基于分形技术的植物形态模拟
摘要
自20世纪70年代Mandelbrot提出分形的概念后,分形几何学作为一门新兴的交叉学科,受到学术界的广泛重视。分形理论主要描述自然界和非线性系统中不光滑和不规则的几何形体,它为植物形态的模拟提供了描述语言和理论基础。虽然自然界中的植物种类繁多、形态千差万别,却大都蕴含着一个同样的、具有自相似性质的物质结构规则:植物体中每一相对独立部分形态结构和整体形态结构都具有很高的相似特征,而分形几何学正是表现这一特征的重要数学工具,它为探讨自然界复杂事物的客观规律及其内在联系提供了新的概念和方法。特别是随着分形理论的发展,植物形态模拟逐渐成为计算机图形学研究的一个重要领域。利用分形技术来进行植物形态模拟已经成为当今时代研究者们的热点问题。
本文首先介绍了分形理论的产生与发展、定义、几何特征和Hansdorff测度与维数;概要介绍了较常用的植物形态模拟方法:L-系统,IFS(迭代函数系统),DLA和粒子系统。接着本文主要使用较流行的L-系统和IFS对植物形态进行模拟,并在这两个方面进行了深入的研究。
在L-系统方面:介绍了L-系统的基本原理、分类、算法设计;通过对一些现有的基于L-系统植物形态模拟方法进行研究和改进,实现了植物枝干随着层次变化而产生粗细变化;为了使构成植物的主干和分枝在粗细、长短、颜色和生长方向等方面有所差异,在L-系统的绘图参数中加入了标志变量和随机函数,并实现了L-系统的参数化控制,从而克服各层次枝干千篇一律的不足,使植物的整体形态更加自然逼真;对L-系统的绘图动作进行了改进,把在两点间绘制直线操作改为绘制其它图形(如圆形,矩形等)的操作,使绘制出的植物形态各异,灵活多变;本文在三维L-系统植物形态模拟也做了一些工作,用圆台模拟植物的枝干,通过引入半径衰减系数和长度衰减系数,可以体现树木的枝干底部粗、顶部细,且越往上生长分枝越短越细这种自然生长特性,从而模拟出真实感很强的三维L-系统植物。
在IFS方面:介绍了IFS的基本原理,仿射变换,IFS的算法实现;研究了迭代概率对分形树木形态的影响,通过在迭代概率中引入随机数发生器对概率进行调节,来表现多种不同的植物形态;对IFS算法的绘图动作进行了分析与改进,把IFS绘制点的操作改为绘制其它图形:如圆形、矩形、三角形或者是两种图形的结合;研究并实现了由L-系统控制的IFS算法进行植物形态模拟的方法:即用L-系统模拟植物的枝干,用IFS算法模拟植物的叶子,将两种方法扬长避短的融合在一起,可以得到比较自然和具有真实感的植物形态模拟效果。
最后,在动态变化的植物形态模拟方面也做了深入的研究,通过动态改变植物的控制参数连续生成植物图像,从而实现对植物形态的动态模拟,使生成的分形植物更加生动逼真,更加符合植物的自然变化规律。
As a new interdisciplinary, fractal geometry has received wide attention by the academia, since Mandelbrot proposed fractal concept in the 1970s. Fractal theory mainly describes the nature and nonlinear system objects which are not smooth and irregular, it also provides description language and theoretical basis for plant simulation. Natural plants have a wide variety, so their forms are varied, however, they mostly contain a same and self-similar physical structure rule: each independent part of form structure and the whole structure of plant possess highly similar characteristics. While fractal geometry is an important mathematical tool which can expresses this characteristics, and it provides a new concept and method for exploring the objective law and the inherent link of the complicated objects of the nature. Especially with the development of fractal theory, natural plant simulation is becoming an important research field of computer graphics. Plant simulation using fractal technology has already been one of the hot issues in modern times.
In the first place, the creation and development history of fractal theory, the definition and the geometric characteristics of fractal as well as Hansdorff measurement and dimension are presented in the paper; and also plant simulation methods that are more commonly used are introduced briefly including L-System, IFS, Diffusion Limited Aggregation and Particle System. In the next place, this paper mainly aims at L-System and IFS, which is much more popular in the aspects of plant simulation, and makes an intensive study of the two plant simulation methods.
