Random Affine Iterated Function Systems: Mixing and Encoding

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 : x → A 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.