On Estimating the Perimeter Using the Alpha-Shape

TL;DR

This paper investigates how well the alpha-shape perimeter estimator approximates the true perimeter of a smooth domain from uniform samples, supported by theoretical analysis and numerical experiments.

Contribution

It provides a theoretical analysis of the alpha-shape perimeter estimator's performance for smooth domains, complemented by numerical validation.

Findings

The alpha-shape estimator converges to the true perimeter under certain conditions.

Numerical experiments confirm the theoretical convergence rates.

The method is effective for estimating perimeters from uniform samples.

Abstract

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.