Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular Boundaries

TL;DR

This paper develops new practical formulae for estimating the fractal dimension of irregular boundaries, improving accuracy and applicability in spatial analysis of complex geographical features with limited data.

Contribution

It introduces two sets of fractal dimension estimation formulae based on geometric measure relations from regular figures, addressing overestimation issues of previous methods.

Findings

Different formulae have distinct merits and application scopes.

The second set relates boundary dimension to shape indexes.

The formulae enable rapid estimation under data scarcity.

Abstract

Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of boundaries systematically when image data is limited. An approximation estimation formulae of boundary dimension based on square is widely applied in urban and ecological studies. However, the boundary dimension is sometimes overestimated. This paper is devoted to developing a series of practicable formulae for boundary dimension estimation using ideas from fractals. A number of regular figures are employed as reference shapes, from which the corresponding geometric measure relations are constructed; from these measure relations, two sets of fractal dimension estimation formulae are derived for describing fractal-like boundaries. Correspondingly, a group of shape indexes can be defined. A finding is that different formulae have different merits and spheres of application, and the second set of boundary dimensions is a function of the shape indexes. Under condition of data shortage, these formulae can be utilized to estimate boundary dimension values rapidly. Moreover, the relationships between boundary dimension and shape indexes are instructive to understand the association and differences between characteristic scales and scaling. The formulae may be useful for the pre-fractal studies in geography, geomorphology, ecology, landscape science, and especially, urban science.