Fractals everywhere
Error analysis and convergence of capacity dimension algorithms
SIAM Journal on Applied Mathematics
Fractal cities: a geometry of form and function
Fractal cities: a geometry of form and function
Advances in the implementation of the box-counting method of fractal dimension estimation
Applied Mathematics and Computation
On calculation of fractal dimension of images
Pattern Recognition Letters
On the Calculation of Fractal Features from Images
IEEE Transactions on Pattern Analysis and Machine Intelligence
An improved box-counting method for image fractal dimension estimation
Pattern Recognition
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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.