Editing Iterated function system

{{Short description|Method for the construction of fractals}}
[[File:Sierpinski1.png|thumb|right|250px|[[Sierpinski triangle]] created using IFS (colored to illustrate self-similar structure)]]
[[File:Chris Ursitti fractal 0000.png|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 [[fractal]]s; the resulting fractals are often [[self-similar]]. IFS fractals are more related to [[set theory]] than fractal geometry.<ref name="picg">{{cite book |title=Progress in Computer Graphics: Volume 1 |last=Zobrist |first= George Winston |author2=Chaman Sabharwal |year=1992 |publisher=Intellect Books |isbn=9780893916510 |page=135 |url=https://play.google.com/store/books/details?id=Ai6Qo0qoE9EC |access-date=7 May 2017}}</ref> 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 [[contraction mapping|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 mapping]]s on a [[complete metric space]].<ref>Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc. {{ISBN|9780120790623}}.</ref> 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 <math>f_i</math> is a contraction on the complete metric space <math>X</math>.

==Properties==
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated)]]
[[File:Ifs-construction.png|thumb|IFS being made with two functions.]]
Hutchinson showed that, for the metric space <math>\mathbb{R}^n</math>, or more generally, for a complete metric space <math>X</math>, such a system of functions has a unique nonempty [[Compact space|compact]] (closed and bounded) fixed set ''S''.<ref name=hutchinson>{{cite journal |last=Hutchinson |first=John E. |title=Fractals and self similarity |journal=Indiana Univ. Math. J. |volume=30 |year=1981 |pages=713–747 |doi=10.1512/iumj.1981.30.30055 |url=https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf |issue=5|doi-access=free }}</ref> One way of constructing a fixed set is to start with an initial nonempty closed and bounded set ''S''<sub>0</sub> and iterate the actions of the ''f''<sub>''i''</sub>, taking ''S''<sub>''n''+1</sub> to be the union of the images of ''S''<sub>''n''</sub> under the ''f''<sub>''i''</sub>; then taking ''S'' to be the [[Closure (topology)|closure]] of the limit <math>\lim_{n \rightarrow \infty} S_n</math>. Symbolically, the unique fixed (nonempty compact) set <math>S\subseteq X</math> has the property
:<math>S = \overline{\bigcup_{i=1}^N f_i(S)}.</math>
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] <math>F: 2^X\to 2^X</math> defined for <math>A\subseteq X</math> via
:<math>F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.</math>

The existence and uniqueness of ''S'' 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 <math>A</math> in <math>X</math>. (For contractive IFS this convergence takes place even for any nonempty closed bounded set <math>A</math>). Random elements arbitrarily close to ''S'' 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 ''X'') can yield attractors.
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.<ref>M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System</ref>

The collection of functions <math>f_i</math> [[Generating set|generates]] a [[monoid]] under [[Function composition|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 ''k'' functions, then one may visualize the monoid as a full [[k-ary tree|''k''-ary tree]], also known as a [[Cayley tree]].

==Constructions==
[[File:Fractal fern explained.png|thumb|upright|[[Barnsley's fern]], an early IFS]]
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS.]]
[[File:Coupe d'arbre.png|thumb|IFS "tree" constructed with non-linear function Julia]]
[[File:HERBO avecTige.png|thumb|]]
Sometimes each function <math>f_i</math> is required to be a [[Linear transformation|linear]], or more generally an [[affine transformation|affine]], transformation, and hence represented by a [[matrix (mathematics)|matrix]]. However, IFSs may also be built from non-linear functions, including [[projective transformation]]s and [[Möbius transformation]]s. 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.<ref>{{cite web |last=Draves |first=Scott |author-link=Scott Draves |author2=Erik Reckase |date=July 2007 |url=http://flam3.com/flame.pdf |title=The Fractal Flame Algorithm |access-date=2008-07-17 |archive-url=https://web.archive.org/web/20080509073421/http://flam3.com/flame.pdf |archive-date=2008-05-09 |url-status=dead }}</ref>

