Architecture and Mathematics from Antiquity to the Future by Kim Williams & Michael J. Ostwald

Author:Kim Williams & Michael J. Ostwald
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Fig. 69.2Third stage grid placed over the east elevation of the Tomek house showing box-counting. Image: authors

Fig. 69.3Log-log diagram of the comparison between the number of boxes counted in a grid and the size of the grid. Image: authors

The wider the lines in the source image, the more chance they have of being counted twice when grid sizes become very small, leading to artificially increased D values. To counter this situation, Archimage software pre-processes images using a line-detection algorithm (Sobel gradient technique) that produces images for analysis that are one pixel wide. Benoit overcomes this problem by allowing the analytical grid to be rotated or resized to minimize the impact of line weight at each scale of observation. In addition to the line width problem, the volume and distribution of white or empty space around the source image can also alter the result. To solve this, Foroutan-Pour et al. (1999) offer an algorithm to optimize the way in which an image is positioned against its background and suggestions on how to derive an ideal analytical grid. A further, related issue is that the proportions of the image being analysed also influence the result. If the original image being analysed is not pre-sized to produce a clear starting grid, then an additional step must be added to ensure that a divisible starting grid is determined. Benoit solves this problem by cropping the image size to achieve a whole-number starting grid.2 In contrast, Archimage enlarges the image by adding small amounts of empty space to the boundaries. While neither of these variations change the elevation in the source image, they produce subtle variations in the resultant D.

A final challenge for any application of the box-counting method is the problem of statistical divergence. The average slope of the log-log graph may be the approximate D value, but the points generating the line are not always consistent with it. The D value is only a reasonable approximation when most of the points in the chart correspond with the resultant average line. The question then becomes, how are divergent points handled? While there is no definitive answer to this question, divergent results tend to occur primarily at the extremes of the graph; with the largest and smallest grid sizes but not normally those in between. Bovill is aware of this problem and solves it by intuitively determining where the practical limits of scaling in an image can be found. Benoit similarly allows the operator to intuitively deactivate certain data points or use a range of algorithms to determine best-fit for the data. While both of these are possible solutions, neither of them are useful for producing a consistent analysis of almost 50 images.

For the present research, similar settings for starting grid proportion and size minimize the number of divergent results associated with the largest grid dimensions. However, for divergences associated with small scale grids different tactics are used. The parameters of Benoit may be set to limit the smallest scale grids uniformly, creating a consistent set of results.

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