Stochastic Methods and Non-Fractal Applications

  • Huw Jones
Conference paper
Part of the Focus on Computer Graphics book series (FOCUS COMPUTER)

Abstract

Our everyday environment abounds with examples of stochastic phenomena, objects or activities that are governed by probabilistic laws [COX65]. These range from observations evident to any observer, such as the number of people in a supermarket checkout queue or the time taken to drive to work on a particular day, to more latent situations, such as the energy levels of subatomic particles. Practitioners of statistics and operational research have, for decades, used computers to create models of such activity in order to gain useful insights into the behaviour of systems that depend on statistical or probabilistic laws. More recently, stochastic methods have been used to model certain natural phenomena in a visually convincing way. Images depicting simulations of the structures of, for example, plants [PRUS90] and other life forms [KAAN91], marble [PERL85], clouds [VOSS85], mountainous tenain [SAUP88] and the boundaries of cities [BATT91] have become familiar. Many researchers use standard “random number generators” to reproduce such effects; not all of them appreciate the implications of what they are doing. The purpose of this tutorial is to describe some of the underlying statistical theory and to show its application to a selection of techniques in computer graphics. The development will be explanatory rather than theoretically rigorous. The intention is to give end users an understanding of the methods they are using without converting them into statistical experts. Statistics is similar to many other technical subjects in that much of its mystique is concerned with its terminology. Many terms will be highlighted when introduced.

Keywords

Sponge Hull Riser Aliasing 

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References

  1. [ANDR72]
    Andrews, D.F., P.J. Bickel, F.R. Hampel, PJ. Huber, W.H. Rogers, J.W. Tukey, (1972) Robust Estimates of Location, Princeton University Press, Princeton, N J.MATHGoogle Scholar
  2. [ATKI80]
    Atkinson A.C., (1980) Tests of pseudo-random numbers, Appl Stats, 29(2), 164–171.MATHCrossRefGoogle Scholar
  3. [BATT91]
    Batty, M., (1991) Cities as Fractals: Simulating Growth and Form, in Fractals and Chaos, Crilly, A J., R.A. Earnshaw, H. Jones (eds), Springer-Verlag, New York, N.Y.Google Scholar
  4. [BURG89]
    Burger, P., D. Gillies, (1989) Interactive Computer Graphics, Addison-Wesley, Wokingham, England.MATHGoogle Scholar
  5. [COHE85]
    Cohen, M.F., D.P. Greenberg, (1985) The Hemi-Cube: A Radiosity Solution for Complex Environments, SIGGRAPH 85, 31–40.CrossRefGoogle Scholar
  6. [COX65]
    Cox, D.R., H.D. Miller, (1965) The Theory of Stochastic Processes, Methuen, London, England.MATHGoogle Scholar
  7. [CRAM46]
    Cramer, H., (1946) Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  8. [DAGP88]
    Dagpunar, J., (1988) Principles of Random Variate Generation, Clarendon Press, Oxford, England.MATHGoogle Scholar
  9. [EGGE79]
    Egger, M.J., (1979) Power transformations to achieve symmetry in quantal bioassay, Technical Report No 47, Division of Biostatistics, Stanford, Cal.Google Scholar
  10. [FOLE90]
    Foley, J.D., A. van Dam, S.K. Feiner, J.F. Hughes, (1990) Computer Graphics: Principles and Practice (2 ed), Addison-Wesley, Reading, Mass.Google Scholar
  11. [HAMM64]
    Hammersley, J.M., D.C. Handscomb, (1964) Monte Carlo Methods, Methuen, London, England.MATHCrossRefGoogle Scholar
  12. [HULL62]
    Hull, T.E., A.R. Dobell, (1962) Random Number Generators, SIAM Rev., 4, 230–254.MathSciNetMATHCrossRefGoogle Scholar
  13. [KAAN91]
    Kaandorp, J., (1991) Modelling Growth Forms of Sponges with Fractal Techniques, in Fractals and Chaos, Crilly, AJ., R.A. Earnshaw, H. Jones (eds), Springer-Verlag, New York, N.Y.Google Scholar
  14. [KNUT81]
    Knuth, D.E., (1981) The Art of computer Programming, Vol 2, Seminumerical Algorithms, Addison-Wesley, Reading, Mass.MATHGoogle Scholar
  15. [MORG84]
    Morgan, B.J.T., (1984) Elements of Simulation, Chapman and Hall, London, England.MATHGoogle Scholar
  16. [PERL85]
    Perlin, K., (1985) “An Image Synthesizer, ” SIGGRAPH 85, 287–296.CrossRefGoogle Scholar
  17. [PRUS90]
    Prusinkiewicz, P., A. Lindenmayer, (1990) The Algorithmic Beauty of Plants, Springer-Verlag, New York, N.Y.MATHCrossRefGoogle Scholar
  18. [SAUP88]
    Saupe, D., (1988) Algorithms for Random Fractals, in The Science of Fractal Images, Peitgen, H.O., D. Saupe (eds), Springer-Verlag, New York, NY.Google Scholar
  19. [TOCH63]
    Tocher, K.D., (1963) The Art of Simulation, Hodder and Stoughton, London, England.Google Scholar
  20. [VOSS85]
    Voss, R., “Fractals in Nature: Characterization, Measurement and Simulation, ” in Course Notes 15 for SIGGRAPH 87, Anaheim, CA July 1987.Google Scholar

Copyright information

© EUROGRAPHICS The European Association for Computer graphics 1994

Authors and Affiliations

  • Huw Jones

There are no affiliations available

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