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		<title>迭代函数系统</title>
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		<updated>2020-12-27T15:23:41Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2初步翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc{{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions 构建==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
 | align = &lt;br /&gt;
 | direction = horizontal&lt;br /&gt;
 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
&amp;lt;gallery mode=&amp;quot;packed&amp;quot; heights=&amp;quot;140&amp;quot;&amp;gt;&lt;br /&gt;
File:Golden_Square_fractal_6.png|[[Golden ratio|Golden square]] fractal&lt;br /&gt;
File:Half_square_fractal_5.svg.png| Half fractal 半分形&lt;br /&gt;
&amp;lt;/gallery&amp;gt;&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:|用户:]]参与编译，[[用户:|用户:]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20586</id>
		<title>迭代函数系统</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20586"/>
		<updated>2020-12-27T15:22:56Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc{{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions 构建==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
 | align = &lt;br /&gt;
 | direction = horizontal&lt;br /&gt;
 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
&amp;lt;gallery mode=&amp;quot;packed&amp;quot; heights=&amp;quot;140&amp;quot;&amp;gt;&lt;br /&gt;
File:Golden_Square_fractal_6.png|[[Golden ratio|Golden square]] fractal&lt;br /&gt;
File:Half_square_fractal_5.svg.png| Half fractal 半分形&lt;br /&gt;
&amp;lt;/gallery&amp;gt;&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:|用户:]]参与编译，[[用户:|用户:]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20584</id>
		<title>迭代函数系统</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20584"/>
		<updated>2020-12-27T15:20:28Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{short description|&lt;br /&gt;
https://en.wikipedia.org/wiki/Iterated_function_system&lt;br /&gt;
https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&lt;br /&gt;
https://www.yuque.com/docs/share/f9207544-7eb4-488e-8aa0-6bd977c0ee4c?# 《迭代函数系统》&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;&amp;lt;/font&amp;gt;&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc{{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions 构建==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
 | align = &lt;br /&gt;
 | direction = horizontal&lt;br /&gt;
 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
&amp;lt;gallery mode=&amp;quot;packed&amp;quot; heights=&amp;quot;140&amp;quot;&amp;gt;&lt;br /&gt;
File:Golden_Square_fractal_6.png|[[Golden ratio|Golden square]] fractal&lt;br /&gt;
File:Half_square_fractal_5.svg.png| Half fractal 半分形&lt;br /&gt;
&amp;lt;/gallery&amp;gt;&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:|用户:]]参与编译，[[用户:|用户:]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20582</id>
		<title>迭代函数系统</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20582"/>
		<updated>2020-12-27T15:18:14Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：/* Constructions */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc{{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions 构建==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
 | align = &lt;br /&gt;
 | direction = horizontal&lt;br /&gt;
 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
&amp;lt;gallery mode=&amp;quot;packed&amp;quot; heights=&amp;quot;140&amp;quot;&amp;gt;&lt;br /&gt;
File:Golden_Square_fractal_6.png|[[Golden ratio|Golden square]] fractal&lt;br /&gt;
File:Half_square_fractal_5.svg.png| Half fractal 半分形&lt;br /&gt;
&amp;lt;/gallery&amp;gt;&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:|用户:]]参与编译，[[用户:|用户:]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20580</id>
		<title>迭代函数系统</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20580"/>
		<updated>2020-12-27T15:13:29Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc{{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
 | align = &lt;br /&gt;
 | direction = horizontal&lt;br /&gt;
 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
&amp;lt;gallery mode=&amp;quot;packed&amp;quot; heights=&amp;quot;140&amp;quot;&amp;gt;&lt;br /&gt;
File:Golden_Square_fractal_6.png|[[Golden ratio|Golden square]] fractal&lt;br /&gt;
File:Half_square_fractal_5.svg.png| Half fractal 半分形&lt;br /&gt;
&amp;lt;/gallery&amp;gt;&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:|用户:]]参与编译，[[用户:|用户:]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20579</id>
		<title>迭代函数系统</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20579"/>
		<updated>2020-12-27T15:06:11Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc {{cite book |isbn=9780120790623}}&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
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Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:趣木木|趣木木]]参与编译，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0%E7%B3%BB%E7%BB%9F&amp;diff=20572</id>
		<title>迭代函数系统</title>
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		<updated>2020-12-27T14:57:45Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;此词条Eatcosmos2翻译。&lt;br /&gt;
[[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）。]]&lt;br /&gt;
[[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。]]&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;picg&amp;quot;&amp;gt;{{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}}&amp;lt;/ref&amp;gt; They were introduced in 1981.&lt;br /&gt;
在[[数学]]中，&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;'''迭代函数系统''' '''iterated function systems'''('''IFS''')&amp;lt;/font&amp;gt;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。&lt;br /&gt;
'''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 &amp;quot;function system&amp;quot;). 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.&lt;br /&gt;
'''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。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分形&amp;lt;/font&amp;gt;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 &amp;quot;函数系统&amp;quot;）。典型的例子是&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[谢尔宾斯基三角形]]Sierpiński triangle&amp;lt;/font&amp;gt;。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。&lt;br /&gt;
==Definition 定义==&lt;br /&gt;
Formally, an [[iterated function]] system is a finite set of [[contraction mapping]]s on a [[complete metric space]].&amp;lt;ref&amp;gt;Michael Barnsley (1988). ''Fractals Everywhere'', p.82. Academic Press, Inc. {{ISBN|9780120790623}}.&amp;lt;/ref&amp;gt; Symbolically,&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
is an iterated function system if each &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; is a contraction on the complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;.&lt;br /&gt;
从形式上看，迭代函数系统是一个完整的度量空间上的有限收缩映射集。符号表示为，&lt;br /&gt;
:&amp;lt;math&amp;gt;\{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N}&amp;lt;/math&amp;gt;&lt;br /&gt;
如果每个&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;都是完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;上的收缩，则是一个迭代函数系统。&lt;br /&gt;
==Properties 性质==&lt;br /&gt;
[[File:Chaosgame.gif|thumb|right|250px|Construction of an IFS by the [[chaos game]] (animated) 通过[[chaos game]] (动画) 构建的一个IFS]]&lt;br /&gt;
[[File:Ifs-construction.png|thumb|IFS being made with two functions. IFS由两个函数构建。]]&lt;br /&gt;
Hutchinson (1981) showed that, for the metric space &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt;, or more generally, for a complete metric space &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;, 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''&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and iterate the actions of the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;, taking ''S''&amp;lt;sub&amp;gt;''n''+1&amp;lt;/sub&amp;gt; to be the union of the images of ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt; under the ''f''&amp;lt;sub&amp;gt;''i''&amp;lt;/sub&amp;gt;; then taking ''S'' to be the [[Closure (topology)|closure]] of the union of the ''S''&amp;lt;sub&amp;gt;''n''&amp;lt;/sub&amp;gt;. Symbolically, the unique fixed (nonempty compact) set &amp;lt;math&amp;gt;S\subseteq X&amp;lt;/math&amp;gt; has the property&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
约翰·哈钦森 Hutchinson(1981)表明，对于度量空间&amp;lt;math&amp;gt;&amp;lt;mathbb{R}^n&amp;lt;/math&amp;gt;，或者更一般地说，对于一个完整的度量空间&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;，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S&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;开始，迭代f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;的作用，取S&amp;lt;sub&amp;gt;n+1&amp;lt;/sub&amp;gt;为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;在f&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;下的图像的联合；然后取S为S&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;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&amp;lt;math&amp;gt;S&amp;lt;subseteq X&amp;lt;/math&amp;gt;具有如下性质:&lt;br /&gt;
:&amp;lt;math&amp;gt;S = \overline{\bigcup_{i=1}^N f_i(S)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The set ''S'' is thus the fixed set of the [[Hutchinson operator]] &amp;lt;math&amp;gt;F: 2^X\to 2^X&amp;lt;/math&amp;gt; defined for &amp;lt;math&amp;gt;A\subseteq X&amp;lt;/math&amp;gt; via&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
集合''S''因此是[[Hutchinson算子]]的固定集。&amp;lt;math&amp;gt;F: 2^X/to 2^X&amp;lt;/math&amp;gt;定义为&amp;lt;math&amp;gt;A/subseteq X&amp;lt;/math&amp;gt;，通过&lt;br /&gt;
:&amp;lt;math&amp;gt;F(A)=\overline{\bigcup_{i=1}^N f_i(A)}.&amp;lt;/math&amp;gt;&lt;br /&gt;
The existence and uniqueness of ''S'' is a consequence of the [[contraction mapping principle]], as is the fact that&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
''S''的存在性和唯一性是[[contraction mapping principle]]的结果，就如&lt;br /&gt;
:&amp;lt;math&amp;gt;\lim_{n\to\infty}F^{\circ n}(A)=S&amp;lt;/math&amp;gt;&lt;br /&gt;
for any nonempty compact set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;. (For contractive IFS this convergence takes place even for any nonempty closed bounded set &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;). Random elements arbitrarily close to ''S'' may be obtained by the &amp;quot;chaos game,&amp;quot; described below.&lt;br /&gt;
对于&amp;lt;math&amp;gt;X&amp;lt;/math&amp;gt;中的任何非空紧集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;。(对于收缩型IFS，即使对于任何非空的闭合有界集&amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt;，也会发生这种收敛。)。任意接近''S''的随机元素可以通过下面描述的 &amp;quot;chaos game &amp;quot;获得。&lt;br /&gt;
Recently it was shown that the IFSs of noncontractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in ''X'') can yield attractors.&lt;br /&gt;
最近，人们证明了非收缩型的IFS(即由相对于''X''中的任何拓扑等价度量不收缩的映射组成)可以产生吸引子。&lt;br /&gt;
These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.&amp;lt;ref&amp;gt;M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
The collection of functions &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; [[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]].&lt;br /&gt;
函数的集合&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;[[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]]。&lt;br /&gt;
==Constructions==&lt;br /&gt;
[[File:Fractal_fern_explained.png|thumb|upright|[[Barnsley's fern]], an early IFS [[Barnsley's fern]]，一个早期的 IFS]]&lt;br /&gt;
[[File:Menger sponge (IFS).jpg|thumb|200px|[[Menger sponge]], a 3-Dimensional IFS. [[Menger sponge]]，一个3维的 IFS.]]&lt;br /&gt;
[[File:Coupe d'arbre.png|thumb|IFS &amp;quot;tree&amp;quot; constructed with non-linear function Julia]]&lt;br /&gt;
[[File:Golden_Square_fractal_with_T-branching_8.svg.png|thumb|[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]]]&lt;br /&gt;
{{multiple image&lt;br /&gt;
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 | width = 150&lt;br /&gt;
 | header = 2012年10月11日&lt;br /&gt;
 | image1 = Golden_Square_fractal_6.svg.png&lt;br /&gt;
 | caption1 = [[Golden ratio|Golden square]] fractal&lt;br /&gt;
 | image2 = Half_square_fractal_5.svg.png&lt;br /&gt;
 | caption2 = Half fractal&lt;br /&gt;
 | footer = 2012年10月22日&lt;br /&gt;
}}&lt;br /&gt;
Sometimes each function &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; 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.&lt;br /&gt;
有时，每个函数&amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt;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。&lt;br /&gt;
The most common algorithm to compute IFS fractals is called the &amp;quot;[[chaos game]]&amp;quot;. 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.&lt;br /&gt;
