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Sierpinski triangle created using IFS (colored to illustrate self-similar structure)

[[使用 IFS 创建的谢尔宾斯基三角形(着色以说明自相似结构)]

Colored IFS designed using Apophysis software and rendered by the Electric Sheep.

Apophysis software and rendered by the Electric Sheep.]]

[软件，并由电子羊渲染]

In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry.^[1] They were introduced in 1981.

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

在数学中，迭代函数系统是构造分形的一种方法; 由此产生的分形通常是自相似的。与分形几何相比，IFS 分形与集合论的关系更大。它们是在1981年推出的。

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

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

IFS 分形，正如它们通常被称为的那样，可以是任意数量的维数，但是通常是用2D 计算和绘制的。分形是由几个自身副本的结合组成的，每个副本由一个函数(因此称为“函数系统”)进行转换。典型的例子是 sierpi 滑雪三角形。这些函数通常是收缩的，这意味着它们使点靠得更近，使形状变小。因此，迭代函数系统分形的形状是由几个可能重叠的自身的小副本组成的，每个副本也是由自身的副本组成的，而且是无限的。这就是其自相似分形特性的来源。

Definition

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.^[2] Symbolically,

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

在形式上，迭代函数系统是完备空间上的有限压缩映射集。象征性地,

[math]\displaystyle{ \{f_i:X\to X\mid i=1,2,\dots,N\},\ N\in\mathbb{N} }[/math]

数学中 x 到 x 中间 i = 1,2，点，n } ，n

is an iterated function system if each [math]\displaystyle{ f_i }[/math] is a contraction on the complete metric space [math]\displaystyle{ X }[/math].

如果每一个迭代函数系统都是完备空间的收缩。

Properties

Construction of an IFS by the chaos game (animated)

通过混沌游戏(动画)构建一个 IFS

IFS being made with two functions.

IFS 具有两个功能。

Hutchinson (1981) showed that, for the metric space [math]\displaystyle{ \mathbb{R}^n }[/math], or more generally, for a complete metric space [math]\displaystyle{ X }[/math], such a system of functions has a unique nonempty 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 S₀ and iterate the actions of the f_i, taking S_n+1 to be the union of the images of S_n under the f_i; then taking S to be the closure of the union of the S_n. Symbolically, the unique fixed (nonempty compact) set [math]\displaystyle{ S\subseteq X }[/math] has the property

Hutchinson (1981)证明，对于度量空间 < math > mathbb { r } ^ n </math > ，或者更广泛地说，对于完备空间 < math > x </math > ，这样一个函数系统有唯一的非空紧(闭和有界)固定集 s。构造固定集的一种方法是从初始非空封闭有界集 s 0 开始，迭代 f i 的作用，将 s n + 1 作为 f i 下 s n 图像的合并，然后将 s n 作为 s n 图像的合并。从符号上看，唯一的固定(非空紧凑)集 < math > s subseteq x </math > 具有该属性

[math]\displaystyle{ S = \overline{\bigcup_{i=1}^N f_i(S)}. }[/math]

< math > s = overline { bigcup _ { i = 1} ^ n f _ i (s)} . </math >

The set S is thus the fixed set of the Hutchinson operator [math]\displaystyle{ F: 2^X\to 2^X }[/math] defined for [math]\displaystyle{ A\subseteq X }[/math] via

因此，集合 s 是哈钦森算子 < math > f: 2 ^ x to 2 ^ x </math > 为 < math > a subseteq x </math > via 定义的固定集合

[math]\displaystyle{ F(A)=\overline{\bigcup_{i=1}^N f_i(A)}. }[/math]

< math > f (a) = overline { bigcup _ { i = 1} ^ n f _ i (a)} . </math >

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

的存在性和唯一性是压缩映射原理的结果，事实上也是如此

[math]\displaystyle{ \lim_{n\to\infty}F^{\circ n}(A)=S }[/math]

< math > lim { n to infty } f ^ { circ n }(a) = s </math >

for any nonempty compact set [math]\displaystyle{ A }[/math] in [math]\displaystyle{ X }[/math]. (For contractive IFS this convergence takes place even for any nonempty closed bounded set [math]\displaystyle{ A }[/math]). Random elements arbitrarily close to S may be obtained by the "chaos game," described below.

