Closed Contour Fractal Dimension Estimation by the Fourier Transform

TL;DR

This paper introduces a new Fourier-based method for accurately estimating the fractal dimension of closed contour objects, demonstrating high precision and robustness compared to classical methods.

Contribution

It presents a novel Fourier transform approach for fractal dimension estimation of closed contours, improving accuracy and robustness over existing techniques.

Findings

High precision in fractal dimension estimation

Robustness against noise and variations

Outperforms classical methods in tests

Abstract

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, it is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e. g., Bouligand-Minkowski, box-couting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique.