Subdivision schemes and irregular grids
type: Item
instance of: scholarly article
title: Subdivision schemes and irregular grids
zbMATH Open document ID: 1062.41002
DOI: 10.1023/B:NUMA.0000016582.07613.38
author: Voichiţa Teodora Maxim
author: Marie-Laurence Mazure
published in: Numerical Algorithms
publication date: 2004-05-27T00:00:00Z
review text: In computer aided geometric design, subdivision schemes are a classical process to construct curves and surfaces. This paper deals only with curves. The principle is simple. A curve in \(\mathbb R^d\) is obtained as the limit of a sequence of polygonal lines in this space. At each level, each vertex of the polygonal line is calculated from a finite number of vertices of the polygonal line of the previous level (actually, as a linear combination in the linear case). The limit curve is expected to be at least continuous, and if possible, to satisfy additional smoothness properties. As an instance, starting from a bi-infinite sequence of initial vertices, suppose that the new vertices are located at one fourth and three fourths respectively on each segment of the initial polygonal line. Repeating this process describes the well-known Chaikin algorithm, and the limit curve it produces is the quadratic \(C^1\) polynomial spline curve associated with regularly spaced knots, the control polygon of which is the initial polygonal line. The latter particular example is naturally related to the regular grid of \(\mathbb R\). The analysis of the convergence of any subdivision schemes is generally done by reference to the regular grid too, that is by considering the convergence of the sequence of piecewise affine functions parametrizing the polygonal line of level \(j\), with knots at the points \(2^{-j}k\), \(k\in\mathbb Z\). However the limitation to regular grids is far too restrictive. The necessity of allowing non regular ones naturally arises, for instance, through the example of spline curves. Another motivation for the consideration of non regular grids comes from wavelets and multiresolution analysis to which subdivision schemes are closely related. For a long time wavelets were considered on regular grids. Yet, the necessity of extending them to the nonregular case appeared in recent years with the introduction of second generation wavelets. The present paper studies t…
Mathematics Subject Classification ID: 41-06
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 65Q05
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 2069206
zbMATH Keywords: computer aided geometric design
zbMATH Keywords: subdivision schemes
zbMATH Keywords: equivalent schemes
zbMATH Keywords: irregular grids
zbMATH Keywords: Lagrange interpolating subdivision schemes
zbMATH Keywords: convergence
zbMATH Keywords: differentiability of the limit functions
zbMATH Keywords: difference schemes
zbMATH Keywords: quasi-regular grids
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
full work available at URL: b:numa.0000016582.07613.38
OpenAlex ID: W2082377208
description: scientific article; zbMATH DE number 2069206