A method for estimating the dimension of asymptotic fractal sets

Abstract

The development of fractal geometry has prompted the use of fractal dimensions of objects as measures of morphological complexity. Many biological specimens show fractal scaling, but within limited scale ranges. Beyond those limits, the specimens are Euclidean. Such objects are called asymptotic fractals and alternative models have been proposed to describe them. These approaches rely on fitting data gathered with length-resolution techniques (for example the ‘yardstick’ method) to theoretical models of asymptotic fractal scaling. Unfortunately, data produced with the so-called ‘box counting’ method cannot be used with these models. We report a new approach to estimate the asymptotic fractal behaviour (at low resolution) of asymptotic fractals based on the single assumption that the specimen approaches a Euclidean object at high resolutions. The procedure described can be applied using length-resolution methods as well as the box counting method and opens the possibility for estimating asymptotic fractal behaviour in both cluster and branching structures.