. [33]
    M. Giona, Vector analysis on fractal curves, in Fractals: Theory and Applications in Engineering, ed. M. Dekking, J. Levy-Vehel, E. Lutton, and C. Tricot, Springer Verlag, London (1999). pp. 307–323.Google Scholar
  36. [34]
    K. Gröchenig and W.R. Madych, Multiresolution analysis, Haar bases and self-similar tilings of Rn, IEEE Trans. Inform. Theory, 39, 556–568 (1992).CrossRefGoogle Scholar
  37. [35]
    B. Guiheneuf and J. Levy Véhel, 2-Microlocal analysis and applications in signal processing (preprint, INRIA Rocquencourt, 1997).Google Scholar
  38. [36]
    J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math. 30, 713–747 (1981).MathSciNetCrossRefMATHGoogle Scholar
  39. [37]
    A. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, IEEE Trans. Image Proc. 1, 18–30 (1992).CrossRefGoogle Scholar
  40. [38]
    S. Jaffard, Multifractal formalism for functions, I, SIAM J. Math. Anal. 28, 944–970 (1997).MathSciNetCrossRefMATHGoogle Scholar
  41. [39]
    S. Karlin, Some random walks arising in learning models, I., Pacific J. Math. 3, 725–756 (1953).MathSciNetMATHGoogle Scholar
  42. [40]
    H. Krupnik, D. Malah and E. Karnin, Fractal representation of images via the discrete wavelet transform, Proc. IEEE 18th Conference on Electrical Engineering (Tel-Aviv, 7–8 March 1995).Google Scholar
  43. [41]
    H.E. Kunze and E.R. Vrscay, Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems, 15, 745–770 (1999).MathSciNetCrossRefMATHGoogle Scholar
  44. [42]
    J. Lévy Véhel, Fractal approaches in signal processing (preprint).Google Scholar
  45. [43]
    J. Lévy Véhel, Introduction to the multifractal analysis of images, in Fractal Image Encoding and Analysis, Y. Fisher, Ed., NATO ASI Series F 159, Springer Verlag, New York (1998).Google Scholar
  46. [44]
    J. Lévy Véhel and B. Guiheneuf, Multifractal image denoising (preprint, INRIA Rocquencourt, 1997).Google Scholar
  47. [45]
    J. Li and C.-C. Jay Kuo, Fractal wavelet coding using a rate-distortion constraint, Proc. ICIP-96, IEEE International Conference on Image Processing, Lausanne, Sept. 1996.Google Scholar
  48. [46]
    N. Lu, Fractal Imaging, Academic Press, NY (1997).MATHGoogle Scholar
  49. [47]
    S.G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. PAMI 11(7), 674–693 (1989).CrossRefMATHGoogle Scholar
  50. [48]
    S. Mallat, A Wavelet Tour of Signal Processing, 2Edition, Academic Press, New York (2001).Google Scholar
  51. [49]
    P. Massopust, Fractal Functions, Fractal Surfaces and Wavelets, Academic Press, New York (1994).MATHGoogle Scholar
  52. [50]
    F. Mendivil and D. Piché, Two algorithms for nonseparable wavelet transforms and applications to image compression, in Fractals: Theory and Applications in Engineering, ed. M. Dekking, J. Levy-Vehel, E. Lutton, and C. Tricot, Springer Verlag, London (1999).Google Scholar
  53. [51]
    F. Mendivil and E.R. Vrscay, Correspondence between fractal-wavelet transforms and Iterated Function Systems with Grey Level Maps, in Fractals in Engineering: From Theory to Industrial Applications, ed. J. Levy-Vehel, E. Lutton and C. Tricot, Springer Verlag, London, pp. 54–64. (1997).Google Scholar
  54. [52]
    F. Mendivil and E.R. Vrscay, Fractal vector measures and vector calculus on planar fractal domains (preprint, 2001).Google Scholar
  55. [53]
    D.M. Monro, A hybrid fractal transform, Proc. ICASSP 5, 162–172 (1993).Google Scholar
  56. [54]
    D.M. Monro and F. Dudbridge, Fractal Block Coding of Images, Electron. Lett. 28, 1053–1054 (1992).CrossRefGoogle Scholar
  57. [55]
    S. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30, 475–488 (1969).MathSciNetMATHGoogle Scholar
  58. [56]
    A.H. Read, The solution of a functional equation, Proc. Roy. Soc. Edin. A 63, 336–345 (1951–1952).MathSciNetMATHGoogle Scholar
  59. [57]
    M. Ruhl and H. Hartenstein, Optimal fractal coding is NP-hard, Proceedings of the IEEE Data Compression Conference, J. Storer and M. Cohn, Eds., Snowbird, Utah 1997.Google Scholar
  60. [58]
    A. Said and W.P. Pearlman, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Trans. Circuits and Systems for Video Tech. 6, 243–250 (1996).CrossRefGoogle Scholar
  61. [59]
    B. Simon, Explicit link between local fractal transform and multiresolution transform, Proc. ICIP-95, IEEE International Conference on Image Processing, Washington D.C., Oct. 1995.Google Scholar
  62. [60]
    D.R. Smart, Fixed Point Theorems, Cambridge University Press, London (1974). p. 3.MATHGoogle Scholar
  63. [61]
    R. Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, Boca Raton (1994).MATHGoogle Scholar
  64. [62]
    A. van de Walle, Relating fractal compression to transform methods, Master of Mathematics Thesis, Department of Applied Mathematics, University of Waterloo (1995).Google Scholar
  65. [63]
    E.R. Vrscay, Iterated function systems: theory, applications and the inverse problem, in Fractal Geometry and Analysis, J. Bélair and S. Dubuc, Eds., NATO ASI Series C, Vol. 346, Kluwer, Dordrecht (1991).Google Scholar
  66. [64]
    E.R. Vrscay, A Generalized Class of Fractal-Wavelet Transforms for Image Representation and Compression, Can. J. Elect. Comp. Eng. 23(1–2), 69–84 (1998).Google Scholar
  67. [65]
    E.R. Vrscay and D. Saupe, “Can one break the ‘collage barrier’ in fractal image coding?” in Fractals: Theory and Applications in Engineering, ed. M. Dekking, J. Levy-Vehel, E. Lutton, and C. Tricot, Springer Verlag, London, (1999). pp. 307–323.Google Scholar
  68. [66]
    R. A. Wannamaker and E.R. Vrscay, Fractal Wavelet Compression of Audio Signals, J. Audio Eng. Soc. 45(7–8), 540–553 (1997).Google Scholar
  69. [67]
    R.F. Williams, Composition of contractions, Bol. Soc. Brasil. Mat. 2, 55–59 (1971).MathSciNetCrossRefMATHGoogle Scholar
  70. [68]
    S.J. Woolley and D.M. Monro, Rate/distortion performance of fractal transforms for image compression, Fractals 2, 395–398 (1994).CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • Edward R. Vrscay
    • 1
  1. 1.Department of Applied Mathematics, Faculty of MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations