Estimation of fractal dimension through morphological decomposition

Abstract

Set theory based morphological transformations have been employed to decompose a binary fractal by means of discrete structuring elements such as square, rhombus and octagon. This decomposition provides an alternative ap- proach to estimate fractal dimensions. The fractal dimensions estimated through this morphological decomposition procedure by employing different structuring elements are considerably similar. A color-coding scheme is adapted to identify the several sizes of decomposed non-overlapping disks (DNDs) that could be fit into a fractal. This exercise facilitates to test the number–radius relationship from which the fractal dimension has been estimated for a Koch Quadric, which yield the significantly similar values of 1.67 ± 0.05 by three structuring elements. In addition to this dimension, by considering the number of DNDs of various orders (radii) and the mean diameter of disks (MDDs) of corresponding order, two topological quantities namely number ratio (RB ) and mean diameter ratio (RL ) are computed, employing which another type of fractal dimension is estimated as log RB log RL . These results are in accord with the fractal dimensions computed through number–radius relationship, and connectivity network of the Koch Quadric that is reported elsewhere.