Iterated function system

thumb|right|250px|[[Sierpiński triangle created using IFS (colored to illustrate self-similar structure)]]

thumb|right|200px|Colored IFS designed using [[Apophysis (software)|Apophysis software and rendered by the Electric Sheep]]

In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.

IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński triangle. The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature.

Definition

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space. Symbolically,

:<math>\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}</math>

is an iterated function system if each is a contraction on the complete metric space .

Properties

thumb|right|250px|Construction of an IFS by the [[chaos game (animated)]]

thumb|IFS being made with two functions

Hutchinson showed that, for the metric space , or more generally, for a complete metric space <math>X</math>, such a system of functions has a unique nonempty compact (closed and bounded) fixed set . One way of constructing a fixed set is to start with an initial nonempty closed and bounded set and iterate the actions of the , taking to be the union of the images of under the ; then taking to be the closure of the limit . Symbolically, the unique fixed (nonempty compact) set has the property

:<math>S = \overline{\bigcup_{i=1}^N f_i(S)}.</math>

The set is thus the fixed set of the Hutchinson operator defined for via

:<math>F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.</math>

The existence and uniqueness of is a consequence of the contraction mapping principle, as is the fact that

:<math>\lim_{n\to\infty}F^{n}(A)=S</math>

for any nonempty compact set in . (For contractive IFS this convergence takes place even for any nonempty closed bounded set ). Random elements arbitrarily close to may be obtained by the "chaos game", described below.

Recently it was shown that the IFSs of non-contractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ) can yield attractors. These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.

The collection of functions generates a monoid under composition. If there are only two such functions, the monoid can be visualized as a binary tree, where, at each node of the tree, one may compose with the one or the other function (i.e. take the left or the right branch). In general, if there are functions, then one may visualize the monoid as a full -ary tree, also known as a Cayley tree.

Constructions

thumb|upright|[[Barnsley's fern, an early IFS]]

thumb|200px|[[Menger sponge, a 3-dimensional IFS]]

thumb|IFS "tree" constructed with non-linear function Julia

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Sometimes each function is required to be a linear, or more generally an affine, transformation, and hence represented by a matrix. However, IFSs may also be built from non-linear functions, including projective transformations and Möbius transformations. The Fractal flame is an example of an IFS with nonlinear functions.

The most common algorithm to compute IFS fractals is called the "chaos game". It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system to transform the point to get a next point. An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, and then to plot the results of applying each of these sequences of functions to an initial point or shape.

Each of these algorithms provides a global construction which generates points distributed across the whole fractal. If a small area of the fractal is being drawn, many of these points will fall outside of the screen boundaries. This makes zooming into an IFS construction drawn in this manner impractical.

Although the theory of IFS requires each function to be contractive, in practice software that implements IFS only require that the whole system be contractive on average.

Partitioned iterated function systems

PIFS (partitioned iterated function systems), also called local iterated function systems,

The inverse problem

Very fast algorithms exist to generate an image from a set of IFS or PIFS parameters. It is faster and requires much less storage space to store a description of how it was created, transmit that description to a destination device, and regenerate that image anew on the destination device, than to store and transmit the color of each pixel in the image.

The inverse problem is more difficult: given some original arbitrary digital image such as a digital photograph, try to find a set of IFS parameters which, when evaluated by iteration, produces another image visually similar to the original. In 1989, Arnaud Jacquin presented a solution to a restricted form of the inverse problem using only PIFS; the general form of the inverse problem remains unsolved.

Notes

References

External links

A Primer on the Elementary Theory of Infinite Compositions of Complex Functions