Fractal Curvature Measures and Image Analysis

TL;DR

This paper explores fractal curvature measures derived from Minkowski functionals of self-similar sets, proposing estimators for fractal dimension and curvatures, and tests their effectiveness on binary images.

Contribution

It introduces new estimators for fractal dimensions and curvatures based on Minkowski functionals, and evaluates their performance on image data.

Findings

Estimators accurately determine fractal dimensions from images.

Fractal curvature estimators show promising results in image analysis.

Method enhances quantitative analysis of self-similar structures.

Abstract

Since the recent dissertation by Steffen Winter, for certain self-similar sets $F$ the growth behaviour of the Minkowski functionals of the parallel sets $F_\varepsilon := \{x\in \mathbb R^d : d(x,F)\leq \varepsilon\}$ as $\varepsilon \downarrow 0$ is known, leading to the notion of fractal curvatures $C_k^f(F)$, $k\in \{0,\ldots,d\}$. The dependence of the growth behaviour on the fractal dimension $s = \dim F$ is exploited, and estimators for $s$ and $C_k^f(F)$ are derived. The performance of these estimators is tested on binary images of self-similar sets.