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Top1. 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.