计算IFS分形最常用的算法叫做&amp;quot;[[chaos game]]&amp;quot;。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。&lt;br /&gt;
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.&lt;br /&gt;
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。&lt;br /&gt;
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.&amp;lt;ref&amp;gt;{{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 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
==Partitioned iterated function systems 分割迭代函数系统==&lt;br /&gt;
PIFS (partitioned iterated function systems), also called local iterated function systems,&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt; 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.&amp;lt;ref name=&amp;quot;SIGGRAPH'92&amp;quot;&amp;gt;{{cite conference&lt;br /&gt;
|url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf&lt;br /&gt;
|last= Fischer&lt;br /&gt;
|first= Yuval&lt;br /&gt;
|title= SIGGRAPH'92 course notes - Fractal Image Compression&lt;br /&gt;
|conference= SIGGRAPH&lt;br /&gt;
|conference-url= http://www.siggraph.org/&lt;br /&gt;
|editor= Przemyslaw Prusinkiewicz&lt;br /&gt;
|publisher= [[ACM SIGGRAPH]]&lt;br /&gt;
|volume= Fractals - From Folk Art to Hyperreality&lt;br /&gt;
|date= 1992-08-12&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;分割迭代函数系统 PIFS(partitioned iterated function systems)&amp;lt;/font&amp;gt;，也叫局部迭代函数系统，给出了令人惊讶的良好的图像压缩效果，即使是对于那些看起来不具有简单的IFS factals所显示的自相似结构的照片。&lt;br /&gt;
==The inverse problem 逆问题==&lt;br /&gt;
{{Main|Fractal compression 分形压缩}}&lt;br /&gt;
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.&amp;lt;ref name=&amp;quot;lacroix&amp;quot;&amp;gt;Bruno Lacroix. [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf &amp;quot;Fractal Image Compression&amp;quot;]. 1998.&amp;lt;/ref&amp;gt;&lt;br /&gt;
有非常快速的算法可以从一组IFS或PIFS参数生成图像。与存储和传输图像中每个像素的颜色相比，存储描述如何创建图像、将该描述传输到目标设备并在目标设备上重新生成图像的速度更快，所需的存储空间也更小。&lt;br /&gt;
The 逆问题 [[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.&lt;br /&gt;
&amp;lt;ref&amp;gt;Dietmar Saupe, Raouf Hamzaoui.&lt;br /&gt;
[https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf &amp;quot;A Review of the Fractal Image Compression Literature&amp;quot;].&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;kominek&amp;quot;&amp;gt;&lt;br /&gt;
John Kominek.&lt;br /&gt;
[https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf &amp;quot;Algorithm for Fast Fractal Image Compression&amp;quot;].&lt;br /&gt;
{{doi|10.1117/12.206368}}.&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;lacroix&amp;quot;/&amp;gt;&lt;br /&gt;
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。&lt;br /&gt;
As of 1995, all [[fractal compression]] software is based on Jacquin's approach.&amp;lt;ref name=&amp;quot;kominek&amp;quot;/&amp;gt;&lt;br /&gt;
截至1995年，所有[[fractal compressio]]软件都是基于Jacquin的方法。&lt;br /&gt;
==Examples 范例==&lt;br /&gt;
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.&lt;br /&gt;
该图显示了两个仿射函数在IFS上的构造。这两个函数用它们对双单位正方形的影响来表示（函数将轮廓正方形转化为阴影正方形）。两个函数的组合形成[[Hutchinson算子]]。图中显示了该算子的三次迭代，然后是定点的最终图像，即最终的分形。&lt;br /&gt;
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.&lt;br /&gt;
IFS可以生成分形的早期例子包括[[Cantor set]]，最早描述于1884年；[[de Rham curve]]，[[Georges de Rham]]在1957年描述的一种自相似曲线。&lt;br /&gt;
==进一步阅读==&lt;br /&gt;
{{Portal|Mathematics}}&lt;br /&gt;
*[[Complex-base system#Base .E2.88.921.C2.B1i|Complex-base system]]&lt;br /&gt;
*[[Collage theorem]]&lt;br /&gt;
*[[Infinite compositions of analytic functions]]&lt;br /&gt;
*[[L-system]]&lt;br /&gt;
*[[Fractal compression]]&lt;br /&gt;
==参考文献==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
==相关链接==&lt;br /&gt;
*{{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 |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 }}&lt;br /&gt;
*{{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] }}&lt;br /&gt;
*{{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|bibcode=2010arXiv1005.0322B }}&lt;br /&gt;
*'''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|url=https://journals.onaft.edu.ua/index.php/geometry/article/view/1485 }}&lt;br /&gt;
 &lt;br /&gt;
{{Fractal software}}&lt;br /&gt;
{{Fractals}}&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
----&lt;br /&gt;
本中文词条由[[用户:趣木木|趣木木]]参与编译，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。&lt;br /&gt;
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''&lt;br /&gt;
[[分类:Iterated function system fractals 迭代函数系统分形]]&lt;br /&gt;
[[分类:1981 introductions]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Half_square_fractal_5.svg.png&amp;diff=20550</id>
		<title>文件:Half square fractal 5.svg.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Half_square_fractal_5.svg.png&amp;diff=20550"/>
		<updated>2020-12-27T14:16:05Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;Half fractal&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Golden_Square_fractal_6.png&amp;diff=20548</id>
		<title>文件:Golden Square fractal 6.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Golden_Square_fractal_6.png&amp;diff=20548"/>
		<updated>2020-12-27T14:15:29Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;[[Golden ratio|Golden square]] fractal&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Golden_Square_fractal_with_T-branching_8.svg.png&amp;diff=20547</id>
		<title>文件:Golden Square fractal with T-branching 8.svg.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Golden_Square_fractal_with_T-branching_8.svg.png&amp;diff=20547"/>
		<updated>2020-12-27T14:14:15Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;[[Golden ratio|Golden squares]] with [[T-square (fractal)|T-branching]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Menger_sponge_(IFS).jpg&amp;diff=20545</id>
		<title>文件:Menger sponge (IFS).jpg</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Menger_sponge_(IFS).jpg&amp;diff=20545"/>
		<updated>2020-12-27T14:12:52Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;[[Menger sponge]], a 3-Dimensional IFS.&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Fractal_fern_explained.png&amp;diff=20543</id>
		<title>文件:Fractal fern explained.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Fractal_fern_explained.png&amp;diff=20543"/>
		<updated>2020-12-27T14:11:04Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;[[Barnsley's fern]], an early IFS&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Ifs-construction.png&amp;diff=20541</id>
		<title>文件:Ifs-construction.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Ifs-construction.png&amp;diff=20541"/>
		<updated>2020-12-27T14:08:45Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;IFS being made with two functions. IFS由两个函数构建。&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Chaosgame.gif&amp;diff=20540</id>
		<title>文件:Chaosgame.gif</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Chaosgame.gif&amp;diff=20540"/>
		<updated>2020-12-27T14:07:55Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;Construction of an IFS by the [[chaos game]] (animated)&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Chris_Ursitti_fractal_0000.png&amp;diff=20531</id>
		<title>文件:Chris Ursitti fractal 0000.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Chris_Ursitti_fractal_0000.png&amp;diff=20531"/>
		<updated>2020-12-27T13:21:57Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;Colored IFS designed using [[Apophysis (software)|Apophysis]] software and rendered by the [[Electric Sheep]].&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Sierpinski_triangle.png&amp;diff=20529</id>
		<title>文件:Sierpinski triangle.png</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:Sierpinski_triangle.png&amp;diff=20529"/>
		<updated>2020-12-27T13:03:33Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;图片1&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%B3%BB%E7%BB%9F%E5%8A%A8%E5%8A%9B%E5%AD%A6%E8%BD%AF%E4%BB%B6%E5%AF%B9%E6%AF%94&amp;diff=20072</id>
		<title>系统动力学软件对比</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%B3%BB%E7%BB%9F%E5%8A%A8%E5%8A%9B%E5%AD%A6%E8%BD%AF%E4%BB%B6%E5%AF%B9%E6%AF%94&amp;diff=20072"/>
		<updated>2020-12-13T14:53:11Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;This is a comparison of various aspects of software offering [[system dynamics]] features:&lt;br /&gt;
这是对提供&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[系统动力学]] system dynamics&amp;lt;/font&amp;gt;功能的软件各方面的比较：&lt;br /&gt;
&lt;br /&gt;
''Due to concerns over commercial postings on the system dynamics main topic, commercial hyperlinks are specifically NOT active on this list.''&lt;br /&gt;
&amp;quot;由于考虑到系统动力学主要专题的商业性帖子的性质，商业性超链接在本清单上是特殊的无法点击的。&amp;quot;&lt;br /&gt;
&lt;br /&gt;
==Table of system dynamics software==&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable sortable&amp;quot;&lt;br /&gt;
|+ Sortable table&lt;br /&gt;
|-&lt;br /&gt;
! Package name 包名&lt;br /&gt;
! [[Software license|Licensing]] 软件许可证&lt;br /&gt;
! Implementation language 实现语言&lt;br /&gt;
! Last update (year) 上次更新（年）&lt;br /&gt;
! More info 更多信息&lt;br /&gt;
|-&lt;br /&gt;
| [[AMESim]]&lt;br /&gt;
| [[Proprietary]], commercial 专有的，商业的&lt;br /&gt;
| C&lt;br /&gt;
| 2017&lt;br /&gt;
| Distributed by Siemens. Modelica-models supported. 由西门子开发。支持Modelica模型。&lt;br /&gt;
|-&lt;br /&gt;
|-&lt;br /&gt;
| [[Analytica (software)|Analytica]] 分析软件&lt;br /&gt;
| [[Proprietary]], [[Commercial software|commercial]], free limited version 免费限量版&lt;br /&gt;
| [[C++]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Supports system dynamics, Monte Carlo simulation for uncertainty, array abstraction for handling multidimensional data, linear and non-linear optimization. Uses influence diagrams to define, navigate, and document models. 支持&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;系统动力学 system dynamics&amp;lt;/font&amp;gt;，用于不确定性的 蒙特卡罗模拟 Monte Carlo simulation，处理多维数据的数组抽象，线性和非线性优化。使用&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;影响图 influence diagrams&amp;lt;/font&amp;gt;来定义、导航和记录模型。&lt;br /&gt;
|-&lt;br /&gt;
| [[AnyLogic]]&lt;br /&gt;
| [[Proprietary]], commercial, free Personal Learning Edition (PLE) for education, formal or not&lt;br /&gt;
| [[Java (programming language)|Java]]&lt;br /&gt;
| 2020&lt;br /&gt;
| Supports system dynamics, agent based and discrete event modeling, allows making hybrid models. 支持系统动力学、基于代理和离散事件建模，允许制作&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;混合模型 hybrid models&amp;lt;/font&amp;gt;。&lt;br /&gt;
|-&lt;br /&gt;
| [[ASCEND]]&lt;br /&gt;
| {{Free}}, [[GNU General Public License]] (GPL)&lt;br /&gt;
| [[C (programming language)|C]]&lt;br /&gt;
| 2012&lt;br /&gt;
| For solving small to very large mathematical models, systems of non-linear equations, linear and nonlinear optimisation problems, dynamic systems expressed as differential-algebraic equations. 用于求解小型到特大型数学模型、非线性方程系统、线性和非线性优化问题、用微分代数方程表示的动态系统。&lt;br /&gt;
|-&lt;br /&gt;
| [[Berkeley Madonna]]&lt;br /&gt;
| [[Proprietary]], [[shareware]]&lt;br /&gt;
| C++, Java&lt;br /&gt;
| 2018&lt;br /&gt;
| Developed on Berkeley campus under sponsorship of NSF and NIH, used by academic and commercial institutions to build mathematical models for research and teaching. Solves ordinary differential equations (initial conditions and boundary value problems), difference equations (initial conditions and boundary value problems), multi-dimensional transcendental algebraic equation roots, discrete simulations using conveyors, ovens, and queues. Suitable for large-scale systems, boundary value problems, Monte Carlo models, curve fitting, root finding, batch processes, parameter plots, stiff systems, etc. 在美国国家科学基金会和美国国立卫生研究院的赞助下，在伯克利校园内开发，被学术和商业机构用于建立数学模型，用于研究和教学。解决普通微分方程（初始条件和边界值问题）、差分方程（初始条件和边界值问题）、多维超代数方程根、使用&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;传送带、烘箱&amp;lt;/font&amp;gt;和队列进行离散模拟。适用于大规模系统、边界值问题、蒙特卡洛模型、曲线拟合、寻根、批处理、参数图、刚性系统等。&lt;br /&gt;
|-&lt;br /&gt;
| [[DYNAMO (programming language)|DYNAMO]]&lt;br /&gt;
| [[Proprietary]], no longer distributed commercially&lt;br /&gt;
| AED, [[Pascal (programming language)|Pascal]]&lt;br /&gt;
| 1986&lt;br /&gt;
| Historic DYNAMO models are often available at the MIT system dynamics website. DYNAMO software for microcomputers may be available via eBay or other resale sites. 在麻省理工学院的系统动力学网站上通常可以获得历史上的DYNAMO模型。用于微机的DYNAMO软件可通过eBay或其他转售网站获得。&lt;br /&gt;
|-&lt;br /&gt;
|[[GoldSim]]&lt;br /&gt;
|[[Proprietary]], commercial&lt;br /&gt;
|C++&lt;br /&gt;
|2019&lt;br /&gt;
