On the estimation of the medial axis and inner parallel body

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 λ -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 λ -medial axis M λ ( C ) and the inner parallel body I λ ( C ) of C . The consistency of these estimators follows from two results of stability (i.e. continuity in the Hausdorff metric) of M λ ( C ) and I λ ( C ) obtained under some, not too restrictive, regularity assumptions on C . As a consequence, natural algorithms for the approximation of the λ -medial axis and the λ -inner parallel body can be derived. The whole approach could be useful for some practical problems in image analysis where estimation techniques are needed.