Estimation of fractal dimension and fractal curvatures from digital images

TL;DR

This paper introduces a novel method for estimating fractal dimension and curvatures from digital images by evaluating multiple geometric characteristics, providing a more detailed fractal analysis.

Contribution

It proposes a new approach using all intrinsic volumes of parallel sets to estimate fractal dimension and curvatures, enhancing fractal classification.

Findings

Estimators are consistent based on theoretical analysis.

Method successfully applied to digital images of self-similar sets.

Provides finer classification of fractals beyond just dimension.

Abstract

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e.\ volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.