1.4.3 More General Fractal Dimensions: The Capacity Dimension
The self-similarity dimension requires that each little object formed by dividing the line segments of the whole object into smaller pieces must be an exact copy of the whole object. Thus the self-similarity dimension can only be used to analyze objects that are geometrically self-similar. To determine the dimension of irregularly shaped objects requires a more general form of the fractal dimension. Two such forms are the capacity and the Hausdorff-Besicovitch dimension.
To evaluate the capacity of an object we cover it with "balls" of a certain radius r. We find the smallest number of balls N(r) needed to cover all the parts of an object. We then shrink the radius of the balls and again count the smallest number needed to cover the object. The capacity is the value of Log N(r) / Log (l/r) in the limit as the radius of the balls shrinks to 0.
The capacity is a generalization of the self-similarity dimension. The self-similarity dimension d = Log N / Log M, where N is the number of smaller copies of the whole object seen when each line segment is replaced by M pieces. The spatial resolution in the self-similarity dimension is proportional to 1/M. The spatial resolution in the capacity is proportional to the radius r of the balls used to cover the object. Thus r is proportional to 1/M. To arrive at the capacity from the self-similarity dimension, we first replace M by 1 /r. This leads us to d = Log N / Log (1/r), Second, instead of counting N, the number of smaller copies of the whole object, we count N(r) the number of balls needed to cover the object. This leads us to d = Log N(r) / Log(l/r). Last, we take the limit of Log N(r) / Log(l/r) as the radius of the balls shrinks to 0.