A Fractal Description and Multi-scale Expression Method of Fourier Information Metrics

This paper presents a new method for describing the complexity of geographic line elements. Firstly, the Fourier series is used to transform the geographic line elements from the spatial domain to the frequency domain for analysis. Fourier expansions are performed on different geographic line elements to obtain different Fourier descriptors, then we use the method of setting the area threshold to control the deviation of the geographic elements before and after the Fourier expansion to a certain extent, so that the curve reduced by the Fourier descriptor can replace the original geographic line elements. Secondly, based on Shannon's information entropy theory, the amount of information of the fitted curve after Fourier expansion is calculated. Then, by combining the frequency domain with the fractal theory, the data of the amount of information is further processed based on the Head-tail data break and Koch's Two-Eighth Law, and the concept of the distribution index p is proposed. Finally, the distribution index and the square root model are combined, and the approximate expression of the distribution index of the map at different scales is obtained by the least squares method. We select the contour data of a region to conduct experiments, and experiment results show that the distribution index has a good expression effect on the complexity of geographic line elements at different scales, and can be applied to multi-scale expression of line elements.