|Differs from traditional system dynamics approaches in that 1) it puts much greater emphasis on probabilistic simulation techniques to support representation of uncertain and/or stochastic systems; and 2) it provides a wide variety of specialized model objects (beyond stocks, flows and converters) in order to make models less abstract (and hence more transparent) and help represent processes and events that cannot easily be represented using a traditional system dynamics approach. These differences are due to the fact that GoldSim is primarily used for engineering and scientific applications where quantitative probabilistic predictions of future performance are required. 与传统系统动力学方法的不同之处在于：1）它更加强调概率模拟技术，以支持不确定和/或随机系统的表示；2）它提供了多种专门的模型对象（除&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;库存、流量和转换器&amp;lt;/font&amp;gt;外），以使模型不那么抽象（因此更加透明），并帮助表示那些无法用传统系统动力学方法轻松表示的过程和事件。这些差异是由于GoldSim主要用于需要对未来性能进行定量概率预测的工程和科学应用。&lt;br /&gt;
|-&lt;br /&gt;
|[https://insightmaker.com/ Insight Maker]&lt;br /&gt;
|Free,  Insight Maker Public License (GPL adjusted)&lt;br /&gt;
|JavaScript&lt;br /&gt;
|2020&lt;br /&gt;
|Insight Maker supports System Dynamics modeling: a powerful method for exploring systems on an aggregate level. It is 100% browser based with an open support group using Google mail list. Insight Maker 支持系统动力学建模：一个强大的方法，用于探索系统的总体水平。它是100%基于浏览器的，有一个使用Google邮件列表的开放支持组。&lt;br /&gt;
|-&lt;br /&gt;
| [[iThink]]&lt;br /&gt;
| [[Proprietary]], commercial 专有的，商业的&lt;br /&gt;
| C++, Javascript&lt;br /&gt;
| 2018&lt;br /&gt;
| System dynamics and discrete event modeling with some agent-based capabilities. Drag and drop user interface builder allows simulations to be published online. Includes multilevel hierarchical models, reusable modules, multidimensional arrays, optimization, and Monte Carlo analysis. 系统动力学和离散事件建模，具有一些基于代理的功能。拖放式用户界面构建器允许在线发布模拟结果。包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。&lt;br /&gt;
|-&lt;br /&gt;
|[https://ncase.me/loopy/ LOOPY]&lt;br /&gt;
|Free, CC Zero license&lt;br /&gt;
|JavaScript&lt;br /&gt;
|2019&lt;br /&gt;
|A tool for thinking in systems. Users draw circles and lines to build an interactive simulation of a complex system. 一个系统领域的思考工具。用户通过圆形和线条对复杂系统进行交互式模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[MapleSim]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| Java (GUI), C, [[Maple (software)|Maple]] (engine)&lt;br /&gt;
| 2017&lt;br /&gt;
| [[Modelica]]-based system-level modeling tool. Leverages symbolic computing via tight integration with Maple. 基于[[Modelica]]的系统级建模工具。通过与Maple的紧密集成，使用符号计算。&lt;br /&gt;
|-&lt;br /&gt;
| [[NetLogo]]&lt;br /&gt;
| {{Free}}, GPLv2&lt;br /&gt;
| Java, [[Scala (programming language)|Scala]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Agent-based modeling environment in LOGO; supports system dynamics models as a secondary feature. LOGO中的建模环境是基于代理的；支持系统动力学模型作为辅助功能。&lt;br /&gt;
|-&lt;br /&gt;
| [[OpenModelica]]&lt;br /&gt;
| {{Free}}, OSMC Public License, [[Eclipse Public License]] (EPL), [[GNU General Public License]] (GPL) &lt;br /&gt;
| C, C++, MetaModelica&lt;br /&gt;
| 2018&lt;br /&gt;
| [[Modelica]]-based modeling and simulation environment. Modelica is an object-oriented, declarative, multi-domain modeling language for component-oriented modeling of complex systems; there is a freely available [https://www.modelica.org/libraries System Dynamics library]. 基于[[Modelica]]的建模和仿真环境。Modelica是一种面向对象、声明式、多域建模语言，用于复杂系统的组件化建模；有一个免费版本[https://www.modelica.org/libraries System Dynamics library]。&lt;br /&gt;
|-&lt;br /&gt;
|Powersim Studio&lt;br /&gt;
|Proprietary, commercial, free limited version&lt;br /&gt;
|C++&lt;br /&gt;
|2018&lt;br /&gt;
|Supports system dynamics; building graphical diagrams using stocks and flow, including delays and feedback for non-linear models. Supports units, multi-dimensions running scenario simulations and Monte Carlo simulations. 支持系统动力学；利用存量和流量建立 图形图 graphical diagrams，包括非线性模型的延迟和反馈。支持单位、多维度运行情景模拟和蒙特卡洛模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[Sheetless]]&lt;br /&gt;
| Proprietary, commercial, free online version&lt;br /&gt;
| Rust, Javascript&lt;br /&gt;
| 2020&lt;br /&gt;
| System dynamics with some discrete event and agent-based capabilities. Web-based modelling with drag and drop user interface makes it easy to publish and share. Extensive data integrations and model export as code targeting R, Python, C, Rust, Javascript for Data Science workflows. 系统动力学，具有一些离散事件和基于代理的功能。基于网络的建模和支持拖放的用户界面使其易于发布和共享。广泛的数据集成和模型导出为针对R、Python、C、Rust、Javascript的数据科学工作流的代码。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simcad Pro]]&lt;br /&gt;
| Proprietary, commercial, free full featured demo&lt;br /&gt;
| Visual interface&lt;br /&gt;
| 2019&lt;br /&gt;
| Interactive process simulation software with support for manufacturing, healthcare and supply chain. 2D and 3D visualization with VR capability. 交互式流程模拟软件，支持制造、医疗和供应链。具有VR功能的2D和3D可视化。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simantics System Dynamics]]&lt;br /&gt;
| {{Free}}, [[Eclipse Public License]] (EPL)&lt;br /&gt;
| Java, [[Modelica]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Free and open source system dynamics modelling software with stock and flow modelling, hierarchical models and array variables. 免费和开源的系统动力学建模软件，具有存量和流量建模、分层模型和阵列变量。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simulink]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| ?&lt;br /&gt;
| 2019&lt;br /&gt;
| Tight integration with [[MATLAB]]&lt;br /&gt;
|-&lt;br /&gt;
| [https://statsim.com StatSim]&lt;br /&gt;
| {{Free}}, open-source&lt;br /&gt;
| [[JavaScript]], [[WebAssembly]], WebPPL&lt;br /&gt;
| 2020&lt;br /&gt;
| Free web app for statistical simulations and system dynamics modelling. Supports model fitting with Bayesian inference and mathematical optimization. 用于统计模拟和系统动力学建模的免费网络应用。支持贝叶斯推理和数学优化的模型拟合。&lt;br /&gt;
|-&lt;br /&gt;
| [[STELLA (programming language)|Stella]]&lt;br /&gt;
| [[Proprietary]], commercial, limited free online version&lt;br /&gt;
| C++, Javascript&lt;br /&gt;
| 2020&lt;br /&gt;
| System dynamics and discrete event modeling with some agent-based capabilities. Drag and drop user interface builder in Architect versions allows simulations to be published online. Includes multilevel hierarchical models, reusable modules, multidimensional arrays, optimization, and Monte Carlo analysis. 系统动力学和离散事件建模，具有一些基于代理的功能。在Architect版本中，拖放式的用户界面构建器允许在线发布模拟结果，包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。&lt;br /&gt;
|-&lt;br /&gt;
|[https://www.true-world.com/htm/en/index.html TRUE]&lt;br /&gt;
|Free, proprietary&lt;br /&gt;
|Wlanguage&lt;br /&gt;
|2019&lt;br /&gt;
|The System Dynamics software TRUE (Temporal Reasoning Universal Elaboration), developed by True-World System Dynamics, is a tool for modeling, simulating, analyzing and optimizing multidomain dynamic applications. True-World 系统动力学公司开发的系统动力学软件TRUE(Temporal Reasoning Universal Elaboration)是一个多领域动态应用的建模、仿真、分析和优化工具。&lt;br /&gt;
|-&lt;br /&gt;
| [[Vensim]]&lt;br /&gt;
| [[Proprietary]], commercial, free Personal Learning Edition (PLE) for education and personal use&lt;br /&gt;
| C, C++&lt;br /&gt;
| 2020&lt;br /&gt;
| Continuous simulation with stocks and flows, some discrete delay and discrete event functionality. Flexible array syntax with mapping among dimensions. Extensive support for time series data import and export, with calibration optimization and [[Markov chain Monte Carlo]] for estimation. Monte Carlo and other sensitivity simulation methods. Graphical model construction and interfaces. External functions and compiled simulation. 具有存货和流量的连续模拟，一些离散延迟和离散事件功能。灵活的数组语法与维度之间的映射。广泛支持时间序列数据的导入和导出，具有校准优化和[[马尔可夫链蒙特卡洛]]估计功能。蒙特卡洛和其他敏感性模拟方法。图形化的模型构建和界面。支持外部函数和编译模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[Ventity]]&lt;br /&gt;
| [[Proprietary]], commercial, free academic version&lt;br /&gt;
| C#&lt;br /&gt;
| 2019&lt;br /&gt;
| Entity-based simulation, supporting classic SD as well as detailed and agent simulations with dynamic creation of structure, ad hoc data and sparse matrices. Collaboration and source control friendly. Continuous simulation with stocks and flows and discrete event actions between time steps. Time series and initialization data import and export, with calibration optimization sensitivity simulation. Graphical model construction and interface. Geospatial viewer for GIS-based simulation. 基于实体 entity 的模拟，支持尽可能详细的经典SD以及可动态创建结构、特设数据和稀疏矩阵。协作和源控制友好的代理模拟。连续模拟，支持存货和流量，以及时间步长之间的离散事件动作，支持时间序列和初始化数据的导入和导出。时间序列和初始化数据的导入和导出，具有校准优化灵敏度模拟功能。图形化的模型构建和界面。地理空间查看器，用于基于GIS的模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[VisSim]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| C&lt;br /&gt;
| 2011&lt;br /&gt;
| Accredited education institutions are allowed to site license VisSim v3.0 for free. The latest versions, and add-ons, are available to students and academic institutions at reduced pricing. A read-only version of the software, VisSim Viewer is available for free and provides a way for unlicensed users to run VisSim models. 经认可的教育机构可以免费获得VisSim v3.0。学生和学术机构可以以优惠价格获得最新版本和附加组件。VisSim Viewer是该软件的只读版本，可免费使用，为未获得许可的用户提供运行VisSim模型的方法。&lt;br /&gt;
|-&lt;br /&gt;
| [[Wolfram SystemModeler]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| &lt;br /&gt;
| 2019&lt;br /&gt;
| Supports system dynamics, discrete event modeling, external C-functions, hierarchical models, and the [[Modelica]] modeling language with tight integration with [[Mathematica]]. Models can be exported to run as standalone applications for users without SystemModeler. 支持系统动力学、离散事件建模、外部C函数、分层模型以及[[Modelica]]建模语言，并与[[Mathematica]]紧密集成。模型可以导出为独立的应用程序，供没有SystemModeler的用户运行。&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
:L.[[Note|NoteL]]Graphing library MxGraph is licensed separately under paid commercial licence&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
&lt;br /&gt;
* [[List of computer simulation software]]&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
System dynamics&lt;br /&gt;
系统动力学&lt;br /&gt;
&lt;br /&gt;
[[Category:Systems theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Systems theory&lt;br /&gt;
范畴: 系统论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Comparison of system dynamics software]]. Its edit history can be viewed at [[系统动力学软件对比/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E7%B3%BB%E7%BB%9F%E5%8A%A8%E5%8A%9B%E5%AD%A6%E8%BD%AF%E4%BB%B6%E5%AF%B9%E6%AF%94&amp;diff=20071</id>
		<title>系统动力学软件对比</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E7%B3%BB%E7%BB%9F%E5%8A%A8%E5%8A%9B%E5%AD%A6%E8%BD%AF%E4%BB%B6%E5%AF%B9%E6%AF%94&amp;diff=20071"/>
		<updated>2020-12-13T14:51:16Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;This is a comparison of various aspects of software offering [[system dynamics]] features:&lt;br /&gt;
这是对提供&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;[[系统动力学]] system dynamics&amp;lt;/font&amp;gt;功能的软件各方面的比较：&lt;br /&gt;
&lt;br /&gt;
''Due to concerns over commercial postings on the system dynamics main topic, commercial hyperlinks are specifically NOT active on this list.''&lt;br /&gt;
&amp;quot;由于考虑到系统动力学主要专题的商业性帖子的性质，商业性超链接在本清单上是特殊的无法点击的。&amp;quot;&lt;br /&gt;
&lt;br /&gt;
==Table of system dynamics software==&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable sortable&amp;quot;&lt;br /&gt;
|+ Sortable table&lt;br /&gt;
|-&lt;br /&gt;
! Package name 包名&lt;br /&gt;
! [[Software license|Licensing]] 软件许可证&lt;br /&gt;
! Implementation language 实现语言&lt;br /&gt;
! Last update (year) 上次更新（年）&lt;br /&gt;
! More info 更多信息&lt;br /&gt;
|-&lt;br /&gt;
| [[AMESim]]&lt;br /&gt;
| [[Proprietary]], commercial 专有的，商业的&lt;br /&gt;
| C&lt;br /&gt;
| 2017&lt;br /&gt;
| Distributed by Siemens. Modelica-models supported. 由西门子开发。支持Modelica模型。&lt;br /&gt;
|-&lt;br /&gt;
|-&lt;br /&gt;
| [[Analytica (software)|Analytica]] 分析软件&lt;br /&gt;
| [[Proprietary]], [[Commercial software|commercial]], free limited version 免费限量版&lt;br /&gt;
| [[C++]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Supports system dynamics, Monte Carlo simulation for uncertainty, array abstraction for handling multidimensional data, linear and non-linear optimization. Uses influence diagrams to define, navigate, and document models. 支持&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;系统动力学 system dynamics&amp;lt;/font&amp;gt;，用于不确定性的 蒙特卡罗模拟 Monte Carlo simulation，处理多维数据的数组抽象，线性和非线性优化。使用&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;影响图 influence diagrams&amp;lt;/font&amp;gt;来定义、导航和记录模型。&lt;br /&gt;
|-&lt;br /&gt;
| [[AnyLogic]]&lt;br /&gt;
| [[Proprietary]], commercial, free Personal Learning Edition (PLE) for education, formal or not&lt;br /&gt;
