距离比值迭代分形及复迭代函数系统的研究
摘要
将分形几何与计算机图形学结合,实现分形体的可视化以及利用分形模拟自然景物是计算机图形学中的重要研究方向之一。应用计算机图形学研究分形几何有助于揭示分形本身的结构和性质,还可以进行艺术创作,生成分形艺术图形。本文针对分形图形学中存在的若干问题进行研究,主要的创新工作如下:
1.提出了一种构造分形图的新方法:距离比值迭代法。该方法采用两点迭代,利用其距离比值的迭代收敛速度来绘制分形图。不同于逃逸时间算法,距离比值迭代法绘制的广义M-J集在映射稳定区域内具有丰富的细节,并且能够绘制一些逃逸时间算法无法绘制的映射分形图。
2.详尽研究了复映射f(z)=z~α+c构成的距离比值广义M-J集的性质。讨论了对于不同的参数c和指数α,距离比值广义M-J集构图特征的变化,说明了距离比值广义M-J集与传统M-J集的区别和联系。讨论了一些以往未被研究过的映射所产生的分形集,如复映射f(z)=z~α的距离比值广义J集,复映射f(z)=z~α+c(0<α<1)的距离比值广义M集等。
3.全面系统地考查了距离比值迭代方法可能的扩展。包括:不同初始迭代点对构图的影响;迭代过程中引进其他映射对结果图的影响;各种周期点、周期轨道时距离比值迭代方法的变化等。
4.研究了复映射族f(z)=z~2+c_i的迭代性质和规律,研究了其成为迭代函数系统的条件,分析了其吸引子的空间范围及不动点的分布规律等。
5.提出了一个基于迭代函数系统的模拟干笔飞白效果的纹理模型。该模型采用迭代函数系统的吸引子作为笔迹,能够用于现有毛笔模型中,较好的实现干笔飞白效果。
Since the introduction of fractal by Mandelbrot, fractals have experiencedconsiderable success in quantifying the complex structure exhibited by many naturalpatterns and have captured the imaginations of both scientists and artists. Manyresearchers perform research on the generation method of fractal and draw manybeautiful images. The research of fractal has experienced a slow progressing phase. Itis needed to conclude current rendering methods and introduce new algorithm.
The research of fractal needs the application of computer graphics method. One ofthe reasons why fractal concept can be accepted by people rapidly is the help of thecomputer graphics method. By giving a deep research on the application of computergraphics in fractal rendering, we can increase the comprehension of fractal and extendfractal application region. Meanwhile, the nature of complex structure of fractal needsthe fractal rendering method to be improved.
The basic idea of current fractal rendering algorithm is to treat fractal as point sets,select different color for pixel during iteration process and render it on screen, such aschaos game method for rendering IFS and escape time algorithm for renderingcomplex mapping. These methods have following shortages:
1. Current render method of complex mapping can only render the borderbetween the stable region and escape region, and cannot display the detail of stableregion and escape region and it is difficult to understand fractal image structure.
2.The iteration fractal image of some complex mapping is difficult to rendering.It is needed to modify current method or develop new method to research the iterativeproperties of the mapping.
3. There is no related work about making fractal image with some painting style,so it is possible to extend the application region of fractal image.
