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

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