对于 < math > x </math > 中的任意非空紧集 < math > a </math > 。(对于收缩的 IFS，这种收敛甚至发生在任何非空的封闭集 < math > a </math >)。任意接近 s 的随机元素可以通过下面描述的“混沌游戏”获得。

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.

近年来研究表明，非收缩型 IFSs (即非收缩型 IFSs)与非收缩型 IFSs (即非收缩型 IFSs)之间存在一定的相关性。组成的映射，不收缩方面的任何拓扑等价度量在 x)可以产生吸引子。

These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.^[3]

These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.

这些在射影空间中自然产生，尽管圆上的经典无理角也可以调整。

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

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

函数 < math > f _ i </math > 的集合在 composition 下生成一个 monoid。如果只有两个这样的函数，monoid 可以可视化为一棵二叉树，其中，在树的每个节点上，一个可以与一个或另一个函数(即。采取左侧或右侧分支)。一般来说，如果有 k 个函数，那么我们可以把 monoid 想象成一个完整的 k 元树，也称为 Cayley 树。

Constructions

Barnsley's fern, an early IFS

Menger sponge, a 3-Dimensional IFS.

IFS "tree" constructed with non-linear function Julia

用非线性函数 Julia 构造 IFS“树”

文件:Golden Square fractal with T-branching 8.svg

Golden squares with T-branching

Golden squares with T-branching]]

带有 t 型分支的金色方块]]

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2012年10月11日

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| image1 = Golden Square fractal 6.svg

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| caption1 = Golden square fractal

| caption1 = Golden square fractal

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

有时候每个函数 < math > f _ i </math > 都需要是线性的，或者更一般地说是仿射变换，因此需要用矩阵表示。然而，IFSs 也可以由非线性函数构建，包括射影变换和 m ö bius 变换。分形火焰是具有非线性函数的迭代函数系统的一个例子。

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

计算 IFS 分形最常用的算法叫做“混沌博弈”。它包括在平面上选择一个随机点，然后迭代应用从函数系统中随机选择的函数之一，将该点变换为下一个点。另一种算法是生成一个给定最大长度的函数序列，然后将这些函数序列中的每一个应用到初始点或形状上的结果绘制成图。

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

这些算法中的每一个都提供了一个全局构造，生成分布在整个分形上的点。如果画出一小块分形区域，其中许多点将落在屏幕边界之外。这使得缩小到以这种方式绘制的 IFS 结构是不切实际的。

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.^[4]

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.

虽然迭代函数系统理论要求每个函数都是收缩的，但在实际应用中，实现迭代函数系统的软件只要求整个系统平均是收缩的。

Partitioned iterated function systems

PIFS (partitioned iterated function systems), also called local iterated function systems,^[5] 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.^[6]

The inverse problem

Very fast algorithms exist to generate an image from a set of IFS or PIFS parameters. It is faster and requires much less storage space to store a description of how it was created, transmit that description to a destination device, and regenerate that image anew on the destination device, than to store and transmit the color of each pixel in the image.^[5]

The inverse problem is more difficult: given some original arbitrary digital image such as a digital photograph, try to find a set of IFS parameters which, when evaluated by iteration, produces another image visually similar to the original.

In 1989, Arnaud Jacquin presented a solution to a restricted form of the inverse problem using only PIFS; the general form of the inverse problem remains unsolved.^[7]^[8]^[5]

As of 1995, all fractal compression software is based on Jacquin's approach.^[8]

Examples

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

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

Category:1981 introductions

类别: 1981年引言

This page was moved from wikipedia:en:Iterated function system. Its edit history can be viewed at 迭代函数系统/edithistory

↑ Zobrist, George Winston; Chaman Sabharwal (1992). Progress in Computer Graphics: Volume 1. Intellect Books. p. 135. ISBN 9780893916510. https://play.google.com/store/books/details?id=Ai6Qo0qoE9EC. Retrieved 7 May 2017.

