分形的正交频谱分析

Spectrum Analysis of Fractals Based on Orthogonal Decomposition

为了对几何分形对象进行频域表达,利用具有自相似结构的正交函数系——V-系统对分形做正交分解,提出一种对分形对象的频谱分析方法.该方法利用V-系统对分形对象进行数学表达,得到分形的V-谱;在已知分形的V-谱中引入调节参数,并给出一类通过调节参数的设置,获得分形变体或生成新分形的方法.实验结果表明,分形的V-谱不仅可以展现分形对象的整体轮廓与逐级逼近的复杂细节,还便于对不同分形间的差异进行量化;通过V-谱的调节可以产生意想不到的几何分形.通过较多图例诠释文中方法与效果,最后给出河流变迁的例子,表明分形对象的频谱分析方法具有良好的应用前景.

To represent fractals in frequency domain, this paper proposes a method of fractal frequency analysis based on an orthogonal function system, the V-system, with self-similar structure. Firstly, expand the fractal object into a V-series and the V-spectra are obtained accordingly. Then introduce several adjust-able parameters to the V-spectra of the known fractal. Finally, present a method of generating the fractal va-riants or new fractal curves by changing the values of the adjustable parameters. The experimental results show that the V-spectra of a fractal can not only display the contour of the fractal and its complex details in layers, but also quantify the difference between two different fractals. Many amazing geometric fractals can be produced by adjusting the V-spectra. Plenty of fractal graphs are provided in this paper to interpret the proposed method and show its effect. In addition, fractal curves describing changes in rivers are provided in the end of the paper to illustrate the application prospect of the fractal orthogonal representation.