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On the estimation of the medial axis and inner parallel body. (English) Zbl 1288.62107
Summary: The medial axis and the inner parallel body of a set \(C\) are different formal translations for the notions of the “central core” and the “bulk”, respectively, of \(C\). On the basis of their applications in image analysis, both notions (and especially the first one) have been extensively studied in the literature, from different points of view. A modified version of the medial axis, called \(\lambda\)-medial axis, has been recently proposed in order to get better stability properties. The estimation of these relevant subsets from a random sample of points is a partially open problem which has been considered only very recently.
Our aim is to show that standard, relatively simple, techniques of set estimation can provide natural, consistent, easy-to-implement estimators for both the \(\lambda\)-medial axis \(\mathcal{M}_\lambda(C)\) and the inner parallel body \(I_\lambda(C)\) of \(C\). The consistency of these estimators follows from two results of stability (i.e., continuity in the Hausdorff metric) of \(\mathcal{M}_\lambda(C)\) and \(I_\lambda(C)\) obtained under some, not too restrictive, regularity assumptions on \(C\). As a consequence, natural algorithms for the approximation of the \(\lambda\)-medial axis and the \(\lambda\)-inner parallel body can be derived. The whole approach could be useful for some practical problems in image analysis where estimation techniques are needed.

MSC:
62H35 Image analysis in multivariate analysis
62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
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