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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]].

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]].

函数的集合<math>f_i</math>[[Generating set|generates]]是[[Function composition|composition]]下的[[monoid]]。如果只有两个这样的函数，那么这个单体可以可视化为一棵[[binary tree]]，在树的每一个节点上，我们可以用一个或另一个函数进行合成（''即''取左支或右支）。一般来说，如果有''k''函数，那么可以将单子可视化为一个完整的[[k-ary树|''k''ary树]]，也称为[[Cayley tree]]。

函数的集合<math>f_i</math>[[Generating set|generates]]是[[Function composition|composition]]下的[[monoid]]。如果只有两个这样的函数，那么这个单体可以可视化为一棵[[binary tree]]，在树的每一个节点上，我们可以用一个或另一个函数进行合成（''即''取左支或右支）。一般来说，如果有''k''函数，那么可以将单子可视化为一个完整的[[k-ary树|''k''ary树]]，也称为[[Cayley tree]]。

==Constructions==

==Constructions 构建==

[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]

[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]

[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]

[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]

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 |authorlink=Scott Draves |author2=Erik Reckase |date=July 2007 |url=http://flam3.com/flame.pdf |title=The Fractal Flame Algorithm |format=pdf |accessdate=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>

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 |authorlink=Scott Draves |author2=Erik Reckase |date=July 2007 |url=http://flam3.com/flame.pdf |title=The Fractal Flame Algorithm |format=pdf |accessdate=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>

虽然IFS的理论要求每个功能都是收缩的，但实际上实现IFS的软件只要求整个系统平均是收缩的。

虽然IFS的理论要求每个功能都是收缩的，但实际上实现IFS的软件只要求整个系统平均是收缩的。

==Partitioned iterated function systems 分割迭代函数系统==

==Partitioned iterated function systems 分割迭代函数系统==

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 factals.<ref name="SIGGRAPH'92">{{cite conference

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 factals.<ref name="SIGGRAPH'92">{{cite conference