Box-Counting Dimension of Fractal Urban Form: Stability Issues and Measurement Design

Box-Counting Dimension of Fractal Urban Form: Stability Issues and Measurement Design

Shiguo Jiang (Department of Geography, The Ohio State University, Columbus, OH, USA) and Desheng Liu (Department of Geography and Department of Statistics, The Ohio State University, Columbus, OH, USA)
Copyright: © 2012 |Pages: 23
DOI: 10.4018/jalr.2012070104
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The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes; 3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these issues in the case of Beijing City. The authors propose corresponding improved techniques with modified measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then applied to real city data. The results show that a reliable estimate of box dimension for urban form can be obtained using their method.
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1. Introduction

Fractal dimension is a useful landscape metric in that it can capture the irregularity and complexity of landscape patterns (Hargis et al., 1998; Herold et al., 2005; Imre & Bogaert, 2004; Pincheira-Ulbrich et al., 2009). The usefulness of fractal in characterizing urban form is shown by various researchers (Batty, 1985; Batty & Longley, 1994; Benguigui et al., 2000; Frankhauser, 1994; Thomas et al., 2008; Thomas et al., 2007). Mostly inspired by the studies on coast line (Mandelbrot, 1967; Richardson, 1961), early studies of urban form mainly focus on city boundaries (Batty & Longley, 1987; Batty & Longley, 1988; Longley & Batty, 1989a, 1989b). Later fractal urban form studies extend to urban surface, i.e., urban land use (Batty & Longley, 1994; Benguigui et al., 2000; Shen, 2002; Thomas et al., 2008; White & Engelen, 1993). It seems to be widely accepted that urban boundary as well as urban surface are both fractals at least in certain stages (Batty & Longley, 1994; Batty & Xie, 1996; Benguigui et al., 2000).

Fractal analysis of urban form relies heavily on the calculation of fractal dimension – the main scaling exponent to describe a fractal set. The fractal dimension of a deterministic fractal (e.g., the Vicsek fractal) can usually be estimated analytically. However, the dimension of a stochastic fractal (e.g., urban form) needs to be estimated numerically. Three numerical methods are popular in the research community: the perimeter-area relation method (Batty & Longley, 1988; Batty & Longley, 1994), the area-radius method (Frankhauser, 1994; White & Engelen, 1993), and the box-counting method (Benguigui et al., 2000; Lu & Tang, 2004; Shen, 2002). Due to its simple algorithm and equal effectiveness to point sets, linear features, areas, and volumes, the box-counting method enjoys a wide popularity across various disciplines such as physics (Lovejoy et al., 1987), earth sciences (Walsh and Watterson 1993), biology (Foroutan-pour et al., 1999), ecology (Halley et al., 2004), and urban studies (Benguigui et al., 2000; Feng & Chen, 2010; Lu & Tang, 2004; Shen, 2002; Verbovsek, 2009). In the case of urban studies, box-counting dimension is an indicator of compactness for the distribution of built-up areas. Despite its popularity in the research community, several issues of the box-counting method remain unsolved.

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