| [[Java (programming language)|Java]]&lt;br /&gt;
| 2020&lt;br /&gt;
| Supports system dynamics, agent based and discrete event modeling, allows making hybrid models. 支持系统动力学、基于代理和离散事件建模，允许制作&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;混合模型 hybrid models&amp;lt;/font&amp;gt;。&lt;br /&gt;
|-&lt;br /&gt;
| [[ASCEND]]&lt;br /&gt;
| {{Free}}, [[GNU General Public License]] (GPL)&lt;br /&gt;
| [[C (programming language)|C]]&lt;br /&gt;
| 2012&lt;br /&gt;
| For solving small to very large mathematical models, systems of non-linear equations, linear and nonlinear optimisation problems, dynamic systems expressed as differential-algebraic equations. 用于求解小型到特大型数学模型、非线性方程系统、线性和非线性优化问题、用微分代数方程表示的动态系统。&lt;br /&gt;
|-&lt;br /&gt;
| [[Berkeley Madonna]]&lt;br /&gt;
| [[Proprietary]], [[shareware]]&lt;br /&gt;
| C++, Java&lt;br /&gt;
| 2018&lt;br /&gt;
| Developed on Berkeley campus under sponsorship of NSF and NIH, used by academic and commercial institutions to build mathematical models for research and teaching. Solves ordinary differential equations (initial conditions and boundary value problems), difference equations (initial conditions and boundary value problems), multi-dimensional transcendental algebraic equation roots, discrete simulations using conveyors, ovens, and queues. Suitable for large-scale systems, boundary value problems, Monte Carlo models, curve fitting, root finding, batch processes, parameter plots, stiff systems, etc. 在美国国家科学基金会和美国国立卫生研究院的赞助下，在伯克利校园内开发，被学术和商业机构用于建立数学模型，用于研究和教学。解决普通微分方程（初始条件和边界值问题）、差分方程（初始条件和边界值问题）、多维超代数方程根、使用&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;传送带、烘箱&amp;lt;/font&amp;gt;和队列进行离散模拟。适用于大规模系统、边界值问题、蒙特卡洛模型、曲线拟合、寻根、批处理、参数图、刚性系统等。&lt;br /&gt;
|-&lt;br /&gt;
| [[DYNAMO (programming language)|DYNAMO]]&lt;br /&gt;
| [[Proprietary]], no longer distributed commercially&lt;br /&gt;
| AED, [[Pascal (programming language)|Pascal]]&lt;br /&gt;
| 1986&lt;br /&gt;
| Historic DYNAMO models are often available at the MIT system dynamics website. DYNAMO software for microcomputers may be available via eBay or other resale sites. 在麻省理工学院的系统动力学网站上通常可以获得历史上的DYNAMO模型。用于微机的DYNAMO软件可通过eBay或其他转售网站获得。&lt;br /&gt;
|-&lt;br /&gt;
|[[GoldSim]]&lt;br /&gt;
|[[Proprietary]], commercial&lt;br /&gt;
|C++&lt;br /&gt;
|2019&lt;br /&gt;
|Differs from traditional system dynamics approaches in that 1) it puts much greater emphasis on probabilistic simulation techniques to support representation of uncertain and/or stochastic systems; and 2) it provides a wide variety of specialized model objects (beyond stocks, flows and converters) in order to make models less abstract (and hence more transparent) and help represent processes and events that cannot easily be represented using a traditional system dynamics approach. These differences are due to the fact that GoldSim is primarily used for engineering and scientific applications where quantitative probabilistic predictions of future performance are required. 与传统系统动力学方法的不同之处在于：1）它更加强调概率模拟技术，以支持不确定和/或随机系统的表示；2）它提供了多种专门的模型对象（除&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;库存、流量和转换器&amp;lt;/font&amp;gt;外），以使模型不那么抽象（因此更加透明），并帮助表示那些无法用传统系统动力学方法轻松表示的过程和事件。这些差异是由于GoldSim主要用于需要对未来性能进行定量概率预测的工程和科学应用。&lt;br /&gt;
|-&lt;br /&gt;
|[https://insightmaker.com/ Insight Maker]&lt;br /&gt;
|Free,  Insight Maker Public License (GPL adjusted)&lt;br /&gt;
|JavaScript&lt;br /&gt;
|2020&lt;br /&gt;
|Insight Maker supports System Dynamics modeling: a powerful method for exploring systems on an aggregate level. It is 100% browser based with an open support group using Google mail list. Insight Maker 支持系统动力学建模：一个强大的方法，用于探索系统的总体水平。它是100%基于浏览器的，有一个使用Google邮件列表的开放支持组。&lt;br /&gt;
|-&lt;br /&gt;
| [[iThink]]&lt;br /&gt;
| [[Proprietary]], commercial 专有的，商业的&lt;br /&gt;
| C++, Javascript&lt;br /&gt;
| 2018&lt;br /&gt;
| System dynamics and discrete event modeling with some agent-based capabilities. Drag and drop user interface builder allows simulations to be published online. Includes multilevel hierarchical models, reusable modules, multidimensional arrays, optimization, and Monte Carlo analysis. 系统动力学和离散事件建模，具有一些基于代理的功能。拖放式用户界面构建器允许在线发布模拟结果。包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。&lt;br /&gt;
|-&lt;br /&gt;
|[https://ncase.me/loopy/ LOOPY]&lt;br /&gt;
|Free, CC Zero license&lt;br /&gt;
|JavaScript&lt;br /&gt;
|2019&lt;br /&gt;
|A tool for thinking in systems. Users draw circles and lines to build an interactive simulation of a complex system. 一个系统领域的思考工具。用户通过圆形和线条对复杂系统进行交互式模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[MapleSim]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| Java (GUI), C, [[Maple (software)|Maple]] (engine)&lt;br /&gt;
| 2017&lt;br /&gt;
| [[Modelica]]-based system-level modeling tool. Leverages symbolic computing via tight integration with Maple. 基于[[Modelica]]的系统级建模工具。通过与Maple的紧密集成，使用符号计算。&lt;br /&gt;
|-&lt;br /&gt;
| [[NetLogo]]&lt;br /&gt;
| {{Free}}, GPLv2&lt;br /&gt;
| Java, [[Scala (programming language)|Scala]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Agent-based modeling environment in LOGO; supports system dynamics models as a secondary feature. LOGO中的建模环境是基于代理的；支持系统动力学模型作为辅助功能。&lt;br /&gt;
|-&lt;br /&gt;
| [[OpenModelica]]&lt;br /&gt;
| {{Free}}, OSMC Public License, [[Eclipse Public License]] (EPL), [[GNU General Public License]] (GPL) &lt;br /&gt;
| C, C++, MetaModelica&lt;br /&gt;
| 2018&lt;br /&gt;
| [[Modelica]]-based modeling and simulation environment. Modelica is an object-oriented, declarative, multi-domain modeling language for component-oriented modeling of complex systems; there is a freely available [https://www.modelica.org/libraries System Dynamics library]. 基于[[Modelica]]的建模和仿真环境。Modelica是一种面向对象、声明式、多域建模语言，用于复杂系统的组件化建模；有一个免费版本[https://www.modelica.org/libraries System Dynamics library]。&lt;br /&gt;
|-&lt;br /&gt;
|Powersim Studio&lt;br /&gt;
|Proprietary, commercial, free limited version&lt;br /&gt;
|C++&lt;br /&gt;
|2018&lt;br /&gt;
|Supports system dynamics; building graphical diagrams using stocks and flow, including delays and feedback for non-linear models. Supports units, multi-dimensions running scenario simulations and Monte Carlo simulations. 支持系统动力学；利用存量和流量建立 图形图 graphical diagrams，包括非线性模型的延迟和反馈。支持单位、多维度运行情景模拟和蒙特卡洛模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[Sheetless]]&lt;br /&gt;
| Proprietary, commercial, free online version&lt;br /&gt;
| Rust, Javascript&lt;br /&gt;
| 2020&lt;br /&gt;
| System dynamics with some discrete event and agent-based capabilities. Web-based modelling with drag and drop user interface makes it easy to publish and share. Extensive data integrations and model export as code targeting R, Python, C, Rust, Javascript for Data Science workflows. 系统动力学，具有一些离散事件和基于代理的功能。基于网络的建模和支持拖放的用户界面使其易于发布和共享。广泛的数据集成和模型导出为针对R、Python、C、Rust、Javascript的数据科学工作流的代码。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simcad Pro]]&lt;br /&gt;
| Proprietary, commercial, free full featured demo&lt;br /&gt;
| Visual interface&lt;br /&gt;
| 2019&lt;br /&gt;
| Interactive process simulation software with support for manufacturing, healthcare and supply chain. 2D and 3D visualization with VR capability. 交互式流程模拟软件，支持制造、医疗和供应链。具有VR功能的2D和3D可视化。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simantics System Dynamics]]&lt;br /&gt;
| {{Free}}, [[Eclipse Public License]] (EPL)&lt;br /&gt;
| Java, [[Modelica]]&lt;br /&gt;
| 2018&lt;br /&gt;
| Free and open source system dynamics modelling software with stock and flow modelling, hierarchical models and array variables. 免费和开源的系统动力学建模软件，具有存量和流量建模、分层模型和阵列变量。&lt;br /&gt;
|-&lt;br /&gt;
| [[Simulink]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| ?&lt;br /&gt;
| 2019&lt;br /&gt;
| Tight integration with [[MATLAB]]&lt;br /&gt;
|-&lt;br /&gt;
| [https://statsim.com StatSim]&lt;br /&gt;
| {{Free}}, open-source&lt;br /&gt;
| [[JavaScript]], [[WebAssembly]], WebPPL&lt;br /&gt;
| 2020&lt;br /&gt;
| Free web app for statistical simulations and system dynamics modelling. Supports model fitting with Bayesian inference and mathematical optimization. 用于统计模拟和系统动力学建模的免费网络应用。支持贝叶斯推理和数学优化的模型拟合。&lt;br /&gt;
|-&lt;br /&gt;
| [[STELLA (programming language)|Stella]]&lt;br /&gt;
| [[Proprietary]], commercial, limited free online version&lt;br /&gt;
| C++, Javascript&lt;br /&gt;
| 2020&lt;br /&gt;
| System dynamics and discrete event modeling with some agent-based capabilities. Drag and drop user interface builder in Architect versions allows simulations to be published online. Includes multilevel hierarchical models, reusable modules, multidimensional arrays, optimization, and Monte Carlo analysis. 系统动力学和离散事件建模，具有一些基于代理的功能。在Architect版本中，拖放式的用户界面构建器允许在线发布模拟结果，包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。包括多级分层模型、可重用模块、多维阵列、优化和蒙特卡洛分析。&lt;br /&gt;
|-&lt;br /&gt;
|[https://www.true-world.com/htm/en/index.html TRUE]&lt;br /&gt;
|Free, proprietary&lt;br /&gt;
|Wlanguage&lt;br /&gt;
|2019&lt;br /&gt;
|The System Dynamics software TRUE (Temporal Reasoning Universal Elaboration), developed by True-World System Dynamics, is a tool for modeling, simulating, analyzing and optimizing multidomain dynamic applications. True-World 系统动力学公司开发的系统动力学软件TRUE(Temporal Reasoning Universal Elaboration)是一个多领域动态应用的建模、仿真、分析和优化工具。&lt;br /&gt;
|-&lt;br /&gt;
| [[Vensim]]&lt;br /&gt;
| [[Proprietary]], commercial, free Personal Learning Edition (PLE) for education and personal use&lt;br /&gt;
| C, C++&lt;br /&gt;
| 2020&lt;br /&gt;
| Continuous simulation with stocks and flows, some discrete delay and discrete event functionality. Flexible array syntax with mapping among dimensions. Extensive support for time series data import and export, with calibration optimization and [[Markov chain Monte Carlo]] for estimation. Monte Carlo and other sensitivity simulation methods. Graphical model construction and interfaces. External functions and compiled simulation. 具有存货和流量的连续模拟，一些离散延迟和离散事件功能。灵活的数组语法与维度之间的映射。广泛支持时间序列数据的导入和导出，具有校准优化和[[马尔可夫链蒙特卡洛]]估计功能。蒙特卡洛和其他敏感性模拟方法。图形化的模型构建和界面。支持外部函数和编译模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[Ventity]]&lt;br /&gt;
| [[Proprietary]], commercial, free academic version&lt;br /&gt;
| C#&lt;br /&gt;
| 2019&lt;br /&gt;
| Entity-based simulation, supporting classic SD as well as detailed and agent simulations with dynamic creation of structure, ad hoc data and sparse matrices. Collaboration and source control friendly. Continuous simulation with stocks and flows and discrete event actions between time steps. Time series and initialization data import and export, with calibration optimization sensitivity simulation. Graphical model construction and interface. Geospatial viewer for GIS-based simulation. 基于实体 entity 的模拟，支持尽可能详细的经典SD以及可动态创建结构、特设数据和稀疏矩阵。协作和源控制友好的代理模拟。连续模拟，支持存货和流量，以及时间步长之间的离散事件动作，支持时间序列和初始化数据的导入和导出。时间序列和初始化数据的导入和导出，具有校准优化灵敏度模拟功能。图形化的模型构建和界面。地理空间查看器，用于基于GIS的模拟。&lt;br /&gt;
|-&lt;br /&gt;
| [[VisSim]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| C&lt;br /&gt;
| 2011&lt;br /&gt;
| Accredited education institutions are allowed to site license VisSim v3.0 for free. The latest versions, and add-ons, are available to students and academic institutions at reduced pricing. A read-only version of the software, VisSim Viewer is available for free and provides a way for unlicensed users to run VisSim models. 经认可的教育机构可以免费获得VisSim v3.0。学生和学术机构可以以优惠价格获得最新版本和附加组件。VisSim Viewer是该软件的只读版本，可免费使用，为未获得许可的用户提供运行VisSim模型的方法。&lt;br /&gt;
|-&lt;br /&gt;
| [[Wolfram SystemModeler]]&lt;br /&gt;
| [[Proprietary]], commercial&lt;br /&gt;
| &lt;br /&gt;
| 2019&lt;br /&gt;
| Supports system dynamics, discrete event modeling, external C-functions, hierarchical models, and the [[Modelica]] modeling language with tight integration with [[Mathematica]]. Models can be exported to run as standalone applications for users without SystemModeler. 支持系统动力学、离散事件建模、外部C函数、分层模型以及[[Modelica]]建模语言，并与[[Mathematica]]紧密集成。模型可以导出为独立的应用程序，供没有SystemModeler的用户运行。&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
:L.{{Note|NoteL}}Graphing library MxGraph is licensed separately under paid commercial licence&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
&lt;br /&gt;
* [[List of computer simulation software]]&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
System dynamics&lt;br /&gt;
系统动力学&lt;br /&gt;
&lt;br /&gt;
[[Category:Systems theory]]&lt;br /&gt;
&lt;br /&gt;
Category:Systems theory&lt;br /&gt;
范畴: 系统论&lt;br /&gt;
&lt;br /&gt;
