Improved box-counting methods to directly estimate the fractal dimension of a rough surface

Rough surfaces exist widely in nature, and the measurement of these surfaces’ fractal dimension is beneficial for understanding their formation mechanism and effect. The basic principles of previous cubic covering methods are described, and then the stable cubic covering method (SCCM) and stable differential cubic covering method (SDCCM) are proposed based on the box-counting method. Afterward, a real sandstone fracture surface and Takagi surfaces are adopted to compare these methods’ differences. The results show that different measurement methods are needed to measure surfaces with different theoretical fractal dimensions. Meanwhile, the SCCM and SDCCM can obtain more stable measurement results than previous methods. Then, the recommended methods and their corresponding scale factors are summarized. Finally, the potential of these cubic covering methods to measure a thick film is described. The research is of great significance to the fractal dimension measurement of a rough surface or a thick film.