Abstract
Let $\mathcal{X}=\{X_1,\ldots X_n\}\subset \mathbb{R}^d$ be a random sample of observations drawn with a probability distribution supported on $S$ satisfying that both $S$ and $\overline{S^c}$ are $r_0$-convex ($r_0>0$). In this paper we propose an estimator of the medial axis of $S$ based on the $\lambda$-medial axis and the $r$-convex hull. Its convergence rate is derived. An heuristic to tune the parameters of the estimator is given and a small simulation study is performed.
Highlights
Let S ⊂ Rd be a compact set, its medial axis, introduced in [6] as the set of points in Rd that has at least two different projections on ∂S has been initially proposed as a tool for biological shape recognition
When dealing with compact sets we can only focus on M(S) the inner part of the medial axis
When S is compact the knowledge of S is equivalent to the knowledge of medial axis transform that is the medial axis and the function r(x) = d(x, ∂S) because we have: S=
Summary
Let S ⊂ Rd be a compact set, its medial axis, introduced in [6] as the set of points in Rd that has at least two different projections on ∂S (see Figure 1) has been initially proposed as a tool for biological shape recognition. The medial axis is difficult to estimate because it is not continuous with respect to the Hausdorff distance dh (recall that for A and B two sets dh(A, B) = max{supa∈A d(a, B), supb∈B d(b, A)}) This is detailed in [19] (see pages 217 − 238) and illustrated in Figure 2 part a)). Given a sample point Xn drawn on S (instead of “near ∂S”), it is proved in [11], under no more shape hypothesis than regularity, that given a support estimator Sn such that dh(Sn, S) → 0 a.s. and dh(∂Sn, ∂S) → 0 a.s. dh(Mλ(Sn), Mλ(S)) → 0 a.s. We are going to introduce a new medial axis estimator Mλ(Xn) that is morally very close to the one introduced in [8].
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