MULTIFRACTAL ANALYSES OF GRAYSCALE AND BINARY SOIL THIN SECTION IMAGES

Abstract

Multifractal analyses of binary images of soil thin sections (STS) are widely used to characterize pore structure. However, no geometrical model is known to exist for a binary multifractal. Thus, the multifractality of binary images, and the accuracy of multifractal parameters estimated from them, need to be carefully evaluated. We captured 8-bit depth resolution digital grayscale images of three STS images with dimensions of 1024 × 1024 pixels and a pixel length of 1.9 μm. Random grayscale geometrical multifractal fields (GMF) with similar dimensions and known multifractal parameters were constructed using generators extracted from the STS images. The STS and GMF grayscale images were objectively thresholded to give six binary images. The method of moments was used to compute the log-transformed partition function, log (χ(q, δ)) versus log(δ) where δ is box size, for each grayscale image and its binary counterpart. Consistent linearity was observed in the resulting functions for the grayscale images, indicating, by definition, multifractal behavior. In contrast, the log (χ(q, δ)) versus log(δ) plots for the binary images exhibited a two-region response, with a flat plateau at small scales and linearity at larger scales, indicating they were not true multifractals. Generalized dimensions (Dq) computed from the linear portions of the binary log-transformed partition functions were significantly over estimated for q ≪ 0 and underestimated for q ≫ 0 relative to corresponding Dq values for the grayscale images. Based on these results we contend that binary images are not mathematical multifractals, and that generalized dimensions estimated from them cannot be used to quantify pore space geometry. Instead we encourage further exploration of the use of grayscale images for multifractal characterization of soil structure. This direct approach is theoretically sound and does not require any intermediate thresholding step, which is known to influence the results of multifractal analyses.