A nonparametric approach to the estimation of lengths and surface areas

Abstract

The Minkowski content $L_0(G)$ of a body $G\subset{\mathbb{R}}^d$ represents the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A method for estimating $L_0(G)$ is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing $G$) in which we are able to identify whether every point is inside or outside $G$. Some theoretical properties concerning strong consistency, $L_1$-error and convergence rates are obtained. A practical application to a problem of image analysis in cardiology is discussed in some detail. A brief simulation study is provided.