Random Affine Iterated Function Systems: Mixing and Encoding

  • Marc A. Berger
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

Hull Convolution Kelly 

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Marc A. Berger
    • 1
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlanta

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