From above discussion we can see there are many problems in fractal technology.This dissertation tries to innovate and break through on these problems, introduce new
fractal rendering method to rendering process and improve fractal image rendering andutilize it to a new level. The innovations of this dissertation are listed as follows:1. Propose the distance ratio iteration method to construct fractal image.The distance ratio iteration method takes two points to perform iteration andutilizes its convergence speed to render fractal image of complex mapping. Thegeneralized Mandelbrot and Julia sets based on distance ratio have rich detail in stableregion. The distance ratio iteration method can render fractal image of some complexmapping which cannot be rendered by escape time algorithm. This dissertation sturdieson the iteration properties of distance ratio and proves its convergence theorem. Basedon these theorems, we introduce three methods for rendering distance ratio fractal,including convergence time method, inverse iteration layer method and mix method.These algorithms can render many fractal images of complex mapping, such aspolynomial mapping, exponent mapping, logarithm mapping and 3x+1 mapping.The escape time algorithm treat all convergence points as a same point, so itcannot render structure of stable region. However, distance ratio method useconvergence time to classify point set and can get complex structure.The fractal image rendered by distance ratio method is the two dimensionprojection of complete four dimension point sets because distance ratio takes twopoints to perform iteration. Distance ratio iteration method provide a general methodfor rendering 2d-4d fractal object.2.Research on the generalized Julia set based on distance ratio for f(z)=zα+c.Firstly we research the visual structure of DRJ for nonparameter mapping f(z)=z2and analysis its buds distribute rule, give formula of each level width. Secondly, weresearch the evolution rule of DRJ for mapping f(z)=zα(1<α<2) and rendering variousexponent mapping ‘s fractal image.We study the generalized Julia set based on distance ratio for f(z)=z2+c anddiscuss the image properties for mapping with attractive fixed point, two period orbitand p period orbit. We state the symmetry theorem of DRJ and prove the border ofDRJ is the mapping‘s Julia set. So the distance ratio iteration method can be a methodfor rendering the inner structure of Julia set. We state the reason of two phase structureexist in DRJ when mapping has two period orbit and give a new algorithm to renderDRJ when the mapping has p period orbit.This dissertation discusses the chooson of z2 for DRJ. Firstly, we research thesituation when z2 is a fixed complex number and discuss image structure for differentlocation of z2. Then we research the situation which z2=g(z1) and give the notation oftwo mapping compose distance ratio Julia set. We discuss some complex mapping
composed distance ratio Julia set and state the extend ability of distance ratio iterationmethod.We research the generalized Julia set based on distance ratio for f(z)=zα+c andstate the shortage of escape time algorithm in this situation and the advantage ofdistance ratio iteration method. The distance ratio Julia set is a fine Julia set and it canrender the detail that not in Julia set.3.Research on generalized Mandelbrot set based on distance ratio for f(z)=zα+c.Taking complex mapping f(z)=zα+c as an example, we generates images of thegeneralized Mandelbrot set based on distance ratio and states conjectures on theirvisual characteristics. Comparing with escape time algorithm, it consists of abundantdetails in the convergent region and more complex structure can be generated.Therefore, distance ratio iteration method can be used as a new method for renderinginner structure of M-set when α>1.DRM for α<0 is composed of two parts: outer curve polygon and innerconstellation. The curve polygon has |α|+1 edges and is covered by petal structures inouter layer;Inner constellation is composed of a central planet and surroundingsatellites. The central planet has |α|+1 protuberances and tangent with curve edge ofoutside polygon in the middle points. The DRM is the complement of M-set incomplex plane.When 0<α<1, M-set generated by escape time algorithm is a kind of simplegeometry image, because the attractive region of the mapping changes.4.Research on the complex iterated function system f(z) =z2+ciThe dissertation researches the iteration properties of complex mapping setf(z)=z2+c by taking them as an complex iterated function system. Firstly we discuss thecondition of the complex mapping family to become an NIFS. Then the dissertationresearches some common problems in NIFS, such as the relationship between attractorand the fixed point of mapping, the selection of mapping parameter, and the bound ofattractor.Based on above discussion, we can make such conclusion: the complex mappingset is a iteration function system, only if |ci|≤0.25 and the initial iteration point z locatesin the intersection of the mappings of the Julia set or |z|<0.5. The attractor generated bythe complex IFS satisfying this condition converge to the cycle whose center is theorigin and radius is(1 ? 1 ? 4 R)/ 2. The equations for computing fixed points are listedin the former part.
5.Propose a model to render dry brush effect of Chinese brush work.Some article states that there is fractal character in Chinese paint and calligraphy.So the fractal image can be used to Chinese painting and calligraphy. We propose acalligraphy model based on iteration function system construct by complex mappingfamily f(z)=z2+ci. The models construct iterated function system based on user's brushspeed and contrail renders the attractor by chaos game algorithm as final dry brushstroke. It is easy to combine this model to current brush model.In conclusion, this dissertation has academic significance and value of application,and it enriches the research of fractal geometry. It also provides constructive methodand techniques for research of fractal geometry.
