双相钢定向柱体视学模型的非参数推断
Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel
摘要：
不透明介质中的定向圆柱体用于表示钢中的某些微观结构物体。不透明介质平行于圆柱体对称轴线切片，并且切割平面包含圆柱体的可观察的矩形轮廓。建立了三维圆柱体的半径和高度平方的联合密度与可观测二维矩形的半宽和高度平方的联合密度之间的一一对应关系。我们提出了一个非参数估计程序来估计分布和期望的各种数量的利益，如圆柱体半径，高度，长宽比，表面积和体积从观察到的二维矩形的宽度和高度。此外，圆柱体的半径和高度之间的协方差估计。建立这些估计量的渐近行为以产生期望的逐点置信区间和感兴趣的量的分布的逐点置信集。这些量中的许多可以与材料的机械性能相关联，因此对工业有用。我们举例说明的数学模型和估计程序，使用带状显微组织，近90 \textmu米的深度已通过连续切片观察。
英文摘要：
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.
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