Robust estimation of multifractal dimensions of hierarchical structures in heterogeneous materials

The article addresses methodological issues in applying multifractal analysis to images of material microstructures, including those with different degrees of hierarchy. A method is proposed for the automatic localization of the scale invariance range (the range of fractality) based on a sliding analysis of the linearity of the graph showing the dependence of the logarithm of the generalized statistical sum on the logarithm of box size. This method relies on the stability of the linear approximation slope and the value of the coefficient of determination . The behavior of the generalized statistical sum at different values of the exponent q is examined, allowing the identification of nested scale levels within the structure. The paper presents results from modeling complex nested structures and their multifractal analysis. The obtained data demonstrate the importance of a comprehensive approach to interpreting multifractal spectra in order to improve the accuracy of quantitative characterization of inhomogeneous microstructures. • The use of normalized plots of versus , highlighting characteristic plateau regions at both small and large scales, greatly facilitates the identification of scale-invariant intervals. • The method for localizing linear regions in plots based on the Student’s t -statistic for the slope of least-squares regression is proposed. • Structures exhibiting hierarchical (nested) organization are characterized by multiple fractality ranges, manifesting as a sequence of linear regions with distinct slopes.