# Random Affine Iterated Function Systems: Mixing and Encoding

• Marc A. Berger
Chapter
Part of the Progress in Probability book series (PRPR, volume 27)

## Abstract

This paper is concerned with a probabilistic algorithm for image generation. The simplest form of the algorithm is illustrated in Fig. 1. The leaf is generated as follows. Pick any point X 0 ∈ ℝ2. There are four affine transformations T : xA x + b listed on top of this Fig., and four probabilities p i underneath them. Choose one of these transformations at random, according to the probabilities p i — say T k is chosen, and apply it to X 0, thereby obtaining X 1 = T k X 0. Then choose a transformation again at random, independent of the previous choice, and apply it to X 1, thereby obtaining X 2. Continue in this fashion, and plot the orbit {X n }. The result is the leaf shown. By tabulating the frequencies with which the points X n fall into the various pixels of the graphics window, one can actually plot the empirical distribution $${1 \over {n + 1}}\sum\nolimits_{k = 0}^n {{\delta _{{X_k}}}}$$, using a grey scale to convert statistical frequency to color. The darker portions of the leaf correspond to high probability density.

## Keywords

Markov Chain Stationary Distribution Affine Transformation Coupling Condition Supporting Hyperplane
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.

## References

1. [1]
Barnsley, M. F., Fractals Everywhere, Academic Press, New York, 1988.
2. [2]
Barnsley, M. F., Berger, M. A. and Soner, H. M., Mixing Markov chains and their images, Prob. Eng. Inf. Sci. 2 (1988), 387–414.
3. [3]
Barnsley, M. F., Demko, S. G., Elton, J. and Geronimo, J. S., Invariant measures for Markov processes arising from function iteration with place-dependent probabilities, Ann. Inst. H. Poincaré, 24 (1988), 367–394.
4. [4]
Barnsley, M. F., Elton, J. H. and Hardin, D. P., Recurrent iterated function systems, Const. Approx. 5 (1989), 3–31.
5. [5]
Berger, M. A. and Soner, H. M., Random walks generated by affine mappings, J. Theor. Prob. 1 (1988), 239–254.
6. [6]
Billingsley, P., Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1965.Google Scholar
7. [7]
8. [8]
Furstenberg, H. and Kesten, H., Products of random matrices, Ann. Math. Stat. 31 (1960) 457–469.
9. [9]
Kelly, F. P., Reversibility and Stochastic Networks, John Wiley & Sons, Inc., New York, 1979.
10. [10]
Kingman, J. F. C, Subadditive ergodic theory, Ann. Prob. 1 (1973), 883–909.