Extraction method to ensure the uniqueness of scale-free region in surface fractal PSD equation

Abstract

Scale-free region is a specific frequency region only in which the fractal characteristics for surface topography exists. The uniqueness in fractal characterization exists in theory; however, by incorporating the conventional methods, various scale-free regions can be obtained for the same surface profile, resulting in the non-uniqueness of fractal characterization and reconstruction. Therefore, this paper aims to solve the non-uniqueness problem of the scale-free region. Firstly, the origins of such non-uniqueness are revealed, including random components introduced by manual selection under single-scale and information distortion affected by measuring frequency under multi-scales. Then, a filtering method based on an equal PSD amplitude Weierstrass-Mandelbrot function is proposed to extract the length and particular fractal components of the scale-free region. The proposed method is verified by acquiring the unique scale-free region length and calculating the unique fractal dimension D with the extracted fractal components. Additionally, measurement influences are discussed, including measuring frequency and length. Further application on surfaces with different grinding processes and roughness levels is also conducted to test the practicability of the method. In summary, the necessary conditions for measuring the scale-free region are revealed. On this basis, obtaining method for the scale-free region is proposed.