Spline subdivision schemes for convex compact sets
Subdivision schemes are recursive methods for the generation of smooth functions from discrete data. By these methods at each recursion step, new discrete values on a finer grid are computed by weighted sums of the already existing discrete values. In the limit of this recursive process, set-valued functions are generated, which can be expressed explicitly in terms of linear combinations of integer shifts of B-splines with the initial data as coefficients. The subdivision techniques are used to show that these limit set-valued spline functions have shape-preserving properties similar to those of the usual spline functions. The mathematical tools used for analyzing set-valued functions include the support function technique for describing convex compact sets and methods of embedding the cone of convex compact subsets of \(\mathbb{R}^n\) in a linear normed space, with an addition defined as the Minkowsky sum of sets. The extension of subdivision methods from the scalar setting to the set-valued case is applied in the approximate reconstruction of 3-D bodies from finite collections of their parallel cross-sections.
scientific article; zbMATH DE number 1491253
scientific article