Fractal analysis of urban sprawl

Urban sprawl picture shown at the Biennale of Venecia. Picture from the Biennale web page
It is not easy to analyze urban morphology with a case of Urban sprawl like this. I propose here some tools from image analysis softwares that are helpful when we do not have a grid or urban quadrille or any other recognizable form. The first step, and this is pretty much for all softwares is to convert the jpg image into a binary file.
A profile plot of the picture will show the division between urban sprawl and a standard urban pattern. The peaks will show the grey intensity. For the purposes of the exercise, I did not take the time to adjust the greys from the original image to accurately show the empty and fill areas. Anyway, a white area means empty, and we have the option to reverse it.
The depression is showing the freeway. On the left side of it, the peaks are pretty even, it means the urban sprawl is like a uniform shape in its complexity. On the right side, the buildings are not uniform.
I can also count the particles. It is important to notice that the softwares conjoin particles that are very close, so if  infinite particles are seen, it is not considered like this. Because when houses are touching each other, they will conform a conglomerate, we cannot count one by one any more. The red spelling are numbers with the account of particles.
The surface plot is a digital analysis of rugosity in 3D. Notice the difference on the buildings side.
The exercise above is a quick way to compare two different morphologies in the same regional area, or in the same city. If we are interested exclusively in the urban sprawl, we focus strictly in this area of the picture.
I remove the freeway and tall buildings and look for edges to find a definition of the form. The process begins again. If edges are shown, particles, plots of profile and surface will be different. That is why it is so important to select adequate filters. Adding noise is a good option to fill the emptiness, to a certain degree.
Do I find a structure here? Well, it is a fractal urban pattern with fractal dimension D=1.9396, when D is close to 2, the plane (space) is occupied almost completely. This is true for urban sprawl.
Is there any hidden structure? An analysis based on Fourier transform will confirm it.
This Fourier transform of the urban sprawl image tells us that there are no buildings or streets of particular interest, the urban morphology is absolutely uniform, it must be read as a conglomerate. This last image is also a fractal, with D=1.9862. The similar result to the previous image shows that the analysis of fractality, even in the analogy of the FFT image is correct.
All images of analysis by Myriam B. Mahiques.

3 responses to “Fractal analysis of urban sprawl

  1. Hi Miriam,
    Would your analysis show greater difference in fractality between the sprawl and the designed high-rise neighborhood if you used an aerial view, rather than the bird’s eye view? Is 1.9 the boundary dimension (convoluted to almost 2) or the space-filling dimension of the white space (lacy, and therefore less than 2)?

    • Hello Jill, I´ve done many analysis of different urban morphologies and I can tell you that the results of D fractal dimension is higher in latino neighborhoods in CA, for example, or great concentrations like Buenos Aires. It triggers a value of 1.8 approx, and usually a little more, always below 1.9 because of the streets. But this is not the boundary, this is filled areas vs empty. For boundaries, when I apply the filters, the values usually drop to 1.7 or 1.6. Depending on the objective, sometimes I erase the trees, the cars, etc. If this is the case, I work first in AutoCad to obtain a clear image of occupied cells, so the result is more accurate. What you see in this post is just an exercise. You mention the high rise neighborhoods; a better analysis would be to cut the urban layout in layers, but to do this, you need to go to the place and measure; I mean , the result on the 1st level would be different for the 4th level and so on. Best regards, hope my comment could help.

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