基于三维计盒法的城市空间形态分维计算和分析

A three dimensional box-counting method for estimating fractal dimension of urban form

将城市形态分形维数的二维盒维数方法扩展到三维空间,提出了一种三维盒维数计算方法;并且针对以往城市分形研究中人工判定无尺度区不精确的问题,借鉴了二阶导数的方法进行分形无尺度区的自动识别;以扬州市中心城区2003年与2012年两期数据为例,详细介绍了城市三维空间形态分维计盒法的计算过程。实验结果显示,通过二阶导数方法能够有效的自动识别无尺度区;2003年与2012年数据无尺度区线性拟合确定系数R2均在0.996以上,扬州市三维空间形态具有明确的分形特征;2012年较2003年三维分维值的增加表明扬州市城市三维空间利用总体上趋于有效与紧凑。

Urban fractal dimension analysis is an important method for quantifying the measurement of urban morphology. With rapid urban development in China in recent two decades,the three-dimensional characteristic has been the essential feature of urban morphology. However the research of urban fractal dimension are mostly focused on the two-dimensional space,few studies have been conducted from a 3D perspective. In this paper, a three-dimensional boxcounting method is proposed to estimate fractal dimension of urban form. The scale-free region is usually determined by experience in the classical urban fractal research. In the ln r-ln N(r)curve, the researchers find out a line segment with good linear relationship over a range of length scales with their experience as the scaling range. This identification method is subjective, and cannot accurately identify and calculate the fractal dimension. Moreover, for the same set of data, different scale ranges could be attained by different observers. In this paper, a method based on the second-order derivative of log-log curves is used to automatically identify the fractal scale-free range. The central urban area of Yangzhou city is selected as the case area.Two high resolution remote sensing images of 2003 and 2012, which are respectively the Quick Bird data with 0.61 m resolution and the Geo Eye data with 0.5 m resolution, are used to acquire Yangzhou three-dimensional morphology data. Then based on the data, the detailed process of the three-dimensional box-counting method is described. The experimental results show that: the scaling range is the basis of calculating city fractals, the second-order derivative method can automatically identify fractal scale-free range accurately; The linear regression coefficients of scale-free range R2 are all over 0.996 in the two years, which demonstrates that Yangzhou three-dimensional morphology is fractal; The fractal dimension of urban morphology has increased significantly in 2012 compared with the year 2003, which indicates that the utilization of 3D spaces has become more efficient and compact; The three-dimensional box-counting algorithm proposed in this paper not only provides a new methodology in quantifying the measurement of urban spatial structure and the evolution of urban morphology, but also extends the scope of urban studies to higher dimensions.