1.提出了一种构造分形图的新方法:距离比值迭代法。该方法采用两点迭代,利用其距离比值的迭代收敛速度来绘制分形图。不同于逃逸时间算法,距离比值迭代法绘制的广义M-J集在映射稳定区域内具有丰富的细节,并且能够绘制一些逃逸时间算法无法绘制的映射分形图。
2.详尽研究了复映射f(z)=z~α+c构成的距离比值广义M-J集的性质。讨论了对于不同的参数c和指数α,距离比值广义M-J集构图特征的变化,说明了距离比值广义M-J集与传统M-J集的区别和联系。讨论了一些以往未被研究过的映射所产生的分形集,如复映射f(z)=z~α的距离比值广义J集,复映射f(z)=z~α+c(0<α<1)的距离比值广义M集等。
3.全面系统地考查了距离比值迭代方法可能的扩展。包括:不同初始迭代点对构图的影响;迭代过程中引进其他映射对结果图的影响;各种周期点、周期轨道时距离比值迭代方法的变化等。
4.研究了复映射族f(z)=z~2+c_i的迭代性质和规律,研究了其成为迭代函数系统的条件,分析了其吸引子的空间范围及不动点的分布规律等。
5.提出了一个基于迭代函数系统的模拟干笔飞白效果的纹理模型。该模型采用迭代函数系统的吸引子作为笔迹,能够用于现有毛笔模型中,较好的实现干笔飞白效果。
Since the introduction of fractal by Mandelbrot, fractals have experiencedconsiderable success in quantifying the complex structure exhibited by many naturalpatterns and have captured the imaginations of both scientists and artists. Manyresearchers perform research on the generation method of fractal and draw manybeautiful images. The research of fractal has experienced a slow progressing phase. Itis needed to conclude current rendering methods and introduce new algorithm.
The research of fractal needs the application of computer graphics method. One ofthe reasons why fractal concept can be accepted by people rapidly is the help of thecomputer graphics method. By giving a deep research on the application of computergraphics in fractal rendering, we can increase the comprehension of fractal and extendfractal application region. Meanwhile, the nature of complex structure of fractal needsthe fractal rendering method to be improved.
The basic idea of current fractal rendering algorithm is to treat fractal as point sets,select different color for pixel during iteration process and render it on screen, such aschaos game method for rendering IFS and escape time algorithm for renderingcomplex mapping. These methods have following shortages:
1. Current render method of complex mapping can only render the borderbetween the stable region and escape region, and cannot display the detail of stableregion and escape region and it is difficult to understand fractal image structure.
2.The iteration fractal image of some complex mapping is difficult to rendering.It is needed to modify current method or develop new method to research the iterativeproperties of the mapping.
3. There is no related work about making fractal image with some painting style,so it is possible to extend the application region of fractal image.
