We consider the problem of estimating the perimeter of a smooth domain in the plane based on a sample from the uniform distribution over the domain. We study the performance of the estimator defined as the perimeter of the alpha-shape of the sample. Some numerical experiments corroborate our theoretical findings.
The problem of estimating the Minkowski content L 0 (G) of a body G C R d is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L 0 (G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator. The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, [Ann. Statist. 35 (2007) 1031―1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3-dimensional objects) via the consideration of shots at a box containing the body. The statistics considered by these authors, as well as those in subsequent papers, are based on the estimation of Minkowski content and depend on a smoothing parameter which must be carefully chosen. For the same sampling scheme, we introduce a new approach which bypasses this issue, providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions, provided one can collect the function values at the sample points. Examples arise in experiments in which the density of the body can be measured by physical properties of the impacts, or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors. Our method is based on random Delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body. We obtain basic asymptotics of the estimator, perform simulations and discuss, via Google Earth's data, an application to the image analysis of the Aral Sea coast and its cliffs. The Minkowski content L 0 (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 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. The problem of estimating the surface area, L 0, of a set G⊂ℝ d has been extensively considered in several fields of research. For example, stereology focuses on the estimation of L 0 without needing to reconstruct the set G. From a more geometrical point of view, set estimation theory is interested in estimating the shape of the set. Thus, surface area estimation can be seen as a further step where the emphasis is placed on an important geometric characteristic of G. The Minkowski content is an attractive way to define L 0 that has been previously used in the literature on surface area estimation. Pateiro-Lopez and Rodriguez-Casal [B. Pateiro-Lopez and A. Rodriguez-Casal, Length and surface area estimation under smoothness restrictions, Adv. Appl. Prob. 40(2) (2008), pp. 348–358] proposed an estimator, L n , for L 0 under convexity type assumptions. In this paper, we obtain the L 1-convergence rate of L n .