In aspect of L–System: the basic principle, the classification and the algorithm design of L-System are introduced in this paper; by studying and improving the exsiting methods of the plant simulation based on L-System, the paper implements the thickness change of the trunk and branches with the level degree change. In order to make the trunk and branches which compose the plant more diverse in thickness, length, color and growth direction, flag variable and random function are added to the drawing parameters of L-System, and parameterized control of L-System are implemented, so it can make the integral form more natural and lifelike because of overcoming the deficiency of no difference of each level branches. Some work is also carried out in three-dimentional L-System using frustum of a cone to simulate the trunk and branches; by introducing the radius attenuation coefficient and the length of the attenuation coefficient,the natural characteristics of the branch that is thick at the bottom and thin at the top as well as shorter and thiner with the increase of the level are embodied, which makes three-dimentional L-System plant simulation more realistic.
In aspect of IFS: the basic principle of IFS, the affine transformation,and implementation of the IFS algorithm are introduced. Adjusting the iterative probability by using random number generator, more varied plant form are embodied. The IFS algorithm in aspect of the drawing operation are studid and improved, and then the drawing line operation are midified to other graphics operation such as drawing circle, rectangle, triangle or combining of the two graphics; the method of plant simulation using IFS algorithm controlled by L-system are also studied and implented: the trunk and branches are simulated by L-System and the leaves are simulated by IFS algorithm, that is to say, the two methods are made together effectively, which can produces more natural and realistic simulation effect.
At last, a thorough study has been carried out on dynamic changes in plant simulation. By making continuous changes on graphics parameters, some of the graphics parameters are continuously controlled, and then plant simulation that changes dynamically is implemented. As a result, fractal plants are generated more vividly and more in line with natural plants.
本文首先介绍了分形理论的产生与发展、定义、几何特征和Hansdorff测度与维数;概要介绍了较常用的植物形态模拟方法:L-系统,IFS(迭代函数系统),DLA和粒子系统。接着本文主要使用较流行的L-系统和IFS对植物形态进行模拟,并在这两个方面进行了深入的研究。
在L-系统方面:介绍了L-系统的基本原理、分类、算法设计;通过对一些现有的基于L-系统植物形态模拟方法进行研究和改进,实现了植物枝干随着层次变化而产生粗细变化;为了使构成植物的主干和分枝在粗细、长短、颜色和生长方向等方面有所差异,在L-系统的绘图参数中加入了标志变量和随机函数,并实现了L-系统的参数化控制,从而克服各层次枝干千篇一律的不足,使植物的整体形态更加自然逼真;对L-系统的绘图动作进行了改进,把在两点间绘制直线操作改为绘制其它图形(如圆形,矩形等)的操作,使绘制出的植物形态各异,灵活多变;本文在三维L-系统植物形态模拟也做了一些工作,用圆台模拟植物的枝干,通过引入半径衰减系数和长度衰减系数,可以体现树木的枝干底部粗、顶部细,且越往上生长分枝越短越细这种自然生长特性,从而模拟出真实感很强的三维L-系统植物。
在IFS方面:介绍了IFS的基本原理,仿射变换,IFS的算法实现;研究了迭代概率对分形树木形态的影响,通过在迭代概率中引入随机数发生器对概率进行调节,来表现多种不同的植物形态;对IFS算法的绘图动作进行了分析与改进,把IFS绘制点的操作改为绘制其它图形:如圆形、矩形、三角形或者是两种图形的结合;研究并实现了由L-系统控制的IFS算法进行植物形态模拟的方法:即用L-系统模拟植物的枝干,用IFS算法模拟植物的叶子,将两种方法扬长避短的融合在一起,可以得到比较自然和具有真实感的植物形态模拟效果。
最后,在动态变化的植物形态模拟方面也做了深入的研究,通过动态改变植物的控制参数连续生成植物图像,从而实现对植物形态的动态模拟,使生成的分形植物更加生动逼真,更加符合植物的自然变化规律。
As a new interdisciplinary, fractal geometry has received wide attention by the academia, since Mandelbrot proposed fractal concept in the 1970s. Fractal theory mainly describes the nature and nonlinear system objects which are not smooth and irregular, it also provides description language and theoretical basis for plant simulation. Natural plants have a wide variety, so their forms are varied, however, they mostly contain a same and self-similar physical structure rule: each independent part of form structure and the whole structure of plant possess highly similar characteristics. While fractal geometry is an important mathematical tool which can expresses this characteristics, and it provides a new concept and method for exploring the objective law and the inherent link of the complicated objects of the nature. Especially with the development of fractal theory, natural plant simulation is becoming an important research field of computer graphics. Plant simulation using fractal technology has already been one of the hot issues in modern times.
In the first place, the creation and development history of fractal theory, the definition and the geometric characteristics of fractal as well as Hansdorff measurement and dimension are presented in the paper; and also plant simulation methods that are more commonly used are introduced briefly including L-System, IFS, Diffusion Limited Aggregation and Particle System. In the next place, this paper mainly aims at L-System and IFS, which is much more popular in the aspects of plant simulation, and makes an intensive study of the two plant simulation methods.
