A nonparametric approach to the estimation of lengths and surface areas

Abstract

The Minkowski content L₀(G) of a body G⊂ℝ^d represents the boundary length (for d=2) or the surface area (for d=3) of G. A method for estimating L₀(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, L₁-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.