&amp;lt;noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Comparison of system dynamics software]]. Its edit history can be viewed at [[系统动力学软件对比/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:待整理页面]]&lt;/div&gt;</summary>
		<author><name>Eatcosmos2</name></author>
	</entry>
	<entry>
		<id>https://wiki.swarma.org/index.php?title=%E6%B6%8C%E7%8E%B0%E8%BF%9B%E5%8C%96&amp;diff=20034</id>
		<title>涌现进化</title>
		<link rel="alternate" type="text/html" href="https://wiki.swarma.org/index.php?title=%E6%B6%8C%E7%8E%B0%E8%BF%9B%E5%8C%96&amp;diff=20034"/>
		<updated>2020-12-13T13:18:37Z</updated>

		<summary type="html">&lt;p&gt;Eatcosmos2：&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{short description|&lt;br /&gt;
[Emergent evolution \- Wikipedia](https://en.wikipedia.org/wiki/Emergent_evolution#:~:text=Emergent%20evolution%20was%20the%20hypothesis,of%20the%20already%20existing%20entities.)&lt;br /&gt;
[涌现进化 \- 集智百科](https://wiki.swarma.org/index.php?title=%E6%B6%8C%E7%8E%B0%E8%BF%9B%E5%8C%96)&lt;br /&gt;
}}&lt;br /&gt;
此词条暂由Eatcosmos2初步翻译。&lt;br /&gt;
&lt;br /&gt;
{{short description|The hypothesis that, in the course of evolution, some entirely new properties, such as mind and consciousness, appear at certain critical points}}&lt;br /&gt;
&lt;br /&gt;
'''Emergent evolution''' was the [[hypothesis]] that, in the course of [[evolution]], some entirely new properties, such as mind and [[consciousness]], appear at certain critical points, usually because of an unpredictable rearrangement of the already existing entities. The term was originated by the psychologist [[C. Lloyd Morgan]] in 1922 in his [[Gifford Lectures]] at St. Andrews, which would later be published as the 1923 book ''Emergent Evolution''.&amp;lt;ref&amp;gt;{{Cite book|url=http://archive.org/details/emergentevolutio00morg_0|title=Emergent evolution : the Gifford lectures, delivered in the University of St. Andrews in the year 1922|last=Morgan|first=Conway Lloyd|date=1923|publisher=New York : Henry Holt and Company ; London : William and Norgate|others=MIT Libraries}}&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;Bowler 2001&amp;quot;&amp;gt;Bowler, Peter J. (2001). ''Reconciling Science and Religion: The Debate in Early-Twentieth-Century Britain''. University of Chicago Press. pp. 140-142, pp. 376-384. {{ISBN|0-226-06858-7}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Emergent evolution was the hypothesis that, in the course of evolution, some entirely new properties, such as mind and consciousness, appear at certain critical points, usually because of an unpredictable rearrangement of the already existing entities. The term was originated by the psychologist C. Lloyd Morgan in 1922 in his Gifford Lectures at St. Andrews, which would later be published as the 1923 book Emergent Evolution.&lt;br /&gt;
 &lt;br /&gt;
&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;涌现进化&amp;lt;/font&amp;gt; Emergent evolution 是一种假设，即在进化过程中，一些全新的特性，如心智和意识，在某些关键点出现，通常是由于已经存在的实体发生了不可预测的重新排列。这个术语是心理学家 康韦·劳埃德·摩根 C. Lloyd Morgan 于1922年在圣安德鲁斯的《吉福德讲座》中提出的，该讲座后来作为1923年的《涌现进化论》一书出版。&lt;br /&gt;
&lt;br /&gt;
The hypothesis was widely criticized for providing no mechanism to how entirely new properties emerge, and for its historical roots in [[teleology]].&lt;br /&gt;
&amp;lt;ref name=&amp;quot;Bowler 2001&amp;quot;/&amp;gt;&lt;br /&gt;
&amp;lt;ref&amp;gt;McLaughlin, Brian P. (1992). ''The Rise and Fall of British Emergentism''. In A. Beckerman, H. Flohr, and J. Kim, eds., ''Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism''. Berlin: Walter de Gruyter. pp. 49–93. {{ISBN|3-11-012880-2}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;ref&amp;gt;Baylis, Charles A. (1929). ''The Philosophic Functions of Emergence''. ''[[The Philosophical Review]]''. Vol. 38, No. 4. pp. 372-384.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The hypothesis was widely criticized for providing no mechanism to how entirely new properties emerge, and for its historical roots in teleology.&lt;br /&gt;
&lt;br /&gt;
该假说被广泛批评，因为它没有为全新的属性如何涌现提供机制，而且它的历史根源于&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;目的论 teleology&amp;lt;/font&amp;gt;。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Historical context ==&lt;br /&gt;
&lt;br /&gt;
The term ''[[emergence|emergent]]'' was first used to describe the concept by [[George Lewes]] in volume two of his 1875 book ''Problems of Life and Mind'' (p.&amp;amp;nbsp;412).  [[Henri Bergson]] covered similar themes in his popular 1907 book ''[[Creative Evolution (book)|Creative Evolution]]'' on the [[Élan vital]].  Emergence was further developed by [[Samuel Alexander]] in his [[Gifford Lectures]] at [[University of Glasgow|Glasgow]] during 1916–18 and published as ''Space, Time, and Deity'' (1920).  The related term ''emergent evolution'' was coined by [[C. Lloyd Morgan]] in his own Gifford lectures of 1921–22 at [[University of St. Andrews|St. Andrews]] and published as ''Emergent Evolution'' (1923). In an appendix to a lecture in his book, Morgan acknowledged the contributions of [[Roy Wood Sellars]]'s ''Evolutionary Naturalism'' (1922).&lt;br /&gt;
&lt;br /&gt;
The term emergent was first used to describe the concept by George Lewes in volume two of his 1875 book Problems of Life and Mind (p.&amp;amp;nbsp;412).  Henri Bergson covered similar themes in his popular 1907 book Creative Evolution on the Élan vital.  Emergence was further developed by Samuel Alexander in his Gifford Lectures at Glasgow during 1916–18 and published as Space, Time, and Deity (1920).  The related term emergent evolution was coined by C. Lloyd Morgan in his own Gifford lectures of 1921–22 at St. Andrews and published as Emergent Evolution (1923). In an appendix to a lecture in his book, Morgan acknowledged the contributions of Roy Wood Sellars's Evolutionary Naturalism (1922).&lt;br /&gt;
&lt;br /&gt;
刘易士 George Lewes 在其1875年出版的《生命和心灵问题  Problems of Life and Mind》一书的第二卷中首次使用了涌现一词来描述这一概念(p.&amp;amp;nbsp;412)。 亨利·柏格森  Henri Bergson 在他1907年流行的《创造进化论 Creative Evolution》（Élan vital）一书中也涉及了类似的主题。 1916-1918年间，塞缪尔·亚历山大 Samuel Alexander 在格拉斯哥的《吉福德讲座 Gifford lectures》中进一步发展了涌现这个概念，并以《空间、时间和神性 Space, Time, and Deity》（1920）的形式出版。 相关的术语&amp;quot;涌现进化&amp;quot;是由 C. Lloyd Morgan 于1921-1922年在自己在圣安德鲁斯的吉福德讲座中创造的，并以《涌现进化 Emergent Evolution》（1923）出版。C. Lloyd Morgan 在其书中的一篇讲座的附录中，承认了 塞拉斯 Roy Wood Sellars的《进化的自然主义 Evolutionary Naturalism》（1922）的贡献。&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
== Origins ==&lt;br /&gt;
&lt;br /&gt;
=== Response to Darwin's ''Origin of Species'' ===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
[[Charles Darwin]] and [[Alfred Russel Wallace]]'s presentation of [[natural selection]], coupled to the idea of evolution in Western thought, had gained acceptance due to the wealth of observational data provided and the seeming replacement of divine law with natural law in the affairs of men.&amp;lt;ref name=Reid&amp;gt;{{cite book|last=Reid|first=Robert G.B.|title=Biological Emergences: Evolution by Natural Experiment|year=2007|publisher=MIT Press|location=Cambridge, Massachusetts|isbn=978-0262182577|pages=[https://archive.org/details/degreesthatmatte00rapp_0/page/30 30]|url=https://archive.org/details/degreesthatmatte00rapp_0/page/30}}&amp;lt;/ref&amp;gt; However, the mechanism of [[natural selection]] described at the time only explained how organisms adapted to variation. The cause of genetic variation was unknown at the time.&lt;br /&gt;
&lt;br /&gt;
Charles Darwin and Alfred Russel Wallace's presentation of natural selection, coupled to the idea of evolution in Western thought, had gained acceptance due to the wealth of observational data provided and the seeming replacement of divine law with natural law in the affairs of men. However, the mechanism of natural selection described at the time only explained how organisms adapted to variation. The cause of genetic variation was unknown at the time.&lt;br /&gt;
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查尔斯·达尔文 Charles Darwin 和 阿尔弗雷德·拉塞尔·华莱士 Alfred Russel Wallace 对自然选择的介绍，结合西方思想中的进化论思想，由于提供了大量的观察数据，以及在人的事务中似乎用自然法则取代了神的法则，因此得到了人们的认可。然而，当时所描述的自然选择机制只解释了生物体如何适应变异。遗传变异 genetic variation 的原因当时还不清楚。&lt;br /&gt;
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quote|&lt;br /&gt;
Darwin knew that nature had to produce variations before natural selection could act …The problem had been caught by other evolutionists almost as soon as ''[[The Origin of Species]]'' was first published. [[Sir Charles Lyell]] saw it clearly in 1860 before he even became an evolutionist…(Reid, p.3)&amp;lt;ref name=Reid /&amp;gt;}}&lt;br /&gt;
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[[St. George Jackson Mivart|St. George Jackson Mivart's]] ''On the Genesis of Species'' (1872) and [[Edward Drinker Cope|Edward Cope]]'s ''Origin of the Fittest'' (1887) raised the need to address the origin of variation between members of a species. [[William Bateson]] in 1884 distinguished between the origin of novel variations and the action of natural selection (''Materials for the Study of Variation Treated with Especial Regard to Discontinuity in the Origin of Species'').&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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In examining this aspect, excluded ab initio by Darwin, Wallace came to the conclusion that Life itself cannot be understood except by means of a theory that includes &amp;quot;an organising and directive Life-Principle.&amp;quot; These necessarily involve a &amp;quot;Creative Power&amp;quot;, a &amp;quot;directive Mind&amp;quot; and finally &amp;quot;an ultimate Purpose&amp;quot; (the development of Man). It supports the view of John Hunter that &amp;quot;life is the cause, not the consequence&amp;quot; of the organisation of matter. Thus, life precedes matter and when it infuses matter, forms living matter (protoplasm).&lt;br /&gt;
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在研究这个被 Darwin 从一开始就排除在外的方面时，Wallace 得出的结论是，除非通过一个包括 &amp;quot;&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;一个有组织性和指导性的生命准则&amp;lt;/font&amp;gt;&amp;quot; 的理论，否则无法理解生命本身。这些都必然涉及到 &amp;quot;创造性的力量 Creative Power&amp;quot;、&amp;quot;&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;指导性的心灵 directive Mind&amp;lt;/font&amp;gt;&amp;quot;以及最后的 &amp;quot;终极目的 an ultimate Purpose&amp;quot;（人的发展）。它支持 约翰·亨特 John Hunter  的观点，即 &amp;quot;生命是物质组织的原因，而不是结果&amp;quot;。因此，生命先于物质，当它注入物质时，形成了有生命的物质（&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;原生质 protoplasm&amp;lt;/font&amp;gt;）。&lt;br /&gt;
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=== Wallace's further thoughts ===&lt;br /&gt;
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Wallace throughout his life continued to support and extend the scope of Darwin's theory of evolution via the mechanism of natural selection. One of his works, ''Darwinism'' , was often cited in support of Darwin's theory. He also worked to elaborate and extend Darwin and his ideas on natural selection. However, Wallace also realized that the scope and claim of the theory was limited. Darwin himself had limited it.&lt;br /&gt;
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Wallace 一生都在支持和扩展 Darwin 的基于自然选择机制的进化论的范围。他提出的  ''达尔文主义'' 经常被引用来支持 Darwin 的理论。他还致力于阐述和扩展 Darwin 及其关于自然选择的思想。然而，Wallace 也意识到，该理论的范围和主张是有限的。Darwin 本人也对其进行了限制。&lt;br /&gt;
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quote|&lt;br /&gt;
the most prominent feature is that I enter into popular yet critical examination of those underlying fundamental problems which Darwin purposely excluded from his works as being beyond the scope of his enquiry. Such are the nature and cause of Life itself, and more especially of its most fundamental and mysterious powers - growth and reproduction …&lt;br /&gt;
最突出的特点是，我对 Darwin 有意从他的作品中排除的那些根本性问题&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;进行了通俗而批判的考察&amp;lt;/font&amp;gt;，因为这些问题超出了他的考察范围。如生命本身的性质和原因，尤其是生命最基本和最神秘的力量 - 生长和繁殖的性质和原因 …...&lt;br /&gt;
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Darwin always ... adduced the &amp;quot;laws of Growth with Reproduction,&amp;quot; and of &amp;quot;Inheritance with Variability,&amp;quot; as being fundamental facts of nature, without which Natural Selection would be powerless or even non-existent …&lt;br /&gt;
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Darwin 总是 ...... 引证 &amp;quot;生长与繁殖的&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;规律 laws&amp;lt;/font&amp;gt;&amp;quot; 和 &amp;quot;遗传与变异的规律&amp;quot; 是自然界的基本事实，没有这些规律，自然选择就无能为力，甚至不存在 ...... 。&lt;br /&gt;
