A simple method for estimating the fractal dimension from digital images: The compression dimension

TL;DR

This paper introduces a straightforward method called the compression dimension for estimating the fractal dimension directly from digital images, utilizing accessible software and suitable for educational purposes.

Contribution

It presents a novel, simple approach for estimating fractal dimensions from digital images, with validated accuracy comparable to established methods.

Findings

The method works well with limited-quality images.

It achieves accuracy comparable to traditional estimation techniques.

The approach is easy to implement with free software.

Abstract

The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.