Open Access
2019 Convergence rate for the $\lambda $-Medial-Axis estimation under regularity conditions
Catherine Aaron
Electron. J. Statist. 13(2): 2686-2716 (2019). DOI: 10.1214/19-EJS1581


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.


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Catherine Aaron. "Convergence rate for the $\lambda $-Medial-Axis estimation under regularity conditions." Electron. J. Statist. 13 (2) 2686 - 2716, 2019.


Received: 1 June 2018; Published: 2019
First available in Project Euclid: 21 August 2019

zbMATH: 07104728
MathSciNet: MR3995008
Digital Object Identifier: 10.1214/19-EJS1581

Primary: 62G05 , 62H35
Secondary: 62-07 , 62-09 , 62H05

Keywords: $r_{0}$-convexity , Geometric inference , Medial-Axis , skeleton

Vol.13 • No. 2 • 2019
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