Spectrum Analysis of Fractals Based on Orthogonal Decomposition

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 adjustable parameters to the V-spectra of the known fractal. Finally, present a method of generating the fractal variants 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.