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

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 oneto-one relation between the joint density of the squared radius and height of the 3D cylinders and the joint density of the squared halfwidth 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 µm of depth have been observed via serial sectioning.

Smooth estimation of size distributions in an oriented cylinder model

“…4, choosing this constant optimally is often a delicate matter. Another notable fact is the value of the asymptotic MSE in relation to asymptotic distribution results of the empirical (non-smoothed) estimator N n and the isotonic inverse estimator studied by McGarrity et al (2014). Both estimators are asymptotically unbiased, and normal with variance τ q (t) and τ q (t)/2, respectively (both rescaled with rate √ n/ ln n).…”

“…In the process of representing microstructural features of interest like those mentioned in Sect. 1, a first (simple) model was proposed in McGarrity et al (2014). We describe this model briefly here.…”

“…The height of the rectangle is equal to the height of the cut cylinder, and the width of the rectangle is a fraction of the diameter of the cylinder. Taking into account that the probability of cutting a cylinder by the plane depends on the radius of the cylinder, the relationship between the joint density g of the observed rectangle pairs (Z , H ), the squared half-width Z and height H , and the joint density f of (X , H ) is derived in McGarrity et al (2014): Here, m…”

“…Smooth kernel estimate for the distribution function F of the squared radius (left, b n = 30) and volume (right, b n = 900). The step functions are the isotonic estimates studied inMcGarrity et al (2014). Data are those ofFig.…”

Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem

“…\]\end{document} For another example, the distribution function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$V$\end{document} , the volume of an ellipse, can be estimated by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\[ \hat{F}_V(v) = \int_{z=0}^1 \int_{w=0}^{h(v,z)} \hat{F}_{b,x^2}(\hbox{d}w,\hbox{d}z), \quad v > 0, \]\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$h(v,z) = \{3/(4\pi )\}^{1/3}(1 - z)^{1/6}v^{1/3}$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$v > 0$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$0\leq z\leq 1$\end{document} . McGarrity et al (2014) considered the estimation of univariate summary measures that are functions of radii and heights of cylinders. Their method is specially designed for the case where the height of the cylinders does not suffer from an indirect measurement problem, and cannot be directly extended to estimate the axial ratio distribution or the volume distribution of ellipsoids where a bivariate measurement problem is present.…”