Fractal dimension is useful for characterizing objects with tortuous boundaries, providing an additional measurement to characterize shape beyond the commonly used area, perimeter, caliper diameter and their combinations. The fractal dimension of the boundary describes the increase of the measured value of the perimeter of an object resulting from increasing measurement resolution, i.e., decreasing step size. We desire to characterize and classify shapes such as soot particles, as shown in the TEM micrograph in Fig. 1a, and crystalline grains of metal alloys, as shown in the SEM micrograph of Raney nickel in Fig. 4.

This investigation involves evaluation of the Minkowski-sausage or dilation technique (Fig. 1) using the histogram of the Euclidean distance map of the perimeter of the object. The distance map gives bands of various widths that follow the perimeter - greater widths lose detail. Perimeter length is sausage (band) area divided by sausage width, w.