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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)
##### Software:
2D Apollonius Graphs; alphahull; CGAL; R
Full Text:
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