Constructive Approximation

, Volume 12, Issue 1, pp 31–65 | Cite as

Mathematical modeling of real-world images

  • Bl. Sendov


The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR 3, and exploring theHausdorff distance between these point sets, a metric spaceIM D of functions is defined. The main purpose is to show that the functionsf∈IM D, defined on the squareD=[0,1]2, are appropriate mathematical models of real world images.

The properties of the metric spaceIM D are studied and methods of approximation for the purpose of image compression are presented.

The metric spaceIM D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM D may be digitized and represented by a pixel functionp n, withn pixels, in such a way that the distance betweenf andp n is no greater than 2n −1/2.

It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.

AMS classification

Primary 41A46, 68U10 Secondary 41A99, 54D35 

Key words and phrases

Hausdorff distance Hausdorff continuity Completed graph Pixel function Image compression Fractal transform operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. F. Barnsley (1988): Fractals Everywhere. Boston: Academic Press.Google Scholar
  2. 2.
    M. F. Barnsley, L. P. Hurd (1993): Fractal Image Compression. Wellesley, MA: Peters.Google Scholar
  3. 3.
    T. P. Boyanov, V. A. Popov (1970):On the widths of the space of continuous functions in the metric of Hausdorff. Godishnik Sofia Univ. Mat. Fak.63:167–185 (in Bulgarian).MATHGoogle Scholar
  4. 4.
    P. M. Diaconis, M. Shashahani (1986): Products of Random Matrices and Computer Image Generation. Contemporary Mathematics, vol. 50. Providence, RI: American Mathematical Society, pp. 173–182.Google Scholar
  5. 5.
    F. Hausdorff (1927). Mengenlehre. Berlin: Gruyter.Google Scholar
  6. 6.
    J. Hutchinson (1981):Fractals and self-similarity. Indiana Univ. Math. J.,30:713–747.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    T. Kaiser (1978):A limit theorem for Markov chains in compact metric spaces with applications on products of random matrices. Duke Math. J.,45:311–349.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. N. Kolmogorov (1936):Über die beste annaherung von functionen einer functionsklasse. Math. Ann.,37:107–111.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. N. Kolmogorov (1956):Asymptotic characteristics of certain totally bounded metric spaces. Dokl. Acad. Nauk SSSR,108:385–389 (in Russian).MATHMathSciNetGoogle Scholar
  10. 10.
    P. P. Korovkin (1969):An axiomatic approach to certain questions in the constructive approximation theory of functions of one variable. Uncen. Zap. Kaliningr. Gos. Ped. Inst.,69:91–109 (in Russian).MathSciNetGoogle Scholar
  11. 11.
    B. B. Mandelbrot (1982): Form, Chance, and Dimension. New York: Freeman.Google Scholar
  12. 12.
    Bl. Sendov (1962):Approximation of functions by algebraic polynomials with respect to a metric of Hausdorff type. Godishnik Sofia Univ. Mat. Fak.,55:1–39 (in Bulgarian).MATHMathSciNetGoogle Scholar
  13. 13.
    Bl. Sendov (1969):Certain questions in the theory of approximation of functions and sets in the Hausdorff metric, Uspekhi Mat. Nauk,24(5):141–178 (English translation (1969): Russian Math. Surveys,24(5):143–183).MATHMathSciNetGoogle Scholar
  14. 14.
    Bl. Sendov (1990): Hausdorff Approximations. Dordrecht: Kluwer.Google Scholar
  15. 15.
    Bl. Sendov (1993):Integral Hausdorff distance. C. R. Acad. Bulgare Sci.,46(10):21–24.MATHMathSciNetGoogle Scholar
  16. 16.
    Bl. Sendov, V. A. Popov (1970):On certain properties of the Hausdorff metric. Mathematica (Cluj),8:163–172 (in Russian).MathSciNetGoogle Scholar
  17. 17.
    V. Veselinov (1988): Approximation of Functions by Linear Operators with Respect to Hausdorff and Uniform Metrics. Ph.D. Thesis, Sofia University (in Bulgarian).Google Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • Bl. Sendov
    • 1
  1. 1.Center of Informatics and Computer TechnologyBulgarian Academy of Sciences “Acad. G. Bonchev”SofiaBulgaria

Personalised recommendations