LACUNARITY ANALYSES OF MULTIFRACTAL AND NATURAL GRAYSCALE PATTERNS

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, D-2, by: d log(L)/d log(r) = D-2 - 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. D-2 values computed from the d log(L)/d log(r) analysis gave a similar to 1:1 relationship with the known D-2 values. Lacunarity analysis was also employed in discriminating between multifractal grayscale patterns with the same D-2 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, D-q (-infinity < q < +infinity) 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.