A nonparametric approach to the estimation of lengths and surface areas

TL;DR

This paper introduces a nonparametric estimator for boundary length and surface area of bodies in Euclidean space, with theoretical guarantees and practical applications in medical image analysis.

Contribution

It proposes a novel nonparametric method for estimating Minkowski content using random sample data, with proven consistency and convergence properties.

Findings

Estimator is strongly consistent.

Convergence rates are established.

Application demonstrated in cardiology image analysis.

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.