Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel

TL;DR

This paper develops a nonparametric method to infer 3D microstructural features of steel from 2D observations, enabling better understanding of material properties relevant to industry applications.

Contribution

It introduces a novel nonparametric estimation approach linking 2D observed profiles to 3D cylinder distributions in a stereological model.

Findings

Estimation procedures for cylinder radius, height, and aspect ratio are proposed.

Confidence intervals and sets for the estimated quantities are established.

Application to real microstructure data demonstrates practical utility.

Abstract

Oriented circular cylinders in an opaque medium are used to represent certain microstructural objects in steel. The opaque medium is sliced parallel to the cylinder axes of symmetry and the cut-plane contains the observable rectangular profiles of the cylinders. A one-to-one relation between the joint density of the squared radius and height of the 3D cylinders and the joint density of the squared half-width and height of the observable 2D rectangles is established. We propose a nonparametric estimation procedure to estimate the distributions and expectations of various quantities of interest, such as the cylinder radius, height, aspect ratio, surface area and volume from the observed 2D rectangle widths and heights. Also, the covariance between the radius and height of a cylinder is estimated. The asymptotic behavior of these estimators is established to yield point-wise confidence intervals for the expectations and point-wise confidence sets for the distributions of the quantities of interest. Many of these quantities can be linked to the mechanical properties of the material, and are, therefore, useful for industry. We illustrate the mathematical model and estimation procedures using a banded microstructure for which nearly 90 \textmu m of depth have been observed via serial sectioning.