From above discussion we can see there are many problems in fractal technology.This dissertation tries to innovate and break through on these problems, introduce new
fractal rendering method to rendering process and improve fractal image rendering andutilize it to a new level. The innovations of this dissertation are listed as follows:1. Propose the distance ratio iteration method to construct fractal image.The distance ratio iteration method takes two points to perform iteration andutilizes its convergence speed to render fractal image of complex mapping. Thegeneralized Mandelbrot and Julia sets based on distance ratio have rich detail in stableregion. The distance ratio iteration method can render fractal image of some complexmapping which cannot be rendered by escape time algorithm. This dissertation sturdieson the iteration properties of distance ratio and proves its convergence theorem. Basedon these theorems, we introduce three methods for rendering distance ratio fractal,including convergence time method, inverse iteration layer method and mix method.These algorithms can render many fractal images of complex mapping, such aspolynomial mapping, exponent mapping, logarithm mapping and 3x+1 mapping.The escape time algorithm treat all convergence points as a same point, so itcannot render structure of stable region. However, distance ratio method useconvergence time to classify point set and can get complex structure.The fractal image rendered by distance ratio method is the two dimensionprojection of complete four dimension point sets because distance ratio takes twopoints to perform iteration. Distance ratio iteration method provide a general methodfor rendering 2d-4d fractal object.2.Research on the generalized Julia set based on distance ratio for f(z)=zα+c.Firstly we research the visual structure of DRJ for nonparameter mapping f(z)=z2and analysis its buds distribute rule, give formula of each level width. Secondly, weresearch the evolution rule of DRJ for mapping f(z)=zα(1<α<2) and rendering variousexponent mapping ‘s fractal image.We study the generalized Julia set based on distance ratio for f(z)=z2+c anddiscuss the image properties for mapping with attractive fixed point, two period orbitand p period orbit. We state the symmetry theorem of DRJ and prove the border ofDRJ is the mapping‘s Julia set. So the distance ratio iteration method can be a methodfor rendering the inner structure of Julia set. We state the reason of two phase structureexist in DRJ when mapping has two period orbit and give a new algorithm to renderDRJ when the mapping has p period orbit.This dissertation discusses the chooson of z2 for DRJ. Firstly, we research thesituation when z2 is a fixed complex number and discuss image structure for differentlocation of z2. Then we research the situation which z2=g(z1) and give the notation oftwo mapping compose distance ratio Julia set. We discuss some complex mapping
composed distance ratio Julia set and state the extend ability of distance ratio iterationmethod.We research the generalized Julia set based on distance ratio for f(z)=zα+c andstate the shortage of escape time algorithm in this situation and the advantage ofdistance ratio iteration method. The distance ratio Julia set is a fine Julia set and it canrender the detail that not in Julia set.3.Research on generalized Mandelbrot set based on distance ratio for f(z)=zα+c.Taking complex mapping f(z)=zα+c as an example, we generates images of thegeneralized Mandelbrot set based on distance ratio and states conjectures on theirvisual characteristics. Comparing with escape time algorithm, it consists of abundantdetails in the convergent region and more complex structure can be generated.Therefore, distance ratio iteration method can be used as a new method for renderinginner structure of M-set when α>1.DRM for α<0 is composed of two parts: outer curve polygon and innerconstellation. The curve polygon has |α|+1 edges and is covered by petal structures inouter layer;Inner constellation is composed of a central planet and surroundingsatellites. The central planet has |α|+1 protuberances and tangent with curve edge ofoutside polygon in the middle points. The DRM is the complement of M-set incomplex plane.When 0<α<1, M-set generated by escape time algorithm is a kind of simplegeometry image, because the attractive region of the mapping changes.4.Research on the complex iterated function system f(z) =z2+ciThe dissertation researches the iteration properties of complex mapping setf(z)=z2+c by taking them as an complex iterated function system. Firstly we discuss thecondition of the complex mapping family to become an NIFS. Then the dissertationresearches some common problems in NIFS, such as the relationship between attractorand the fixed point of mapping, the selection of mapping parameter, and the bound ofattractor.Based on above discussion, we can make such conclusion: the complex mappingset is a iteration function system, only if |ci|≤0.25 and the initial iteration point z locatesin the intersection of the mappings of the Julia set or |z|<0.5. The attractor generated bythe complex IFS satisfying this condition converge to the cycle whose center is theorigin and radius is(1 ? 1 ? 4 R)/ 2. The equations for computing fixed points are listedin the former part.
5.Propose a model to render dry brush effect of Chinese brush work.Some article states that there is fractal character in Chinese paint and calligraphy.So the fractal image can be used to Chinese painting and calligraphy. We propose acalligraphy model based on iteration function system construct by complex mappingfamily f(z)=z2+ci. The models construct iterated function system based on user's brushspeed and contrail renders the attractor by chaos game algorithm as final dry brushstroke. It is easy to combine this model to current brush model.In conclusion, this dissertation has academic significance and value of application,and it enriches the research of fractal geometry. It also provides constructive methodand techniques for research of fractal geometry.
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[95] UltraFractal software. http://www.ultrafractal.com/
[96] Fractint software. http://spanky.triumf.ca/www/fractint/fractint.html
[97] XaoS software. http://wmi.math.u-szeged.hu/~kovzol/xaos/doku.php
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