↑ Michael Barnsley (1988). Fractals Everywhere, p.82. Academic Press, Inc.

.

↑ M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System
↑ Draves, Scott; Erik Reckase (July 2007). "The Fractal Flame Algorithm" (PDF). Archived from the original (pdf) on 2008-05-09. Retrieved 2008-07-17.
↑ ^5.0 ^5.1 ^5.2 Bruno Lacroix. "Fractal Image Compression". 1998.
↑ {{cite conference PIFS (partitioned iterated function systems), also called local iterated function systems, 分区的迭代函数系统，也称为本地迭代函数系统, |url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf |last= Fischer |first= Yuval |title= SIGGRAPH'92 course notes - Fractal Image Compression |conference= SIGGRAPH 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. 现有的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，它更快，并且需要更少的存储空间来存储它是如何创建的描述，将该描述传输到目标设备，并在目标设备上重新生成该图像。 |conferenceurl= http://www.siggraph.org/ |editor= Przemyslaw Prusinkiewicz The inverse problem is more difficult: given some original arbitrary digital image such as a digital photograph, try to find a set of IFS parameters which, when evaluated by iteration, produces another image visually similar to the original. 反问题更加困难: 给定一些原始的任意数字图像，如一张数码照片，试图找到一组 IFS 参数，通过迭代计算，生成另一张与原始图像视觉上相似的图像。 |publisher= ACM SIGGRAPH 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. and popularized by Michael Barnsley's book Fractals Everywhere. 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 |volume= Fractals - From Folk Art to Hyperreality |date= 1992-08-12 }}
↑ Dietmar Saupe, Raouf Hamzaoui. "A Review of the Fractal Image Compression Literature".
↑ ^8.0 ^8.1 John Kominek. "Algorithm for Fast Fractal Image Compression". doi:10.1117/12.206368.

[picg-1] Zobrist, George Winston; Chaman Sabharwal (1992). Progress in Computer Graphics: Volume 1. Intellect Books. p. 135. ISBN 9780893916510. https://play.google.com/store/books/details?id=Ai6Qo0qoE9EC. Retrieved 7 May 2017.

[2] Michael Barnsley (1988). Fractals Everywhere, p.82. Academic Press, Inc.
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[3] M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System

[4] Draves, Scott; Erik Reckase (July 2007). "The Fractal Flame Algorithm" (PDF). Archived from the original (pdf) on 2008-05-09. Retrieved 2008-07-17.

[lacroix-5] 5.0 ^5.1 ^5.2 Bruno Lacroix. "Fractal Image Compression". 1998.

[SIGGRAPH.2792-6] {{cite conference PIFS (partitioned iterated function systems), also called local iterated function systems, 分区的迭代函数系统，也称为本地迭代函数系统, |url= https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf |last= Fischer |first= Yuval |title= SIGGRAPH'92 course notes - Fractal Image Compression |conference= SIGGRAPH 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. 现有的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，它更快，并且需要更少的存储空间来存储它是如何创建的描述，将该描述传输到目标设备，并在目标设备上重新生成该图像。 |conferenceurl= http://www.siggraph.org/ |editor= Przemyslaw Prusinkiewicz The inverse problem is more difficult: given some original arbitrary digital image such as a digital photograph, try to find a set of IFS parameters which, when evaluated by iteration, produces another image visually similar to the original. 反问题更加困难: 给定一些原始的任意数字图像，如一张数码照片，试图找到一组 IFS 参数，通过迭代计算，生成另一张与原始图像视觉上相似的图像。 |publisher= ACM SIGGRAPH 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. and popularized by Michael Barnsley's book Fractals Everywhere. 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 |volume= Fractals - From Folk Art to Hyperreality |date= 1992-08-12 }}

[7] Dietmar Saupe, Raouf Hamzaoui. "A Review of the Fractal Image Compression Literature".

[kominek-8] 8.0 ^8.1 John Kominek. "Algorithm for Fast Fractal Image Compression". doi:10.1117/12.206368.

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迭代函数系统

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