Fractal Modeling and Fractal Dimension Description of Urban Morphology

Abstract

The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.

Fractal Characteristic Analysis of Urban Land-Cover Spatial Patterns with Spatiotemporal Remote Sensing Images in Shenzhen City (1988–2015)

“…The urban land-cover fractal study is based on the following theoretical assumptions [43]: To measure urban land cover by certain r scale, there is a measure M in the corresponding scale, when the scale r is changed, the measure M(r) is also changed, which can be described as Equation ( 2)…”

Between 2D and 3D: Studying Structural Complexity of Urban Fabric Using Voxels and LiDAR-Derived DSMs

“…The box-counting method was applied on various data forms extracted from urban environments including points, lines and 2D images or 3D models by tracing or converting data from one form to another. A recent review conducted by Chen [18] reaffirmed the relevance of using fractal geometry to analyse urban landscapes and indicated that a 2D analysis was more appropriate for urban analysis as it was simpler and more ef-fective. However, exponential technological advancements now allow the production of increasingly realistic digital surrogates [61].…”

“…Qin et al [17] developed a three-dimensional box-counting method to estimate the fractal dimension of urban form change using remote sensing images. Chen [18] explained cities as pre-fractals rather than real fractals as they have a limited hierarchy, indicating a fractal-like geometric form.…”

“…This transformation allows finding a characteristic dimension to describe the complexity of an urban settings that can be studied through the lens of a fractal dimension. Fractal geometry is a powerful tool for scaling the analysis of urban forms through DSMs [18]. The box-counting method reconceptualised in 3D voxels provides new possibilities for estimating fractal dimensions by considering the three-dimensionality of the urban fabric using DSM.…”

Problems of the Grid Size Selection in Differential Box-Counting (DBC) Methods and an Improvement Strategy

“…The irregularity of a fractal set can be computed using the fractal dimension. The fractal dimension is now widely utilized in computer vision applications such as texture analysis [ 2 ], image segmentation [ 3 ], shape recognition [ 4 ], pattern recognition [ 5 ], time series analysis [ 6 ], complex network analysis [ 7 ], landslide susceptibility assessment [ 8 ], topography analysis of thin films [ 9 ], cement-based materials analysis [ 10 ], and description of urban morphology [ 11 ].…”