Lacunarity definition for ramified data sets based on optimal cover

Lacunarity is a measure of how data fills space. It complements fractal dimension, which measures how much space is filled. This paper discusses the limitations of the standard gliding box algorithm for calculating lacunarity, which leads to a re-examination of what lacunarity is meant to describe. Two new lacunarity measures for ramified data sets are then presented that more directly measure the gaps in a ramified data set. These measures are rigorously defined. An algorithm for estimating the new lacunarity measure, using Fuzzy-C means clustering algorithm, is developed. The lacunarity estimation algorithm is used to analyze two- and three-dimensional Cantor dusts. Applications for these measures include biological modeling and target detection within ramified data sets.