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... even if it were proved to be an exact representation of the facts, it would not be an explanation... because it would not account for the forces, the directive agency, and the organising power which are essential features of growth …&amp;lt;ref name=Wallace&amp;gt;{{cite book|last=Wallace|first=Alfred Russel|title=Theory of Life|year=1911|publisher=Moffat, Yard and Co.|url=https://archive.org/details/worldlifeamanif00wallgoog}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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...... 即使它被证明是对事实的准确表述，它也不能作为一种解释。&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;因为它不能说明作为增长的必要特征的力量、指导机构和组织力量&amp;lt;/font&amp;gt; ......&lt;br /&gt;
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In examining this aspect, excluded ''ab initio'' by Darwin, Wallace came to the conclusion that ''Life'' itself cannot be understood except by means of a theory that includes &amp;quot;an organising and directive Life-Principle.&amp;quot; These necessarily involve a &amp;quot;Creative Power&amp;quot;, a &amp;quot;directive Mind&amp;quot; and finally &amp;quot;an ultimate Purpose&amp;quot; (the development of Man). It supports the view of [[John Hunter (surgeon)|John Hunter]] that &amp;quot;life is the cause, not the consequence&amp;quot; of the organisation of matter. Thus, life precedes matter and when it infuses matter, forms living matter (protoplasm).&lt;br /&gt;
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在研究这个被 Darwin 从一开始就排除在外的方面时，Wallace 得出的结论是，&amp;quot;生命&amp;quot; 本身不能被理解，除非通过一种包括 &amp;quot;组织和指导生命原则 &amp;quot;的理论。这些都必然涉及到 &amp;quot;创造性的力量&amp;quot;、&amp;quot;指导性的心灵&amp;quot; 和最后的 &amp;quot;终极目的&amp;quot;（人的发展）。它支持[[John Hunter (surgeon)|John Hunter]]的观点，即 &amp;quot;生命是物质组织的原因，而不是结果&amp;quot;。因此，生命先于物质，当它注入物质时，形成了有生命的物质（原生质）。&lt;br /&gt;
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quote|&lt;br /&gt;
a very well-founded doctrine, and one which was often advocated by John Hunter, that life is the cause and not the consequence of organisation ... if so, life must be antecedent to organisation, and can only be conceived as indissolubly connected with spirit and with thought, and with the cause of the directive energy everywhere manifested in the growth of living things ... endowed with the mysterious organising power we term life ...&amp;lt;ref name=&amp;quot;Wallace&amp;quot;/&amp;gt;&lt;br /&gt;
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一个非常有根据的学说，也是 John Hunter 经常倡导的学说，即生命是组织的原因而不是结果 ...... 如果是这样，生命一定是组织的前奏，而且只能被认为是与精神和思想密不可分地联系在一起的，而且是与在生物的生长中处处体现的指导性能量的原因联系在一起的 ...... 被赋予了我们称之为生命的这样一种神秘的组织力量 ......&lt;br /&gt;
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Wallace then refers to the operation of another power called &amp;quot;mind&amp;quot; that utilizes the power of life and is connected with a higher realm than life or matter:&lt;br /&gt;
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然后，Wallace 提到了另一种叫做 &amp;quot;心灵&amp;quot; 的力量的运作，它利用生命的力量，与比生命或物质更高的境界相联系：&lt;br /&gt;
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quote|&lt;br /&gt;
evidence of a foreseeing mind which...so directed and organised that life, in all its myriad forms, as, in the far-off future, to provide all that was most essential for the growth and development of man's spiritual nature ...&amp;lt;ref name=&amp;quot;Wallace&amp;quot;/&amp;gt;&lt;br /&gt;
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证据表明，一个先知先觉的头脑 ...... 如此指导和组织生活，以其无数的形式，在遥远的未来，为人类精神本质的成长和发展提供所有最重要的东西 ......&lt;br /&gt;
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Proceeding from Hunter's view that Life is the directive power above and behind living matter, Wallace argues that logically, Mind is the cause of [[consciousness]], which exists in different degrees and kinds in living matter.&lt;br /&gt;
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Wallace 从 Hunter 的观点出发，认为生命是高于和落后于生命物质的指导性力量 directive power，他认为从逻辑上讲，心灵 Mind 是 意识 [[consciousness]]的原因，它在生命物质中以不同程度和形式存在。&lt;br /&gt;
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quote|&lt;br /&gt;
If, as John Hunter, T.H. Huxley, and other eminent thinkers have declared, &amp;quot;life is the cause, not the consequence, of organisation,&amp;quot; so we may believe that mind is the cause, not the consequence, of brain development. ... So there are undoubtedly different degrees and probably also different kinds of mind in various grades of animal life&amp;amp;nbsp;... And&amp;amp;nbsp;... so the mind-giver&amp;amp;nbsp;... enables each class or order of animals to obtain the amount of mind requisite for its place in nature&amp;amp;nbsp;...&amp;lt;ref name=&amp;quot;Wallace&amp;quot;/&amp;gt;&lt;br /&gt;
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如果像 John Hunter、T.H. Huxley 和其他著名思想家所宣称的那样，&amp;quot;生命是组织的原因，而不是结果&amp;quot;，那么我们可以相信，心智是大脑发育的原因，而不是结果。... 所以，在不同等级的动物生命中，无疑存在着不同程度的心智，也可能存在着不同种类的心智。而 ...... 所以心智的给予者 ......使每一类或每一阶的动物都能获得它在自然界中的地位所必需的心智量 ...... &amp;lt;ref name=&amp;quot;Wallace&amp;quot;/&amp;gt;&lt;br /&gt;
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== Emergent evolution ==&lt;br /&gt;
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=== Early roots ===&lt;br /&gt;
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The issue of how change in nature 'emerged' can be found in classical Greek thought -  order coming out of chaos and whether by chance or necessity. [[Aristotle]] spoke of wholes that were greater than the sum of their parts because of emergent properties. The second-century anatomist and physiologist [[Galen]] also distinguished between the resultant and emergent qualities of wholes. (Reid, p.&amp;amp;nbsp;72)&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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关于自然界的变化如何 &amp;quot;出现&amp;quot; 的问题可以在古典希腊思想中找到 - 从混沌中产生秩序，以及是偶然还是必然。[[Aristotle]]谈到，由于涌现的特性，整体大于部分之和。二世纪的解剖学家和生理学家[[Galen]]也区分了整体的结果性和涌现性。(Reid, p. 72)&amp;lt;ref name=Reid /&amp;gt;。&lt;br /&gt;
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[[Hegel]] spoke of the revolutionary progression of life from non-living to conscious and then to the spiritual and Kant perceived that simple parts of an organism interact to produce a progressively complex series of emergences of functional forms, a distinction that carried over to [[John Stuart Mill]] (1843), who stated that even chemical compounds have novel features that cannot be predicted from their elements. [Reid, p.&amp;amp;nbsp;72]&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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[[Hegel]]谈到了生命从非生命到有意识再到精神的革命性进展，康德 Kant 则认为有机体的简单部分相互作用，产生了一系列逐渐复杂的功能形式的出现，这种区别延续到[[John Stuart Mill]]（1843），他指出，即使是化学化合物也有无法从元素中预测的新特征。Reid，p.&amp;amp;nbsp;72]&amp;lt;ref name=Reid /&amp;gt;。&lt;br /&gt;
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The idea of an emergent quality that was something new in nature was further taken up by [[George Henry Lewes]] (1874–1875), who again noted, as with Galen earlier, that these evolutionary &amp;quot;emergent&amp;quot; qualities are distinguishable from adaptive, additive &amp;quot;resultants.&amp;quot; [[Henry Drummond (evangelist)|Henry Drummond]] in ''The Descent of Man'' (1894) stated that emergence can be seen in the fact that the laws of nature are different for the organic or vital compared to the inertial inorganic realm.&lt;br /&gt;
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[[George Henry Lewes]]（1874-1875）进一步提出了自然界中的新事物乃是一种涌现特征的观点，他再次指出，与前面的盖伦一样，这些进化的 &amp;quot;涌现 &amp;quot;特质与适应性、加成性的 &amp;quot;结果 &amp;quot;是有区别的。[[Henry Drummond (evangelist)|Henry Drummond]]在 ''人类的后裔''（1894）中指出，涌现性可以看出，自然规律对于有机界或生命界来说，与惯性的无机界相比是不同的。&lt;br /&gt;
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quote|&lt;br /&gt;
When we pass from the inorganic to the organic we come upon a new set of laws - but the reason why the lower set do not seem to operate in the higher sphere is not that they are annhilated, but that they are overruled. (Drummond 1883, p. 405, quoted in Reid)&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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当我们从无机物到有机物的时候，我们遇到了一套新的规律 - 但是，低级的那套规律之所以在高级的领域里似乎没有发挥作用，不是因为它们被消灭了，而是因为它们被推翻了。&lt;br /&gt;
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As Reid points out, Drummond also realized that greater complexity brought greater adaptability. (Reid. p.&amp;amp;nbsp;73)&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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正如 Reid 所指出的那样，Drummond 也意识到，更大的复杂性带来了更大的适应性。&lt;br /&gt;
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[[Samuel Alexander]] took up the idea that emergences had properties that overruled the demands of the lower levels of organization. And more recently, this theme is taken up by John Holland (1998):&lt;br /&gt;
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[[Samuel Alexander]]提出了这样的观点，即涌现性具有凌驾于低级组织需求之上的特性。最近，John Holland（1998）又提出了这一主题。&lt;br /&gt;
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quote|&lt;br /&gt;
If we turn reductionism on its head we add levels. More carefully, we add new laws that satisfy the constraints imposed by laws already in place. Moreover these new laws apply to complex phenomena that are consequences of the original laws; they are at a new level.&amp;lt;ref name=Holland&amp;gt;{{cite book|last=Holland|first=John|title=Emergence. From Chaos to Order|year=1998|publisher=Addison-Wesley|location=Reading, Massachusetts}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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如果我们把还原论颠倒过来，我们就增加层次。更仔细地说，我们增加了新的法则，这些法则满足已经存在的法则所施加的约束。此外，这些新的法则适用于复杂的现象，这些现象是原有定律的结果；它们处于一个新的层次。&lt;br /&gt;
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=== C. Lloyd Morgan and emergent evolution ===&lt;br /&gt;
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Another major scientist to question natural selection as the motive force of evolution was [[C. Lloyd Morgan]], a zoologist and student of [[T.H. Huxley]], who had a strong influence on Samuel Alexander. His ''Emergent Evolution'' (1923) established the central idea that an emergence might have the appearance of [[Saltation (biology)|saltation]] but was best regarded as &amp;quot;a qualitative change of direction or critical turning point.&amp;quot;(quoted in Reid, p.&amp;amp;nbsp;73-74)&amp;lt;ref name=Reid /&amp;gt;  Morgan, due to his work in animal psychology, had earlier (1894) questioned the  continuity view of mental evolution, and held that there were various discontinuities in cross-species mental abilities. To offset any attempt to read [[anthropomorphism]] into his view, he created the famous, but often misunderstood methodological canon:&lt;br /&gt;
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另一位质疑自然选择是进化的动力的主要科学家是[[C. Lloyd Morgan]]，他是动物学家，也是[[T.H. Huxley]]的学生，他对 Samuel Alexander 影响很大。他的''涌现进化''（1923）确立了中心思想，即涌现可能具有[[盐化(生物学)|盐化]]的外观，但最好被视为 &amp;quot;方向的质变或关键的转折点。&amp;quot;(引自Reid，p.&amp;amp;nbsp;73-74)&amp;lt;ref name=Reid /&amp;gt; Morgan 由于他在动物心理学方面的工作，早些时候（1894）就对心理进化的连续性观点提出了质疑，并认为跨物种的心理能力存在着各种不连续性。为了避免任何将 拟人化[[anthropomorphism]] 引入他的观点的企图，他创造了著名的、但经常被误解的方法论典籍：&lt;br /&gt;
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quote|&lt;br /&gt;
In no case may we interpret an action as the outcome of the exercise of a higher psychical faculty, if it can be interpreted as the outcome of the exercise of one which stands lower in the psychological scale.|Morgan, 1894, p. 53&lt;br /&gt;
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在任何情况下，我们都不能把一个行动解释为行使较高的心理能力的结果，如果它可以被解释为行使一个较低的心理等级的结果。|Morgan, 1894, p. 53&lt;br /&gt;
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However, Morgan realizing that this was being misused to advocate reductionism (rather than as a general methodological caution), introduced a qualification into the second edition of his ''An Introduction to Comparative Psychology'' (1903):&lt;br /&gt;
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然而，Morgan 意识到这一点被滥用于鼓吹还原论（而不是作为一般方法论的告诫），在他的 ''比较心理学导论''（1903）第二版中引入了一个限定条件：&lt;br /&gt;
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quote|&lt;br /&gt;
To this, however, it should be added, lest the range of the principle be misunderstood, that the canon by no means excludes the interpretation of a particular activity in terms of the higher processes, if we already have independent evidence of the occurrence of these higher processes in the animal under observation.|Morgan, 1903, p. 59&lt;br /&gt;
