A new two-dimensional extension of the generalized Higuchi estimator for multifractal data: Theory and application to geoscience problems

In 1988, Tomoyuki Higuchi introduced an algorithmic approximation of the box-counting dimension of the graph of a real-valued univariate function or time series (Higuchi, 1988). This Higuchi fractal dimension has since become very popular as a simple fractal dimension estimator allowing the characterization of the scaling behavior of univariate time series. Besides numerous applications across various fields of science, several extensions of the classical framework have been developed during the past years, including a recent generalization to numerically estimating multifractal spectra from time series (Carrizales-Velazquez et al., 2022).

Here, we propose a novel extension of this multifractal Higuchi dimension analysis (MF-HDA) from one-dimensional time-series to two-dimensional image objects. We start by analyzing the properties of a recent two-dimensional generalization of the classical monofractal Higuchi method (Spasic, 2014), revealing some potentially misinterpreted geometric aspects of that original work. A minor modification is proposed to replace the concept of area by a new quantity that has a straightforward connection with the one-dimensional version of the Higuchi fractal dimension and thus provides the basis for a scaling analysis. Subsequently, we present a general mathematical framework for one- and two-dimensional Higuchi fractal dimension estimates and their generalizations to multifractal spectra, following the ideas underlying our previous one-dimensional MF-HDA.

To demonstrate the appropriate behavior of our new two-dimensional MF-HDA, we numerically estimate the multifractal spectra of different paradigmatic examples of mono- as well as multifractal two-dimensional model systems. For the special case of the two-dimensional binomial multifractal cascade model, we show that the results obtained by our new approach are largely consistent with the analytical multifractal spectra. Moreover, we find that our new approach does not exhibit an artificial widening of the multifractal spectra that is observed when applying a two-dimensional multifractal detrended fluctuation analysis as a numerical benchmark algorithm. Finally, we present some selected examples of applications of our approach to different two-dimensional geoscientific and environmental datasets like satellite images, illustrating the potential of systematic applications of our new two-dimensional MF-HDA method.

C. Carrizales-Vazquez, R.V. Donner & L. Guzman-Vargas, Generalization of Higuchi’s fractal dimension for multifractal analysis of time series with limited length, Nonlinear Dynamics, 108, 417-431, 2022.
T. Higuchi, Approach to an irregular time series on the basis of the fractal theory, Physica D, 31, 277-283, 1988.
S. Spasic, On 2D generalization of Higuchi’s fractal dimension, Chaos, Solitons & Fractals, 69, 179-187, 2014.