论文标题
表面积的估计
Estimation of surface area
论文作者
论文摘要
我们研究了足够光滑的设置$ s \ subset \ mathbb {r}^d $的边界$ \部分s $表面积的问题。我们提出了两个估计器。首先利用了Devroye - 支持估计器，并基于Crofton的公式，该公式大致说明，这是$（d-1）$ - 尺寸 - 尺寸足够光滑的表面积是随机选择线的平均值的平均数量。为此，我们提出了基于Devroye的支持估计量的此类线相交数量的估计器。第二个表面积估算器利用$ \ x $的$α$ -CONVEX船体，该壳体用$C_α（\ x）$表示。更确切地说，它是$c_α（\ x）$的$（d-1）尺寸表面积，如$ |c_α（\ x）| _ {d-1} $表示，事实证明，它会融合到$（D-1）$ - 尺寸表面积为$ \ partial s $。此外，可以使用Crofton的公式计算$ |C_α（\ x）| _ {D-1} $。 我们的结果取决于Devroye的$ S $和$ \ x $之间的Hausdorff距离 - 估计器，以及第二个估算器的$ \ partial s $和$ \ partial s $和$ \ partial s $和$ \ partial s $之间的距离。
We study the problem of estimating the surface area of the boundary $\partial S$ of a sufficiently smooth set $S\subset\mathbb{R}^d$ when the available information is only a finite subset $\X\subset S$. We propose two estimators. The first makes use of the Devroye--Wise support estimator and is based on Crofton's formula, which, roughly speaking, states that the $(d-1)$-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye--Wise support estimators. The second surface area estimator makes use of the $α$-convex hull of $\X$, which is denoted by $C_α(\X)$. More precisely, it is the $(d-1)$-dimensional surface area of $C_α(\X)$, as denoted by $|C_α(\X)|_{d-1}$, which is proven to converge to the $(d-1)$-dimensional surface area of $\partial S$. Moreover, $|C_α(\X)|_{d-1}$ can be computed using Crofton's formula. Our results depend on the Hausdorff distance between $S$ and $\X$ for the Devroye--Wise estimator, and the Hausdorff distance between $\partial S$ and $\partial C_α(\X)$ for the second estimator.
