Nonlinear stereology: Estimation of morphological features of anisotropic ellipsoids with various spatial distributions

In materials science, stereology is a fundamental technique for estimating three-dimensional (3D) microstructural characteristics from two-dimensional (2D) sections. However, conventional geometrical approaches, such as the Schwartz–Saltykov method, rely on implicit assumptions of particle isotropy, random orientation, and negligible impingement, which limits their applicability to realistic microstructures containing anisotropic and clustered particles. In this study, we constructed a virtual model of 3D microstructure consisting of triaxial anisotropic ellipsoidal particles and systematically investigated the relationship between 2D observables and 3D morphological characteristics under different spatial distributions (uniform, random, and clustered). Based on extensive simulation data, we propose a nonlinear stereological framework that integrates multiple 2D morphological features using machine learning–based regression models. The results confirm that Delesse’s principle remains valid for volume fraction estimation regardless of particle anisotropy and spatial arrangement. In contrast, particle size and shape cannot be reliably inferred from single 2D descriptors. Nonlinear regression models, particularly Random Forest and XGBoost, notably outperformed the modified Schwartz–Saltykov method in estimating the 3D equivalent sphere diameter and principal axis lengths of anisotropic particles. Thus, the proposed nonlinear stereological strategy is a physically interpretable and computationally efficient framework for the quantitative characterization of 3D microstructure from limited 2D observations, offering a practical extension of classical stereology for complex material systems. • A virtual 3D microstructure model of anisotropic ellipsoidal particles was constructed • Classical stereology fails under particle anisotropy and spatial clustering • Multiple 2D morphological features were integrated using nonlinear regression models • Random Forest and XGBoost enable accurate 3D morphology estimation from 2D sections • The framework extends stereology beyond isotropic and non-overlapping assumptions