Lacunarity (L) is a scale (r)-dependent parameter that was developed for quantifying clustering in fractals and has subsequently been employed to characterize various natural patterns. For multifractals it can be shown analytically that L is related to the correlation dimension, D2, by: d log (L)/d log (r) = D2 - 2. We empirically tested this equation using two-dimensional multifractal grayscale patterns with known correlation dimensions. These patterns were analyzed for their lacunarity using the gliding-box algorithm. D2 values computed from the d log (L)/d log (r) analysis gave a ~1:1 relationship with the known D2 values. Lacunarity analysis was also employed in discriminating between multifractal grayscale patterns with the same D2 values, but different degrees of scale-dependent clustering. For this purpose, a new lacunarity parameter, 〈L〉, was formulated based on the weighted mean of the log-transformed lacunarity values at different scales. This approach was further used to evaluate scale-dependent clustering in soil thin section grayscale images that had previously been classified as multifractals based on standard method of moments box-counting. Our results indicate that lacunarity analysis may be a more sensitive indicator of multifractal behavior in natural grayscale patterns than the standard approach. Thus, multifractal behavior can be checked without having to compute the whole spectrum of non-integer dimensions, Dq(-∞ < q < +∞) that typically characterize a multifractal. The new 〈L〉 parameter should be useful to researchers who want to explore the correlative influence of clustering on flow and transport in grayscale representations of soil aggregates and heterogeneous aquifers.
LACUNARITY ANALYSES OF MULTIFRACTAL AND NATURAL GRAYSCALE PATTERNS