==Partitioned iterated function systems==

PIFS (partitioned iterated function systems), also called local iterated function systems,<ref name="lacroix"/> give surprisingly good image compression, even for photographs that don't seem to have the kinds of self-similar structure shown by simple IFS fractals.<ref name="SIGGRAPH'92">{{cite conference
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf
|last= Fischer
|first= Yuval
|title= SIGGRAPH'92 course notes - Fractal Image Compression
|conference= SIGGRAPH
|conference-url= http://www.siggraph.org/
|editor= Przemyslaw Prusinkiewicz
|publisher= [[ACM SIGGRAPH]]
|volume= Fractals - From Folk Art to Hyperreality
|date= 1992-08-12
|access-date= 2017-06-30
|archive-date= 2017-09-12
|archive-url= https://web.archive.org/web/20170912012035/https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf
|url-status= dead
}}</ref>

==The inverse problem==
{{Main|Fractal compression}}

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.<ref name="lacroix">Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf "Fractal Image Compression"]. 1998.</ref>

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.<ref>
Dietmar Saupe, Raouf Hamzaoui.
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf "A Review of the Fractal Image Compression Literature"].
</ref><ref name="kominek">
John Kominek.
[https://web.archive.org/web/20181123231446/https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf "Algorithm for Fast Fractal Image Compression"].
{{doi|10.1117/12.206368}}.
</ref><ref name="lacroix"/>

As of 1995, all [[fractal compression]] software is based on Jacquin's approach.<ref name="kominek"/>

==Examples==
The diagram shows the construction on an IFS from two affine functions. The functions are represented by their effect on the bi-unit square (the function transforms the outlined square into the shaded square). The combination of the two functions forms the [[Hutchinson operator]]. Three iterations of the operator are shown, and then the final image is of the fixed point, the final fractal.

Early examples of fractals which may be generated by an IFS include the [[Cantor set]], first described in 1884; and [[de Rham curve]]s, a type of self-similar curve described by [[Georges de Rham]] in 1957.

==History==
IFSs were conceived in their present form by [[John E. Hutchinson]] in 1981<ref name="hutchinson" /> and popularized by [[Michael Barnsley]]'s book ''Fractals Everywhere''. <!-- In 1992 [[Scott Draves]] developed the [[Fractal flame]] algorithm.-->

{{Quote|IFSs provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature.|Michael Barnsley ''et al.''<ref name=V-variable>[[Michael Barnsley]], ''et al.'',{{cite web |url= http://www.maths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf |title=V-variable fractals and superfractals}}&nbsp;{{small|(2.22&nbsp;MB)}}</ref>}}

==See also==
{{Portal|Mathematics}}
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]
*[[Collage theorem]]
*[[Infinite compositions of analytic functions]]
*[[L-system]]
*[[Fractal compression]]

==Notes==
{{Reflist|30em}}

==References==
*{{cite web |last=Draves |first=Scott |author-link=Scott Draves |author2=Erik Reckase |date=July 2007 |url=http://flam3.com/flame.pdf |title=The Fractal Flame Algorithm |access-date=2008-07-17 |archive-url=https://web.archive.org/web/20080509073421/http://flam3.com/flame.pdf |archive-date=2008-05-09 |url-status=dead }}
*{{cite book |last=Falconer |first=Kenneth |author-link=Kenneth Falconer (mathematician) |title=Fractal geometry: Mathematical foundations and applications |url=https://archive.org/details/fractalgeometrym0000falc/page/113 |url-access=registration |publisher=John Wiley and Sons |year=1990 |isbn=0-471-92287-0 |pages=[https://archive.org/details/fractalgeometrym0000falc/page/113 113–117, 136] }}
*{{cite journal |last=Barnsley |first=Michael |author-link=Michael Barnsley |author2=Andrew Vince |arxiv=1005.0322 |title=The Chaos Game on a General Iterated Function System |journal=Ergodic Theory Dynam. Systems |volume=31 |year=2011 |issue=4 |pages=1073–1079|doi=10.1017/S0143385710000428 |bibcode=2010arXiv1005.0322B |s2cid=122674315 }}
*'''For an historical overview, and the generalization :''' {{cite journal |first=Claire|last=David  |  title=fractal properties of Weierstrass-type functions |journal=Proceedings of the International Geometry Center |volume=12 |year=2019 |issue=2 |pages=43–61|doi=10.15673/tmgc.v12i2.1485 |s2cid=209964068 |url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 |doi-access=free }}
 
==External links==
*[https://www.researchgate.net/publication/362010262_A_Primer_on_the_Elementary_Theory_of_Infinite_Compositions_of_Complex_Functions_Images A Primer on the Elementary Theory of Infinite Compositions of Complex Functions]

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[[Category:Iterated function system fractals| ]]
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