Convergence rate for the $\lambda $-Medial-Axis estimation under regularity conditions

Abstract

Let $\mathcal{X}_{n}=\{X_{1},\ldots X_{n}\}\subset \mathbb{R}^{d}$ be a iid random sample of observations drawn with a probability distribution supported by $S$ a compact set satisfying that both $S$ and $\overline{S^{c}}$ are $r_{0}$-convex ($r_{0}>0$). In this paper we study some properties of an estimator of the inner medial axis of $S$ based on the $\lambda $-medial axis. The proposed estimator depends on the choices of $\mathcal{Y}\subset \mathcal{X}_{n}$ an estimator of $\partial S$ and $\hat{S}_{n}$ an estimator of $S$. In a first general theorem we prove that our medial axis estimator converges to the medial axis with a rate $O(\max _{y\in \mathcal{Y}}d(y,\partial S),(\max _{y\in \partial S}d(y,\mathcal{Y})^{2})$. A corollary being that the choice of $\mathcal{Y}$ as the intersection of the sample and its $r$-convex hull, $\mathcal{Y}=C_{r}(\mathcal{X}_{n})\cap \mathcal{X}_{n}$, allows to estimate the medial axis with a convergence rate $O((\ln n/n)^{2/(d+1)})$. In a practical point of view, computational aspects are discussed, algorithms are given and a way to tune the parameters is proposed. A small simulation study is performed to illustrate the results.