基于混沌吸引子的对称分形图方法研究

发布时间：2018-03-27 15:34

本文选题：混沌函数　切入点：PSO　出处：《山东师范大学》2013年硕士论文

【摘要】：分形理论是几何学的一个新的研究方向——分形几何学，因为其能够很好的模拟自然景物，又名大自然的几何学。除了具有普通艺术图案的特点之外，，分形图形一般具有自相似性和无限的精细结构等特征。分形与混沌分别在空间和时间标度下表现出自相似性。可以说，混沌与分形具有内在的本质联系。目前，混沌分形图已经被广泛应用于商标防伪、纺织和科幻影片等之中，在图案设计中占有独特的地位。混沌现象是大千世界中存在的一种复杂的物理现象，是非线性动力系统的固有特性。混沌吸引子是由混沌函数反复迭代而成，由无穷多个迭代点集组成，是混沌现象的一种“图形化”描述。这种图形具有明显的分形特征，也常被称为混沌分形图。有一类混沌分形图不仅具有混沌现象，还具有大量内置的对称现象。本文研究的内容主要是针对具有对称现象的混沌分形图，主要包括三个方面：利用PSO实现混沌吸引子参数的快速优化，提出基于色彩控制圆的渲染方案，图形参数的控制及图形的绘制。首先，利用计算机丰富的显示功能，实现对对称分形图在周期、位置、大小、色彩上的参数可控，使普通用户可通过对话框方便地生成各类对称分形图；在此研究的基础上，提出一种改进的配色方案——基于色彩控制圆的渲染方案，可使图案视觉效果由简单变化到绚丽多彩；最终对某个特定的混沌函数系统编程实现一个产生对称性混沌分形图的模拟系统。按实际的实施过程，具体的研究内容如下： (1)利用PSO优化对称分形图的参数向量本项研究将PSO应用于优化混沌吸引子参数向量的问题，构造了特定混沌函数参数向量作为粒子表达方法，建立了此问题的PSO。试验结果表明，PSO可以快速、高效求得各参数向量的最优解，从而解决了在巨大参数空间下生成混沌分形图困难的问题。 (2)确定渲染方案本文结合对称分形图的特点，以敦煌飞天图的色彩组合为基调，提出了一种以蓝、紫、红、黄为主色调的渲染方法，表现出一种似梦似真飘渺虚幻灵动弥漫的感染力。 (3)对称分形图的参数可控性研究查阅大量的参考文献，研究对称分形图的生成原理及方法。本文以带群p1a1模型为例，通过对其生成原理、绘制方法、渲染方案进行研究，提炼出可控参数。普通用户通过对话框方便的控制这些参数，从而生成用户满意的对称分形图案。 (4)模拟系统的开发开发了一个简单实用的模拟系统，实现了对称分形图的参数的优化、及图案的绘制与保存，设计较为方便的颜色对话框，用户可通过指定调色板中的色彩来设置分形图案的主色调，为用户提供了简单易操作的系统界面，同时为后续的研究提供了便捷工具。本文所生成的对称分形图呈现的空间结构精细、形态瑰丽、内涵丰富，不仅具有丰厚的理论意义，而且可广泛应用于社会生活中的图案设计，有望在纺织、印刷、广告、防伪等图案设计领域引起关注。
[Abstract]:Fractal theory is a new research direction of geometry - fractal geometry, because of its natural scenery simulation can be very good, also known as the geometry of nature. In addition to having features of ordinary art patterns, fractal graphics generally have self similarity and unlimited fine structure feature. Fractal and chaos in space and time scale under the self similarity characteristic. It can be said that the chaos and fractal are internally related. At present, the chaotic fractal graph has been widely used in trademarks, textile and other science fiction films, the pattern design has a unique position.
Chaos is a kind of complex physical phenomena exist in the universe of 1,000,000,000 universes are inherent in the nonlinear dynamic system. The chaotic attractor is a chaotic function iteratively and, by an infinite number of iteration point set, is a "graphic" description of the chaotic phenomenon. This figure has obvious fractal characteristics, also often referred to as the chaos and fractal graph. A kind of chaos and fractal graph not only has chaotic phenomenon, also has a large number of built-in symmetricphenomenon.
The content of this paper is mainly based on chaotic fractal images with symmetrical phenomenon, mainly includes three aspects: to realize rapid optimization of chaotic attractor parameters by using PSO, the rendering scheme based on color control circle, drawing graphics and control parameters. First, the use of computer display, to achieve symmetric fractal graph in the period, location, size, color parameters, which can be conveniently through the dialog box to generate various types of symmetrical fractal figure of ordinary users; on the basis of this study, this paper proposes an improved color scheme - rendering scheme based on color control circle, which can make the design of visual effects by a simple change to achieve a bright and colorful; have the symmetry of chaos and fractal graph simulation system for a particular programming iteration function system. According to the actual implementation process, the specific research contents as follows:
(1) the parameter vector using PSO optimization symmetrical fractal figure
This study applies PSO to optimize the parameter of chaotic attractor vector, constructed the specific chaotic function parameter vector as the particle expression method, established the PSO. test results of this problem shows that PSO can quickly and efficiently obtain the optimal parameters of vector solution, so as to solve the difficulty of generating chaotic fractal graph under huge parameter space the problem.
(2)纭畾娓叉煋鏂规
In this paper, combined with the characteristics of symmetrical fractal figure, Dunhuang flying figure color combinations for the tone, presents a blue, purple, red, yellow color rendering method, showing a dream likereally misty illusory facial diffuse appeal.
(3) the controllable parameters about the symmetrical fractal figure
Access to a large number of references, the generation principle and method of the symmetrical fractal figure. This thesis takes p1a1 model as an example, the generation principle of the method of drawing, rendering research plan, refine controllable parameters. Ordinary users to control these parameters through the dialog box is convenient, so as to generate symmetrical fractal patterns of user satisfaction.
(4) the development of simulation system
The development of a simple and practical simulation system, realize the optimization of parameters of symmetrical fractal figure, and pattern drawing and preservation, design more convenient color dialog box, the user can set the main tone of fractal pattern by specifying the palette color, provides the interface of the system is simple and easy to operate for the users, while providing a convenient tool for the follow-up study.
The spatial structure of the symmetrical fractal figure generated by the present form of fine, magnificent, rich in content, not only has theoretical significance but also rich, can be widely used in the social life of the design, is expected in the textile, printing, advertising, has attracted the attention of security and other design fields.

【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：TP391.41;O189

【参考文献】