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Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution
Version 1
: Received: 4 February 2022 / Approved: 8 February 2022 / Online: 8 February 2022 (13:59:05 CET)
A peer-reviewed article of this Preprint also exists.
Cheeseman, A.K.; Vrscay, E.R. Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution. Mathematics 2022, 10, 839. Cheeseman, A.K.; Vrscay, E.R. Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution. Mathematics 2022, 10, 839.
Abstract
Branching patterns are ubiquitous in nature, consequently over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties, however their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log-log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.
Keywords
Fractal dimension; self-similarity; box-counting method; sandbox method; fractal trees; canopies; vascular networks; branching structures
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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