Is a two-dimensional generalization of the Higuchi algorithm really necessary?

Abstract

Fractal dimensions of data series, particularly time series can be estimated very well by using Higuchi's algorithm. Without phase space constructions, the fractal dimension of a one-dimensional data stream is calculated. Higuchi's method is well accepted and widely applied, because it is very reliable and easy to implement. A generalization of the genuine 1D algorithm to two dimensions would be desirable in order to investigate digital images. In this study, we propose several 2D generalization algorithms and evaluate differences between them. Additionally, a comparison to previously published pseudo 2D generalizations, and to the Fourier and the Blanket method are presented. The algorithms were tested on artificially generated grey value and red-green-blue colour images. It turned out that the proposed 2D generalized Higuchi algorithms are very robust, but differences in between the generalizations as well as differences to the pseudo 2D algorithms are astonishingly small.