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然而，我们应该补充一点，以免该原则的范围被误解，如果我们已经有独立的证据表明在被观察的动物中发生了这些更高级的过程，那么该原则绝不排除对某一特定活动的解释。|Morgan, 1903, p. 59&lt;br /&gt;
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As Reid observes,&lt;br /&gt;
正如 Reid 所言，&lt;br /&gt;
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quote|&lt;br /&gt;
While the so-called historiographical &amp;quot;rehabilitation of the canon&amp;quot; has been underway for some time now, Morgan's emergent evolutionist position (which was the highest expression of his attempt to place the study of mind back into such a &amp;quot;wider&amp;quot; natural history) is seldom mentioned in more than passing terms even within contemporary history of psychology textbooks.&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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虽然所谓的史学上的 &amp;quot;恢复正典 &amp;quot;已经进行了一段时间，但 Morgan 的涌现进化论立场（这是他试图将心灵研究放回这样一个 &amp;quot;更广泛&amp;quot; 的自然史中的最高表现），即使在当代心理学史的教科书中也很少被人提及，只是一带而过。&amp;lt;ref name=Reid /&amp;gt;&lt;br /&gt;
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Morgan also fought against the [[Behaviorism|behaviorist school]] and clarified even more his emergent views on evolution:&lt;br /&gt;
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Morgan 还与 行为主义|行为主义学派 [[Behaviorism|behaviorist school]进行了斗争，更澄清了他对进化论的涌现观点：&lt;br /&gt;
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quote|&lt;br /&gt;
An influential school of 'behaviorists' roundly deny that mental relations, if such there be, are in any sense or in any manner effective... My message is that one may speak of mental relations as effective no less 'scientifically' than... physical relations...|Morgan, 1930, p. 72&lt;br /&gt;
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一个有影响力的 &amp;quot;行为主义者&amp;quot; 学派全面否认心理关系（如果有的话）在任何意义上或以任何方式是有效的 ...... 我的信息是，人们可以说心理关系是有效的，不亚于...... 物理关系 ...... |Morgan, 1930, p. 72&lt;br /&gt;
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His ''Animal Conduct'' (1930) explicitly distinguishes between three &amp;quot;grades&amp;quot; or &amp;quot;levels of mentality&amp;quot; which he labeled: 'percipient, perceptive, and reflective.' (p.&amp;amp;nbsp;42)&lt;br /&gt;
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他的 ''动物行为''（1930）明确区分了三个 &amp;quot;等级 &amp;quot;或 &amp;quot;心态水平&amp;quot;，他把这三个 &amp;quot;等级 &amp;quot;标记为：&amp;quot;知觉的、感性的和反思的&amp;quot;。(p.&amp;amp;nbsp;42)&lt;br /&gt;
=== Alexander and the emergence of mind ===&lt;br /&gt;
Morgan's idea of a polaric relationship between lower and higher, was taken up by Samuel Alexander, who argued that the mental process is not reducible to the neural processes on which it depends at the physical-material level. Instead, they are two poles of a unity of function. Further, the neural process that expressed mental process itself possesses a quality (mind) that the other neural processes don’t. At the same time, the mental process, because it is functionally identical to this particular neural process, is also a vital one.&amp;lt;ref name=Alexander&amp;gt;{{cite book|last=Alexander|first=Samuel|title=Space, Time and Deity|year=1916–18|publisher=Gifford Lectures|location=Glasgow|url=https://archive.org/details/spacetimeanddei00alexgoog}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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Morgan 关于低级和高级之间的两极关系的观点，被 Samuel Alexander 所采纳，他认为，心理过程不能还原为它在物理物质层面所依赖的神经过程。相反，它们是功能统一体的两极。而且，表达心理过程的神经过程本身就具有其他神经过程所不具备的品质（心智）。同时，精神过程由于在功能上与这个特殊的神经过程相同，所以也是一个至关重要的过程。&lt;br /&gt;
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And mental process is also &amp;quot;something new, &amp;quot;a fresh creation&amp;quot;, which precludes a psycho-physiological parallelism. Reductionism is also contrary to empirical fact.&lt;br /&gt;
而心理过程也是某种新东西，&amp;quot;&amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;一种全新的创造 a fresh creation&amp;lt;/font&amp;gt;&amp;quot;，这就排除了心理生理学上的平行主义。&amp;lt;font color=&amp;quot;#ff8000&amp;quot;&amp;gt;还原论 Reductionism&amp;lt;/font&amp;gt; 也是违背经验事实的。&lt;br /&gt;
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quote|&lt;br /&gt;
All the available evidence of fact leads to the conclusion that the mental element is essential to the neural process which it is said to accompany...and is not accidental to it, nor is it in turn indifferent to the mental feature. Epiphenomenalism is a mere fallacy of observation.&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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所有现有的事实证据都可以得出这样的结论：精神因素对它所说的伴随着的神经过程是必不可少的 ...... 不是偶然的，也不是反过来对精神特征无动于衷的。表象主义 Epiphenomenalism 只是一种观察的谬误。&lt;br /&gt;
}}&lt;br /&gt;
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At the same time Alexander stated that his view was not one of animism or vitalism, where the mind is an independent entity action on the brain, or conversely, acted upon by the brain. Mental activity is an emergent, a new &amp;quot;thing&amp;quot; not reducible to its initial neural parts.&lt;br /&gt;
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同时 Alexander 表示，他的观点不是万物有灵论或生命论的观点，在这里，精神是一个独立的实体作用于大脑，或者反过来说，被大脑作用。精神活动是一种涌现的，是一种新的 &amp;quot;事物&amp;quot;，不能还原到最初的神经部分。&lt;br /&gt;
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For Alexander, the world unfolds in space-time, which has the inherent quality of motion. This motion through space-time results in new “complexities of motion” in the form of a new quality or emergent. The emergent retains the qualities of the prior “complexities of motion” but also has something new that was not there before. This something new comes with its own laws of behavior. Time is the quality that creates motion through Space, and matter is simply motion expressed in forms in Space, or as Alexander says a little later, “complexes of motion.” Matter arises out of the basic ground of Space-Time continuity and has an element of “body” (lower order) and an element of “mind” (higher order), or “the conception that a secondary quality is the mind of its primary substrate.”&lt;br /&gt;
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对 Alexander 来说，世界是在时空中展开的，时空具有运动的内在品质。这种在时空中的运动导致了新的 &amp;quot;运动的复杂性&amp;quot;，其形式是一种新的质量或涌现。新出现的东西保留了先前 &amp;quot;运动的复杂性 &amp;quot;的品质，但也有一些新的东西，而这些新的东西是先前没有的。这种新的东西有其自身的行为规律。时间是通过空间产生运动的特质，而物质只是在空间中以形式表现出来的运动，或者如 Alexander 稍后所说的 &amp;quot;运动的复杂性&amp;quot;。物质产生于空间-时间连续性的基本基础上，具有 &amp;quot;身体&amp;quot;（低阶）的要素和 &amp;quot;心灵&amp;quot;（高阶）的要素，或者说 &amp;quot;这样一种概念：一个次要品质是其主要基质的心灵&amp;quot;。&lt;br /&gt;
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Mind is an emergent from life and life itself is an emergent from matter. Each level contains and is interconnected with the level and qualities below it, and to the extent that it contains lower levels, these aspects are subject to the laws of that level. All mental functions are living, but not all living functions are mental; all living functions are physico-chemical, but not all physico-chemical processes are living - just as we could say that all people living in Ohio are Americans, but not all Americans live in Ohio. Thus, there are levels of existence, or natural jurisdictions, within a given higher level such that the higher level contains elements of each of the previous levels of existence. The physical level contains the pure dimensionality of Space-Time in addition to the emergent of physico-chemical processes; the next emergent level, life, also contains Space-Time as well as the physico-chemical in addition to the quality of life; the level of mind contains all of the previous three levels, plus consciousness. As a result of this nesting and inter-action of emergents, like fluid Russian dolls, higher emergents cannot be reduced to lower ones, and different laws and methods of inquiry are required for each level.&lt;br /&gt;
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心灵是生命的产物，生命本身是物质的产物。每一个层次都包含着它下面的层次和品质，并与之相互联系，在它包含低层次的范围内，这些方面都受该层次的规律制约。所有的精神功能都是活的，但不是所有的活的功能都是精神的；所有的活的功能都是物理化学的，但不是所有的物理化学过程都是活的 - 就像我们可以说所有生活在俄亥俄州的人都是美国人，但不是所有的美国人都生活在俄亥俄州。因此，在一个给定的较高层次中，存在着一些层次，或者说是自然的管辖范围，这样，较高层次就包含了前几个层次的每一个存在的要素。物理层次除了物理化学过程的出现外，还包含了时空的纯粹维度；下一个出现层次-生命，除了生命质量外，还包含了时空以及物理化学；心灵层次包含了前面三个层次的所有内容，再加上意识。由于这种涌现物的嵌套和相互作用，就像流动的俄罗斯套娃一样，高层次的涌现物不能归结为低层次的涌现物，每个层次都需要不同的规律和探究方法。&lt;br /&gt;
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quote|&lt;br /&gt;
Life is not an epiphenomenon of matter but an emergent from it&amp;amp;nbsp;... The new character or quality which the vital physico-chemical complex possesses stands to it as soul or mind to the neural basis.&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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生命不是物质的外显现象，而是从物质中涌现的 ...... 生命理化复合体所具有的新的特性或品质对它来说，就像灵魂或心灵对神经基础一样。&lt;br /&gt;
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For Alexander, the &amp;quot;directing agency&amp;quot; or entelechy is found &amp;quot;in the principle or plan&amp;quot;.&lt;br /&gt;
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在 Alexander 看来，&amp;quot;指导机构 &amp;quot;或 &amp;lt;font color=&amp;quot;#32CD32&amp;quot;&amp;gt;entelechy&amp;lt;/font&amp;gt; 是 &amp;quot;在原则或计划中&amp;quot; 发现的。&lt;br /&gt;
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quote|&lt;br /&gt;
a given stage of material complexity is characterised by such and such special features…By accepting this we at any rate confine ourselves to noting the facts…and do not invent entities for which there seems to be no other justification than that something is done in life which is not done in matter.&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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物质复杂性的某一阶段具有这样那样的特点 ...... 接受了这一点，我们无论如何就会把自己局限于注意到事实 ...... 而不会发明一些实体，而这些实体除了在生活中做了一些在物质中没有做的事情之外，似乎没有其他理由。&lt;br /&gt;
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While an emergent is a higher complexity, it also results in a new simplicity as it brings a higher order into what was previously less ordered (a new simplex out of a complex). This new simplicity does not carry any of the qualities or aspects of that emergent level prior to it, but as noted, does still carry within it such lower levels so can be understood to that extent through the science of such levels, yet not itself be understood except by a science that is able to reveal the new laws and principles applicable to it.&lt;br /&gt;
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虽然一个涌现是一个较高的复杂性，但它也导致了一个新的简单性，因为它把一个较高的秩序带入了以前不太有序的东西中（一个复杂的新简单性）。这种新的简单性并不带有它之前的那个涌现层次的任何特质或方面，但如前所述，确实还带有它内部的这种较低层次，所以可以通过这种层次的科学来理解，但它本身却不能被理解，除非是由一种能够揭示适用于它的新的规律和原则的科学来理解。&lt;br /&gt;
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Ascent takes place, it would seem, through complexity.[increasing order] But at each change of quality the complexity as it were gathers itself together and is expressed in a new simplicity.&lt;br /&gt;
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上升似乎是通过复杂性来进行的。[increasing order]但在每一个质量的变化中，复杂性就像它本身一样聚集在一起，并以一种新的简单性来表现。&lt;br /&gt;
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Within a given level of emergence, there are degrees of development.&lt;br /&gt;
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在一个给定的涌现水平内，有某种程度的发展。&lt;br /&gt;
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... There are on one level degrees of perfection or development; and at the same time there is affinity by descent between the existents belonging to the level. This difference of perfection is not the same thing as difference of order or rank such as subsists between matter and life or life and mind ...&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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...... 在一个层次上，有完善或发展的程度；同时，属于这个层次的存在者之间也有因血统而产生的亲和性。这种完善性的差异与秩序或等级的差异不同，如存在于物质与生命或生命与心灵之间 ......&lt;br /&gt;
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The concept or idea of mind, the highest emergent known to us, being at our level, extends all the way down to pure dimensionality or Space-Time. In other words, time is the “mind” of motion, materialising is the “mind” of matter, living the “mind” of life. Motion through pure time (or life astronomical, mind ideational) emerges as matter “materialising” (geological time, life geological, mind existential), and this emerges as life “living” (biological time, life biological, mind experiential), which in turn give us mind “minding” (historical time, life historical, mind cognitional). But there is also an extension possible upwards of mind to what we call Deity.&lt;br /&gt;
