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		1.4.6— Example of Determining the Fractal Dimension: Using the Capacity Dimension and Box Counting

		The capacity dimension d = Log N(r) / Log (l/r), in the limit where r approaches 0, where N(r) is the smallest number of balls of radius r needed to cover an object.

		A useful way to evaluate the capacity is to use ''balls" that are the boxes of a rectangular coordinate grid. This method is called box counting.

		For example, we cover an object with a grid and count how many boxes of the grid contain at least some part of the object. We then repeat this measurement a number of times, each time using boxes with sides that are 1/2 the size of the previous boxes.

		The capacity dimension is then the slope of the plot of Log N(r) versus Log (1/r), or equivalently, the negative of the slope of the plot of Log N(r) versus Log (r).

		If an object is self-similar, then the slope of Log N(r) versus Log (l/r) is the same as the limit of Log N(r) / Log (l/r) as r approaches 0. It is much easier to determine the slope than the limit.

		New algorithms make it possible to determine efficiently the number of boxes that contain at least some part of the object. Using these new algorithms, box counting is a particularly good method to evaluate the fractal dimension of images in photographs.

 

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