The Fourier transform and spectral analysis are employed to estimate the fractal dimension and explore the fractal parameter relations of urban growth and form using mathematical experiments and empirical analyses. Based on the models of urban density, two kinds of fractal dimensions of urban form can be evaluated with the scaling relations between the wave number and the spectral density. One is the radial dimension of self-similar distribution indicating the macro-urban patterns, and the other, the profile dimension of self-affine tracks indicating the micro-urban evolution. If a city's growth follows the power law, the summation of the two dimension values may be a constant under certain condition. The estimated results of the radial dimension suggest a new fractal dimension, which can be termed “image dimension”. A dual-structure model named particle-ripple model (PRM) is proposed to explain the connections and differences between the macro and micro levels of urban form. 1. Introduction Measurement is the basic link between mathematics and empirical research in any factual science [1]. However, for urban studies, the conventional measures based on Euclidean geometry, such as length, area, and density, are sometimes of no effect due to the scale-free property of urban form and growth. Fortunately, fractal geometry provides us with effective measurements based on fractal dimensions for spatial analysis. Since the concepts of fractals were introduced into urban studies by pioneers, such as Arlinghaus [2], Batty and Longley [3], Benguigui and Daoud [4], Frankhauser and Sadler [5], Goodchild and Mark [6], and Fotheringham et al. [7], many of our theories of urban geography have been reinterpreted using ideas from scaling invariance. Batty and Longley [8] and Frankhauser [9] once summarized the models and theories of fractal cities systematically. From then on, research on fractal cities has progressed in various aspects, including urban forms, structures, transportation, and dynamics of urban evolution (e.g., [10–20]). Because of the development of the cellular automata (CA) theory, fractal geometry and computer-simulated experiment of cities became two principal approaches to researching complex urban systems (e.g., [21–25]). Despite all the above-mentioned achievements, however, we often run into some difficult problems in urban analysis. The theory on the fractal dimensions of urban space is less developed. We have varied fractal parameters on cities, but we seldom relate them with each other to form a systematic framework. Moreover, the estimation
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