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Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution

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  • Alison K. Cheeseman

    (Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada)

  • Edward R. Vrscay

    (Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada)

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.

Suggested Citation

  • Alison K. Cheeseman & Edward R. Vrscay, 2022. "Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution," Mathematics, MDPI, vol. 10(5), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:839-:d:765515
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    References listed on IDEAS

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    1. Tél, Tamás & Fülöp, Ágnes & Vicsek, Tamás, 1989. "Determination of fractal dimensions for geometrical multifractals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 159(2), pages 155-166.
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    Cited by:

    1. Francisco Martínez & Bastian Sepúlveda & Hermann Manríquez, 2023. "Fractal Organization of Chilean Cities: Observations from a Developing Country," Land, MDPI, vol. 12(2), pages 1-21, January.
    2. Michael Lim & Alit Kartiwa & Herlina Napitupulu, 2023. "Estimation of Citarum Watershed Boundary’s Length Based on Fractal’s Power Law by the Modified Box-Counting Dimension Algorithm," Mathematics, MDPI, vol. 11(2), pages 1-12, January.
    3. Francisco Martinez & Hermann Manriquez & Alberto Ojeda & Gabriel Olea, 2022. "Organization Patterns of Complex River Networks in Chile: A Fractal Morphology," Mathematics, MDPI, vol. 10(11), pages 1-23, May.

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