In aspect of L–System: the basic principle, the classification and the algorithm design of L-System are introduced in this paper; by studying and improving the exsiting methods of the plant simulation based on L-System, the paper implements the thickness change of the trunk and branches with the level degree change. In order to make the trunk and branches which compose the plant more diverse in thickness, length, color and growth direction, flag variable and random function are added to the drawing parameters of L-System, and parameterized control of L-System are implemented, so it can make the integral form more natural and lifelike because of overcoming the deficiency of no difference of each level branches. Some work is also carried out in three-dimentional L-System using frustum of a cone to simulate the trunk and branches; by introducing the radius attenuation coefficient and the length of the attenuation coefficient,the natural characteristics of the branch that is thick at the bottom and thin at the top as well as shorter and thiner with the increase of the level are embodied, which makes three-dimentional L-System plant simulation more realistic.
In aspect of IFS: the basic principle of IFS, the affine transformation,and implementation of the IFS algorithm are introduced. Adjusting the iterative probability by using random number generator, more varied plant form are embodied. The IFS algorithm in aspect of the drawing operation are studid and improved, and then the drawing line operation are midified to other graphics operation such as drawing circle, rectangle, triangle or combining of the two graphics; the method of plant simulation using IFS algorithm controlled by L-system are also studied and implented: the trunk and branches are simulated by L-System and the leaves are simulated by IFS algorithm, that is to say, the two methods are made together effectively, which can produces more natural and realistic simulation effect.
At last, a thorough study has been carried out on dynamic changes in plant simulation. By making continuous changes on graphics parameters, some of the graphics parameters are continuously controlled, and then plant simulation that changes dynamically is implemented. As a result, fractal plants are generated more vividly and more in line with natural plants.
引文
[1] Mandelbrot B B.Fracals:Form,Chance and Dimension[M].San Francisco:freeman 1977.
[2] Mandelbrot B B.The Fractal Geometry of Nature[M].San Fransisco:freeman 1982.
[3]梁士楚,王伯荪.红树植物木榄种群植冠层结构的分形特征[J].海洋通报,2002,21(5):26-31.
[4]梁士楚,王伯荪.红树植物木榄种群高度结构的分形特征[J].植物生态学报,2002, 26(4):408-412.
[5] Przemyslaw Prusinkiewicz.Modeling plant growth and development[J].Current Opinion in Plant Biology,2004,7(1):79-83.
[6] Prusinkiewcz P,Lindenmanyer A.The algorithmic beauty of plants[M].New York: Springer-Verlag,1996,1-46.
[7]赵星,d.R.Philippe,熊范纶等.虚拟植物生长的双尺度自动机模型[J].计算机学报,2001, 24(6):608-615.
[8]马石安,陈传波.基于迭代函数系统的森林景物动态模拟技术研究[J].计算机工程,2005, 29(4)13-16.
[9] J.Feder.Fractals[M].Plenum Press,New York and London,1988.
[10]龙期威.金属中的分形与复杂性[M].上海:上海科技出版社,1999.10.
[11]李后强等.分形理论及其在分子科学中的应用[M].北京:科学出版社,1993.
[12]马新明,杨娟,熊淑萍等.植物模拟研究现状及展望[J].作物研究,2003,17(3):148-151.
[13]石春林,金之庆,葛道阔.植物可视化研究进展[J].江苏农业科学,2004,23(6):8-10.
[14]冯斌,杨培岭.植物根系的分形及计算机模拟[J].中国农业大学学报,2000,(2):96-99.
[15]金明现,王天铎.玉米根系生长及向水性的模拟[J].植物学报,1996,38(5):384-390.
[16]隋首钢.复动力系统的空间分形Julia集[D].山东大学硕士学位论文.2005.5.
[17]文志英,苏维宜,文志雄,范爱华.分形几何理论的应用[M].杭州:浙江科学出版社,1998.
[18]李水根,吴纪桃.分形与小波[M].北京:科学出版社,2002.
[19]齐东旭著.分形及其计算机生成[M].北京:科学出版社,1994.
[20] A.Lindenmayer.Journal of Theoretical[M].Biology,18(1968)280-300.
[21] A.Smith.Computer Graphics[M].1(1984)1.
[22] P.Prusinkiewicz.In processings of Graphics Interface‘86-vision Interface’86,247-253,GIPS(1986).
[23]刘华杰.分形艺术[M].湖南科学技术出版社,1998.
[24]王东生,曹磊.混沌、分形及其应用[M].中国科学技术大学出版社,1995.
