Reliability test of popular fractal techniques applied to small two-dimensional self-affine data sets

The fractal dimensions of five fractional Brownian motion (fBm) surfaces of 257×257 pixel size, with Hurst exponent H ranging from 0.1 to 0.9, were computed by profile, contour, and surface area analyses. A technique was deemed reliable if it demonstrated accuracy, consistency and sensitivity. Of all the techniques examined, surface area analysis methods, namely, two-dimensional pyramid and Peleg methods, were found to be most reliable and efficient for the data size studied. Hence, these were employed in a preliminary fractal analysis of poly(methyl methacrylate) and poly(styrene) fracture surfaces. The surfaces were imaged at scan sizes ranging from 1-6.5 μm by atomic force microscopy (AFM). The images indicated the existence of fractal structure and a high degree of roughness at microstructural scales for both the surfaces. These observations were supported by the results of the two surface area analysis techniques. A more conclusive study was prevented by severe scoring of the surfaces by the AFM tip at smaller scan sizes and the availability of only a narrow range of scan sizes.