Recovering the shape of a point cloud in the plane
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In this work we deal with the problem of estimating the support S of a probability distribution under shape restrictions. The shape restriction we deal with is an extension of the notion of convexity named α-convexity. Instead of assuming, as in the convex case, the existence of a separating hyperplane for each exterior point of S, we assume the existence of a separating open ball with radius α. Given an α-convex set S, the α-convex hull of independent random points in S is the natural estimator of the set. If α is unknown the r n -convex hull of the sample can be considered being r n a sequence of positive numbers. We analyze the asymptotic properties of the r n -convex hull estimator in the bidimensional case and obtain the convergence rate for the expected distance in measure between the set and the estimator. The geometrical complexity of the estimator and its dependence on r n are also obtained via the analysis of the expected number of vertices of the r n -convex hull.
KeywordsConvex set α-convex set Set estimation Distance in measure Image analysis
Mathematics Subject Classification (2000)60D05 62G20
The second author thanks Prof. Luc Devroye for his valuable help. This work has been partially supported by the Grant MTM2008-03010 from the Spanish Ministerio de Ciencia e Innovación.
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