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[[File:Sierpinski_triangle.png|thumb|right|250px|[[Sierpinski triangle]] created using IFS (colored to illustrate self-similar structure) 使用IFS创建的Sierpinski三角形（用颜色来说明自相似结构）。]]

[[File:Sierpinski_triangle.png|thumb|right|250px|[[Sierpinski triangle]] created using IFS (colored to illustrate self-similar structure) 使用IFS创建的Sierpinski三角形（用颜色来说明自相似结构）。]]

[[File:Chris Ursitti fractal 0000.png|thumb|right|200px|Colored IFS designed using [[Apophysis (software)|Apophysis]] software and rendered by the [[Electric Sheep]]. 彩色 IFS 使用[[Apophysis (software)|Apophysis]]设计，使用[[Electric Sheep 电子羊]]渲染。]]

[[File:Chris Ursitti fractal 0000.png|thumb|right|200px|Colored IFS designed using [[Apophysis (software)|Apophysis]] software and rendered by the [[Electric Sheep]]. 彩色 IFS 使用[[Apophysis (software)|Apophysis]]设计，使用[[Electric Sheep 电子羊]]渲染。]]

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 |accessdate=7 May 2017}}</ref> They were introduced in 1981.

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 |accessdate=7 May 2017}}</ref> They were introduced in 1981.

在[[数学]]中，<font color="#ff8000">'''迭代函数系统''' '''iterated function systems'''('''IFS''')</font>是一种构造[[分形几何 Fractal]]的方法；产生的分形通常是自相似的。IFS分形与集论的关系比分形几何更大。它们于1981年被提出。

在[[数学]]中，<font color="#ff8000">'''迭代函数系统''' '''iterated function systems'''('''IFS''')</font>是一种构造[[分形几何 Fractal]]的方法；产生的分形通常是自相似的。IFS分形与集合论的关系比分形几何更大。它们于1981年被提出。

'''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.

'''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.

'''IFS''' 分形，正如他的名字一样，可以是任何维度的，但通常在2维空间计算和绘制。<font color="#ff8000">分形</font>是由其本身的几个副本的结合体组成，每个副本都由一个函数变换而成（因此称为 "函数系统"）。典型的例子是<font color="#ff8000">[[谢尔宾斯基三角形]]Sierpiński triangle</font>。这些函数通常是收缩的，这意味着它们使点更接近，使形状更小。因此，IFS分形的形状是由若干个可能重叠的较小副本组成的，每个副本也是由自己的副本组成的，无穷无尽[[ad infinitum]]。这就是其自相似分形性质的来源。

'''IFS''' 分形，正如他们被通常称呼一样，可以是任何维度的，但通常在2维空间计算和绘制。<font color="#ff8000">分形</font>是由其本身的几个副本的结合体组成，每个副本都由一个函数变换而成（因此称为 "函数系统"）。典型的例子是<font color="#ff8000">[[谢尔宾斯基三角形]]Sierpiński triangle</font>。这些函数通常是收缩的，这意味着它们使点更接近，使形状更小。因此，IFS分形的形状是由若干个可能重叠的较小副本组成的，每个副本也是由自己的副本组成的，无穷无尽[[ad infinitum]]。这就是其自相似分形性质的来源。

==Definition 定义==

==Definition 定义==

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{{cite book |isbn=9780120790623}}</ref> Symbolically,

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{{cite book |isbn=9780120790623}}</ref> Symbolically,

Hutchinson (1981) 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''. One way of constructing a fixed set is to start with an initial nonempty closed and bounded set ''S''₀ and iterate the actions of the ''f''_''i'', taking ''S''_''n''+1 to be the union of the images of ''S''_''n'' under the ''f''_''i''; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''_''n''. Symbolically, the unique fixed (nonempty compact) set <math>S\subseteq X</math> has the property

Hutchinson (1981) 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''. One way of constructing a fixed set is to start with an initial nonempty closed and bounded set ''S''₀ and iterate the actions of the ''f''_''i'', taking ''S''_''n''+1 to be the union of the images of ''S''_''n'' under the ''f''_''i''; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''_''n''. 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>

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

约翰·哈钦森 Hutchinson(1981)表明，对于度量空间<math><mathbb{R}^n</math>，或者更一般地说，对于一个完整的度量空间<math>X</math>，这样的函数系统有一个唯一的非空紧凑(闭合和有界)的固定集S。构造固定集的一种方法是，从初始的非空闭和有界集合S₀开始，迭代f_i的作用，取S_n+1为S_n在f_i下的图像的联合；然后取S为S_n的联合的闭合。用符号可以表示为，唯一的固定(非空紧凑)集合<math>S<subseteq X</math>具有如下性质:

