Spline subdivision schemes for convex compact sets
type: Item
instance of: scholarly article
title: Spline subdivision schemes for convex compact sets
zbMATH Open document ID: 0970.65016
DOI: 10.1016/S0377-0427(00)00375-7
published in: Journal of Computational and Applied Mathematics
publication date: 2001-10-23T00:00:00Z
review text: 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.
reviewed by: Heiner Mühlig
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 1491253
zbMATH Keywords: set-valued functions
zbMATH Keywords: spline subdivision
zbMATH Keywords: shape preservation
zbMATH Keywords: B-splines
zbMATH Keywords: convex compact sets
zbMATH Keywords: Minkowsky sum of sets
zbMATH Keywords: reconstruction of 3-D bodies
author: Nira Dyn
author: Elza M. Farkhi
MaRDI profile type: Publication
cites work: Integrals of set-valued functions
cites work: Q4250848
cites work: A practical guide to splines
cites work: Stationary subdivision
cites work: Q4026009
cites work: Sur la fonction d'appui des ensembles convexes dans un espace localement convexe
cites work: Compact, convex sets inRnand a new banach lattice. I.- Theory.
cites work: An Embedding Theorem for Spaces of Convex Sets
cites work: Q5638112
cites work: Approximation of convex set-valued functions
full work available at URL: s0377-0427(00)00375-7
OpenAlex ID: W1964080408
description: scientific article; zbMATH DE number 1491253