[25] Lindenmayer A.Mathematical models for cellular interaction in development[J].Journal of Theoretical Biology,1996:280-315.
[26] Lindenmayer A.Development systems without cellular interaction their languages and gratmmers[J].Journal of Theoreical Biology,1997:455-458.
[27] M.F.Barnsley.S.Demko,In Pro.Royal Soc[M].London,A 399(1985)243.
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[29]黄艳峰.植物形态的分形模拟[D].西北工业大学硕士学位论文.2004.3.
[30]肯尼思,法尔科内.分形几何-数学基础及应用[M].东北大学出版社,2001.
[31]璩柏青,张黎明.基于IFS代码的动画仿真[J].电子科技,2004(8):37-40.
[32] B.Benes,E.U.Millan.Virtual Climbing Plants Competing for Space[M].Proceedings of the Computer Animation,Geneva,Switzerland,2002:33-43.
[33]孙博文.分形算法与程序设计Visual C++实现[M].北京:科学出版社,2004.
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[2] Mandelbrot B B.The Fractal Geometry of Nature[M].San Fransisco:freeman 1982.
[3]梁士楚,王伯荪.红树植物木榄种群植冠层结构的分形特征[J].海洋通报,2002,21(5):26-31.
[4]梁士楚,王伯荪.红树植物木榄种群高度结构的分形特征[J].植物生态学报,2002, 26(4):408-412.
[5] Przemyslaw Prusinkiewicz.Modeling plant growth and development[J].Current Opinion in Plant Biology,2004,7(1):79-83.
[6] Prusinkiewcz P,Lindenmanyer A.The algorithmic beauty of plants[M].New York: Springer-Verlag,1996,1-46.
[7]赵星,d.R.Philippe,熊范纶等.虚拟植物生长的双尺度自动机模型[J].计算机学报,2001, 24(6):608-615.
[8]马石安,陈传波.基于迭代函数系统的森林景物动态模拟技术研究[J].计算机工程,2005, 29(4)13-16.
[9] J.Feder.Fractals[M].Plenum Press,New York and London,1988.
[10]龙期威.金属中的分形与复杂性[M].上海:上海科技出版社,1999.10.
[11]李后强等.分形理论及其在分子科学中的应用[M].北京:科学出版社,1993.
[12]马新明,杨娟,熊淑萍等.植物模拟研究现状及展望[J].作物研究,2003,17(3):148-151.
[13]石春林,金之庆,葛道阔.植物可视化研究进展[J].江苏农业科学,2004,23(6):8-10.
[14]冯斌,杨培岭.植物根系的分形及计算机模拟[J].中国农业大学学报,2000,(2):96-99.
[15]金明现,王天铎.玉米根系生长及向水性的模拟[J].植物学报,1996,38(5):384-390.
[16]隋首钢.复动力系统的空间分形Julia集[D].山东大学硕士学位论文.2005.5.
[17]文志英,苏维宜,文志雄,范爱华.分形几何理论的应用[M].杭州:浙江科学出版社,1998.
[18]李水根,吴纪桃.分形与小波[M].北京:科学出版社,2002.
[19]齐东旭著.分形及其计算机生成[M].北京:科学出版社,1994.
[20] A.Lindenmayer.Journal of Theoretical[M].Biology,18(1968)280-300.
[21] A.Smith.Computer Graphics[M].1(1984)1.
[22] P.Prusinkiewicz.In processings of Graphics Interface‘86-vision Interface’86,247-253,GIPS(1986).
[23]刘华杰.分形艺术[M].湖南科学技术出版社,1998.
[24]王东生,曹磊.混沌、分形及其应用[M].中国科学技术大学出版社,1995.
[25] Lindenmayer A.Mathematical models for cellular interaction in development[J].Journal of Theoretical Biology,1996:280-315.
[26] Lindenmayer A.Development systems without cellular interaction their languages and gratmmers[J].Journal of Theoreical Biology,1997:455-458.
[27] M.F.Barnsley.S.Demko,In Pro.Royal Soc[M].London,A 399(1985)243.
[28] M.F.Barnsley.S.Demko,Lecture Notes in Math[M].1105(1984)73.
[29]黄艳峰.植物形态的分形模拟[D].西北工业大学硕士学位论文.2004.3.
[30]肯尼思,法尔科内.分形几何-数学基础及应用[M].东北大学出版社,2001.
[31]璩柏青,张黎明.基于IFS代码的动画仿真[J].电子科技,2004(8):37-40.
[32] B.Benes,E.U.Millan.Virtual Climbing Plants Competing for Space[M].Proceedings of the Computer Animation,Geneva,Switzerland,2002:33-43.
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