Accuracy and precision of fractal dimension measured on model surfaces

To develop a method, which is precise, accurate, and insensitive to the angle of inclination for determining the fractal dimensional increment (D*) of a surface.

Brownian interpolation was used to generate simulated ceramic fracture surfaces having known D* values of 0.1, 0.2, 0.3, and 0.4 with 10 surfaces at each D* value. Each surface was inclined at four angles (0°, 3°, 5°, and 7°) from horizontal. The 160 (40×4) surfaces were analyzed by a variety of methods including Minkowski Cover (MC), Root Mean Square Roughness vs. Area (RMS), Kolmogorov Box (KB), Hurst Exponent (HE), Slit Island Box (SIB), and Slit Island Richardson (SIR). The coefficient of variation (CV) and mean error were used to identify the methods with best precision (lowest CV) and accuracy (lowest mean error), respectively, and three-way ANOVA followed by Turkey's HSD (α=0.05) was used to identify significant effects.

CV was significantly affected by fractal dimension (p=0.002) and method (p<0.001) but not by angle of inclination (p=0.765). The CV value for MC was lower than those for other methods (p≤0.05). Mean error was significantly affected by three-way interaction between fractal dimension, method, and angle of inclination (p<0.001). The mean error for KB was higher than those for other methods (p≤0.05) for inclined surfaces.

MC was determined to have the best combination of precision, accuracy, and lack of sensitivity to angle of inclination for Brownian interpolation surfaces having D* values in the range commonly reported for ceramic fracture surfaces.