约翰·哈钦森 Hutchinson(1981)表明，对于度量空间<math><mathbb{R}^n</math>，或者更一般地说，对于一个完整的度量空间<math>X</math>，这样的函数系统有一个唯一的非空紧凑(闭合且有界)的固定集S。构造固定集的一种方法是，从初始的非空的闭且有界集合S₀开始，迭代f_i的作用，取S_n+1为S_n在f_i下的图像的联合；然后取S为S_n的联合的闭合。用符号可以表示为，唯一的固定(非空紧凑)集合<math>S<subseteq X</math>具有如下性质:

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

:<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

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

这些都是在投影空间中自然产生的，尽管经典的圆上的无理旋转也可以适应。

这些都是在投影空间中自然产生的，尽管经典的圆上的无理旋转也可以适应。

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>能产生是运算下的独异点。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个''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 构建==

==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.]]

[[File:Coupe d'arbre.png|thumb|IFS "tree" constructed with non-linear function Julia]]

[[File:Coupe d'arbre.png|thumb|IFS "tree" constructed with non-linear function Julia|链接=Special:FilePath/Coupe_d'arbre.png]]

[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]

[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]

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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.

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.

有时，每个函数<math>f_i</math>都必须是[[Linear transformation|linear]]，或者更一般的[[affine transformation|affine]]，变换，因此用[[matrix (mathematics)|matrix]]表示。然而，IFS也可以由非线性函数建立，包括[[projective transformation]]和[[Möbius transformation]]。[[Fractal flame]]是一个非线性函数的IFS的例子。

有时，每个函数<math>f_i</math>都必须是线性的[[Linear transformation|linear]]，或者更一般的[[affine transformation|affine]]，变换，因此用[[matrix (mathematics)|matrix]]表示。然而，IFS也可以由非线性函数建立，包括[[projective transformation]]和[[Möbius transformation]]。[[Fractal flame]]是一个非线性函数的IFS的例子。

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.

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.

计算IFS分形最常用的算法叫做"[[chaos game]]"。它包括在平面上随机选取一个点，然后反复应用从函数系统中随机选取的一个函数来变换该点，从而得到下一个点。另一种算法是生成每一个可能的函数序列，直到给定的最大长度，然后将这些函数序列中的每一个应用于初始点或形状的结果绘制出来。

计算IFS分形最常用的算法叫做"[[chaos game]]"。它包括在平面上随机选取一个点，然后反复应用从函数系统中随机选取的一个函数来变换该点，去得到下一个点。另一种算法是生成每一个可能的函数序列，直到给定的最大长度，然后将这些函数序列中的每一个应用于初始点或形状的结果绘制出来。

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.

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.

这些算法中的每一种都提供了一个全局结构，生成分布在整个分形上的点。如果正在绘制分形的一小块区域，这些点中的许多将落在屏幕边界之外。这使得放大以这种方式绘制的IFS结构不切实际。

这些算法中的每一种都提供了一个全局结构，生成分布在整个分形上的点。如果正在绘制分形的一小块区域，这些点中的许多将落在屏幕边界之外。这使得放大以这种方式绘制的IFS结构不切实际。

{{doi|10.1117/12.206368}}.

{{doi|10.1117/12.206368}}.

</ref><ref name="lacroix"/>

</ref><ref name="lacroix"/>

逆向问题比较困难：给定一些原始的任意数字图像，如一张数字照片，试图找到一组IFS参数，通过迭代评估后，产生另一幅与原始图像视觉上相似的图像。在1989年，Arnaud Jacquin 提出了一个只用 PIFS 的限制形式的反问题的解，反问题的一般形式仍未解决。并由迈克尔 · 巴恩斯利的著作《分形无处不在》而流行开来。1992年，斯科特 · 德拉维斯发明了分形火焰算法。

逆问题比较困难：给定一些原始的任意数字图像，如一张数字照片，试图找到一组IFS参数，通过迭代评估后，产生另一幅与原始图像视觉上相似的图像。在1989年，Arnaud Jacquin 提出了一个只用 PIFS 的限制形式的反问题的解，反问题的一般形式仍未解决。并由迈克尔 · 巴恩斯利的著作《分形无处不在》而流行开来。1992年，斯科特 · 德拉维斯发明了分形火焰算法。

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

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

==Examples 范例==

==Examples 范例==

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.

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.