Each image is a 512x512 pixel bitmap where the background color is white and the foreground color is black.
There are 6 different grid sizes: S=128, 64, 32, 16, 8, and 4 where S is the number of pixels on each side of each box.
For each grid size S, NS is the number of boxes that contain at least one foreground pixel.
However, in most definitions of the box-counting dimension, NS is the minimum number of boxes of size S that can be used to completely cover the foreground. A grid starting at the upper corner is a naive arrangement of boxes. If the boxes were more cleverly placed, it might take a few less of them to actually cover the foreground.
The box dimension plot includes a point corresponding to each of the 8 box sizes. The horizontal axis is the base 10 logarithm of the inverse of the box size in pixels. The vertical axis is the base 10 logarithm of the box count. The least squares line of best fit is plotted. The slope of this best fit line is the box dimension of the image.