Hausdorff Dimension by the box counting method
A quantitative analysis of perimeter roughness is carried out to illustrate the
degree of roughness of input images. Commonly known as the Hausdorff
Dimension (H.D.), the algorithm shown in figure 1.1 gives the aggregate
perimeter roughness as a fractal dimension. The fractal dimension describes the
complexity of an object; in the case of devices presented here, this algorithm
gives perimeter roughness which implies parasitic emission sites for extremely
rough perimeters [1]. On Hausdorff Dimension scale, a dimension of 1 equates to
a smooth line, while 2 implies fractal complexity like that of a Julia set, and
because the devices presented here are considered truncated fractals, the fractal
dimension calculated is bound by the above limits, i.e. 1 < H.D. < 2. The
algorithm to achieve this starts with an input electron micrograph image
uploaded within Matlab (figure 1.2), then the Canny algorithm [2] is employed to
find the edge within the image and superimposes a grid of N squares over the
edge, while counting the occupied squares that the edge passes through (top
right, N(s)). This is continued for an increasing number of squares and the
fractal dimension (H.D) is given by the gradient of the logarithm of the number
of squares log N, over the number of squares occupied by the edge log N(s), as
indicated by figure 1.1 (bottom) and equation (1):
