Dynamic pyramidal volume correction method for calculating the three-dimensional fractal dimension of machined surfaces

To address the limitations of existing methods for calculating the fractal dimension of 3D surfaces in terms of accuracy and adaptability, this paper proposes a novel approach based on the Dynamic Pyramidal Volume Correction Method (DPVC). The method establishes a coupled volume correction model by introducing a volume correction coefficient that accounts for both the positional offset of asperities and their overlap with the substrate, enabling quantitative characterization of surface complexity. Using the corrected 3D surface volume as the measure and the grid unit length as the scale, a power-law relationship is constructed. The fractal dimension is then directly obtained from the slope of the linear fitting of the log–log curve within the scaling interval. To validate the proposed method, both isotropic and anisotropic fractal surfaces were generated using the Weierstrass–Mandelbrot (W–M) function and analyzed using DPVC. The results were further compared with those obtained from the Differential Box-Counting (DBC), Variational Method (VM), and Triangular Prism Surface Area (TPSA) methods. Additionally, the four-weight product cascade model was introduced for further validation, confirming the applicability of DPVC in multifractal spectrum computation. Furthermore, real machined surfaces are measured using white light interferometry, and DPVC is applied to the acquired data, demonstrating its effectiveness in practical surface profile analysis. The results show that DPVC achieves the highest computational accuracy and superior adaptability among the evaluated methods, effectively capturing the fractal characteristics of 3D surfaces.