# 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} ∈ ℝ^{2}. 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.

## Keywords

Markov Chain Stationary Distribution Affine Transformation Coupling Condition Supporting Hyperplane## Preview

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