| | | | | | 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. | | | | |