Estimation of the Fractal Dimension of Weierstrass–mandelbrot Function Based on Cuckoo Search Methods

The Weierstrass–Mandelbrot randomized function (WMRF) has been widely applied to characterize rough surface profiles with a specified fractal dimension ([Formula: see text]). However, it is difficult to accurately determine the specified [Formula: see text] for the WMRF. In this study, we proposed a novel approach to determine the [Formula: see text]. First, the cuckoo search (CS) algorithm is adopted as an optimization algorithm to extract the trend line of a rough profile instead of a traditional measurement method that measures profiles with a fixed length or a designed lag distance. Then, the total vertical distance ([Formula: see text]) of all the points on a profile to its optimal trend line that is extracted by the CS measurement method is calculated. Finally, it is found that the absolute exponent value of the power law relationship between [Formula: see text] and the number of steps ([Formula: see text]) has a linear relationship to [Formula: see text], and thus, in turn, provides a method to estimate the value of fractal dimension for WMRF by providing more details about the original profile simultaneously. The CS measurement method furnishes a better method for extracting the trend polyline with variable-step lengths and for characterizing the roughness of a rough profile directly as well.