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心灵的概念或观念，是我们已知的最高级的涌现物，在我们的层面上，一直延伸到纯粹的维度或时空。换句话说，时间是运动的 &amp;quot;心&amp;quot;，物化是物质的 &amp;quot;心&amp;quot;，生活是生命的 &amp;quot;心&amp;quot;。通过纯粹的时间（或生命天文、心灵表意）的运动，出现了物质的 &amp;quot;物化&amp;quot;（地质时间、生命地质、心灵存在），而这又出现了生命的 &amp;quot;生活&amp;quot;（生物时间、生命生物、心灵体验），这又让我们有了心灵的 &amp;quot;心智&amp;quot;（历史时间、生命历史、心灵认知）。但是，心智还有一种向上延伸的可能，就是我们所说的德性 Deity。&lt;br /&gt;
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quote|&lt;br /&gt;
let us describe the empirical quality of any kind of finite which performs to it the office of consciousness or mind as its 'mind.' Yet at the same time let us remember that the 'mind' of a living thing is not conscious mind but is life, and has not the empirical character of consciousness at all, and that life is not merely a lower degree of mind or consciousness, but something different. We are using 'mind' metaphorically by transference from real minds and applying it to the finites on each level in virtue of their distinctive quality; down to Space-Time itself whose existent complexes of bare space-time have for their mind bare time in its empirical variations.&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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让我们描述任何一种有限物的经验品质，它对它执行意识或心灵的职务，作为它的'心灵'。然而同时让我们记住，生物的'心智'不是意识心智，而是生命，根本不具有意识的经验性，生命不只是心智或意识的较低程度，而是不同的东西。我们是从真实的心智转移而隐喻性地使用'心智'，并凭借其独特的品质将其应用于每个层次上的有限性；下至时空本身，其存在的裸露时空的复合体以其经验性变化的裸露时间为心智。}}&lt;br /&gt;
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Alexander goes back to the Greek idea of knowledge being “out there” in the object being contemplated. In that sense, there is not mental object (concept) “distinct” (that is, different in state of being) from the physical object, but only an apparent split between the two, which can then be brought together by proper compresence or participation of the consciousness in the object itself.&lt;br /&gt;
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Alexander 又回到了希腊人的观念，即知识是 &amp;quot;存在于&amp;quot; 被思考的对象中的。在这个意义上，并不存在精神客体（概念）与物理客体 &amp;quot;不同&amp;quot;（即存在状态不同），而只是二者之间存在着明显的分裂，然后可以通过意识在客体本身中的适当压缩或参与而使二者结合起来。&lt;br /&gt;
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quote|&lt;br /&gt;
There is no consciousness lodged, as I have supposed, in the organism as a quality of the neural response; consciousness belongs to the totality of objects, of what are commonly called the objects of consciousness or the field of consciousness&amp;amp;nbsp;... Consciousness is therefore &amp;quot;out there&amp;quot; where the objects are, by a new version of Berkleyanism&amp;amp;nbsp;... Obviously for this doctrine as for mine there is no mental object as distinct from a physical object: the image of a tree is a tree in an appropriate form...&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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并没有像我所认为的那样，意识作为神经反应的一种性质而寄居在有机体中；意识属于对象的整体，属于通常所说的意识对象或意识场&amp;amp;nbsp;. 因此新版的 伯里克利安主义 Berkleyanism 认为意识是 &amp;quot;在那里&amp;quot;，也就是对象所在地。显然对于这个学说和我的学说来说，并不存在有别于物理对象的精神对象：树的形象是一棵树，以适当的形式 ......}}&lt;br /&gt;
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Because of the interconnectedness of the universe by virtue of Space-Time, and because the mind apprehends space, time and motion through a unity of sense and mind experience, there is a form of knowing that is intuitive (participative) - sense and reason are outgrowths from it.&lt;br /&gt;
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由于宇宙凭借时空的相互联系，由于心灵通过感觉和心灵经验的统一来认识空间、时间和运动，所以有一种认识形式是直观的（参与性的）- 感觉和理性是它的产物。&lt;br /&gt;
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In being conscious of its own space and time, the mind is conscious of the space and time of external things and vice versa. This is a direct consequence of the continuity of Space-Time in virtue of which any point-instant is connected sooner or later, directly or indirectly, with every other...&lt;br /&gt;
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The mind therefore does not apprehend the space of its objects, that is their shape, size and locality, by sensation, for it depends for its character on mere spatio-temporal conditions, though it is not to be had as consciousness in the absence of sensation (or else of course ideation). It is clear without repeating these considerations that the same proposition is true of Time; and of motion&amp;amp;nbsp;... I shall call this mode of apprehension in its distinction from sensation, intuition.&amp;amp;nbsp;... Intuition is different from reason, but reason and sense alike are outgrowths from it, empirical determinations of it...&amp;lt;ref name=&amp;quot;Alexander&amp;quot;/&amp;gt;&lt;br /&gt;
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在意识到自己的空间和时间的同时，心灵也意识到外部事物的空间和时间，反之亦然。这是时空连续性的直接后果，任何一点一瞬都迟早直接或间接地与其他每一点一瞬相联系 ......&lt;br /&gt;
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因此，心灵并不通过感觉来领会其对象的空间，即它们的形状、大小和位置，因为它的特性取决于单纯的时空条件，尽管在没有感觉（当然也可以说是意念）的情况下，它不能作为意识。不用重复这些考虑，就可以清楚地看到，同样的命题对时间；对运动&amp;amp;nbsp；......也是如此。我将把这种与感觉相区别的理解方式称为直觉&amp;amp;nbsp;......。直觉不同于理智，但理智和感觉都是它的产物，是它的经验决定 ......}}&lt;br /&gt;
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In a sense, the universe is a participative one and open to participation by mind as well so that mind can intuitively know an object, contrary to what Kant asserted. Participation (togetherness) is something that is “enjoyed” (experienced) not contemplated, though in the higher level of consciousness, it would be contemplated.&lt;br /&gt;
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从某种意义上说，宇宙是一个参与性的宇宙，也开放给心灵参与，这样心灵就可以直观地认识一个对象，这与康德的论断相反。参与（togetherness）是一种 &amp;quot;享受&amp;quot;（体验）而非思索的东西，尽管在更高的意识层面，它会被思索。&lt;br /&gt;
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quote|&lt;br /&gt;
The universe for Alexander is essentially in process, with Time as its ongoing aspect, and the ongoing process consists in the formation of changing complexes of motions. These complexes become ordered in repeatable ways displaying what he calls &amp;quot;qualities.&amp;quot; There is a hierarchy of kinds of organized patterns of motions, in which each level depends on the subvening level, but also displays qualities not shown at the subvening level nor predictable from it… On this there sometimes supervenes a further level with the quality called &amp;quot;life&amp;quot;; and certain subtle syntheses which carry life are the foundation for a further level with a new quality. &amp;quot;mind.&amp;quot; This is the highest level known to us, but not necessarily the highest possible level. The universe has a forward thrust, called its &amp;quot;nisus&amp;quot; (broadly to be identified with the Time aspect) in virtue of which further levels are to be expected...&amp;lt;ref name=Emmet&amp;gt;{{cite web|last=Emmet |first=Dorothy |title=Whitehead and Alexander |url=http://www.religion-online.org/article/whitehead-and-alexander/|accessdate=23 July 2012 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
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对 Alexander 来说，宇宙本质上是处于过程中的，时间是其持续的方面，而这个持续的过程包括形成不断变化的运动复合体。这些复合体以可重复的方式变得有序，显示出他所说的 &amp;quot;品质&amp;quot;。有一个运动组织模式的种类层次，其中每一个层次都依赖于次要层次，但也显示出次要层次没有显示的品质，也无法从次要层次预测 ...... 在此基础上，有时还监督着一个被称为 &amp;quot;生命&amp;quot; 的品质的进一步层次；而某些携带生命的微妙综合体，是具有新品质的进一步层次的基础。&amp;quot;心灵&amp;quot; 这是我们已知的最高层次，但不一定是可能的最高层次。宇宙有一个向前的推力，叫做它的 &amp;quot;nisus&amp;quot;（大体上要与时间方面相鉴别），凭借这个推力，进一步的层次是可以期待的 ......&lt;br /&gt;
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=== Robert G. B. Reid ===&lt;br /&gt;
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Emergent evolution was revived by Robert G. B. Reid (March 20, 1939 - May 28, 2016), a biology professor at the University of Victoria (in British Columbia, Canada). In his book Evolutionary Theory: The Unfinished Synthesis (1985), he stated that the modern evolutionary synthesis with its emphasis on natural selection is an incomplete picture of evolution, and emergent evolution can explain the origin of genetic variation. Biologist Ernst Mayr heavily criticized the book claiming it was a misinformed attack on natural selection. Mayr commented that Reid was working from an &amp;quot;obsolete conceptual framework&amp;quot;, provided no solid evidence and that he was arguing for a teleological process of evolution.&lt;br /&gt;
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涌现进化论是由 Robert G. B. Reid（1939年3月20日-2016年5月28日）恢复的，他是维多利亚大学（位于加拿大不列颠哥伦比亚省）的生物学教授。在他的著作《进化论：未竟的综合 The Unfinished Synthesis》 (1985), 他宣称现代进化论总论的重点在于自然选择，这是进化论不完全的图景，涌现进化可以解释遗传变异的原因。生物学家 恩斯特·迈尔 Ernst Mayr 严厉批评了这本书，称这是对自然选择的错误攻击。Mayr 评论说，Reid 从一个 &amp;quot;过时的概念框架&amp;quot; 出发，没有提供任何可靠的证据，而且他主张的是一种心智学的进化过程。&lt;br /&gt;
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Reid later published the book Biological Emergences (2007) with a theory on how emergent novelties are generated in evolution. According to Massimo Pigliucci &amp;quot;Biological Emergences by Robert Reid is an interesting contribution to the ongoing debate on the status of evolutionary theory, but it is hard to separate the good stuff from the more dubious claims.&amp;quot; Pigliucci noted a dubious claim in the book is that natural selection has no role in evolution. It was positively reviewed by biologist Alexander Badyaev who commented that &amp;quot;the book succeeds in drawing attention to an under appreciated aspect of the evolutionary process&amp;quot;. Others have criticized Reid's unorthodox views on emergence and evolution. Biologist Samuel Scheiner stated that Reid's &amp;quot;presentation is both a caricature of evolutionary theory and severely out of date.&amp;quot;&lt;br /&gt;
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Reid 后来出版了《生物学的涌现 Biological Emergences》（2007年）一书，提出了关于在进化中如何产生涌现新事物的理论。马西莫·皮格利乌奇 Massimo Pigliucci 认为：&amp;quot;Robert Reid 所著的《生物学的涌现》是对正在进行的关于进化论地位的争论的一个有趣的贡献，但很难把好的东西和比较可疑的说法分开。&amp;quot; Pigliucci 指出，书中一个可疑的说法是，自然选择在进化中没有作用。生物学家亚历山大·巴佳耶夫 Alexander Badyaev 对该书进行了积极的评论，他评论说 &amp;quot;这本书成功地引起了人们对进化过程中一个未被充分认识的方面的关注&amp;quot;。其他人则批评了 Reid 关于涌现和进化的非正统观点。生物学家 塞缪尔·谢纳 Samuel Scheiner 说 &amp;quot;Reid 的表述既是对进化论的讽刺，又严重过时&amp;quot;。&lt;br /&gt;
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==See also==&lt;br /&gt;
{{Portal|Evolutionary biology}}&lt;br /&gt;
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* [[The eclipse of Darwinism]] 达尔文主义的黯淡&lt;br /&gt;
* [[Emergentism]] is a corresponding belief in emergence. 涌现论是关于涌现的信仰&lt;br /&gt;
* [[Evolutionary biology]] 进化生物学&lt;br /&gt;
* [[Noogenesis]] 心理演化&lt;br /&gt;
* [[Orthogenesis]] 定向演化&lt;br /&gt;
* [[Vitalism]] 生命力论&lt;br /&gt;
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==References==&lt;br /&gt;
{{Reflist|2}}&lt;br /&gt;
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==Further reading==&lt;br /&gt;
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* [[Samuel Alexander|Alexander, Samuel]]. (1920). ''Space, Time, and Deity''. Kessinger Publishing Reprint. {{ISBN|0-7661-8702-0}} &lt;br /&gt;
* [[Henri Bergson|Bergson, Henri]]. (1911). ''Creative Evolution'' (English translation of ''L'Evolution créatrice''). Dover Publications 1998: {{ISBN|0-486-40036-0}}&lt;br /&gt;
* Lewes, George H. (1875). ''Problems of Life and Mind''. First Series: The Foundations of a Creed, vol. II. University of Michigan Library. {{ISBN|1-4255-5578-0}}&lt;br /&gt;
* MacDougall, William. (1929). ''Modern Materialism and Emergent Evolution''. London: Macmillan.&lt;br /&gt;
* [[C. Lloyd Morgan|Morgan, C. Lloyd]]. (1923). [https://archive.org/stream/emergentevolutio032372mbp#page/n5/mode/2up ''Emergent Evolution'']. Henry Holt and Co. {{ISBN|0-404-60468-4}}&lt;br /&gt;
* Patrick, G. T. W. (1923). ''Emergent Evolution by C. Lloyd Morgan''. ''[[The Journal of Philosophy]]''. Vol. 20, No. 26. pp.&amp;amp;nbsp;714–718.&lt;br /&gt;
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Category:Philosophical theories&lt;br /&gt;
范畴: 哲学理论&lt;br /&gt;
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Category:Evolutionary biology&lt;br /&gt;
分类: 进化生物学&lt;br /&gt;
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Category:Holism&lt;br /&gt;
分类: 整体论&lt;br /&gt;
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&amp;lt;small&amp;gt;This page was moved from [[wikipedia:en:Emergent evolution]]. Its edit history can be viewed at [[涌现进化/edithistory]]&amp;lt;/small&amp;gt;&amp;lt;/noinclude&amp;gt;&lt;br /&gt;
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		<author><name>Eatcosmos2</name></author>
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