Binary univariate dual and primal subdivision schemes
type: Item
instance of: scholarly article
title: Binary univariate dual and primal subdivision schemes
zbMATH Open document ID: 1308.65032
author: Yong-Cai Geng
author: Sumit K. Garg
published in: S\(\vec{\text{e}}\)MA Journal
publication date: 2015-01-21T00:00:00Z
review text: Subdivision is a process that for a given coarse polygon (control polygon) recursively refines it using some predefined rules. This is a very popular method of generating curves and is broadly utilized in computer graphics. In the paper, the authors introduce a family of subdivision schemes \(H_l\) for \(l = 0, 1, 2, \dots\), generated using two well-known subdivision schemes. The first scheme is a four point binary interpolating scheme introduced in [\textit{N. Dyn} et al., Comput. Aided Geom. Des. 4, 257--268 (1987; Zbl 0638.65009)], and the second scheme is a relaxed version of the first one and was given in [\textit{K. Hormann} and \textit{M. A. Sabin}, ibid. 25, No. 1, 41--52 (2008; Zbl 1172.65308)]. After the introduction of the family \(H_l\), the authors show that for \(l = 0, 2, 4, \ldots\) the elements of the family are primal subdivision schemes and for \(l = 1, 3, 5, \ldots\) they are dual subdivision schemes. Next, the authors make different analyses of the family. They start with finding the lower and upper bound on the Hölder continuity. Using the bounds the authors show that elements of the proposed family have higher Hölder continuity as compared to the parent schemes. Then, they prove that the reproducing degree of \(H_l\) for \(l = 0, 1, 2, \ldots\) is cubic and show the limit stencil for \(l = 0, 1, 2\). Finally, the authors analyse the magnitude of artifact. They derive the magnitude of artifact for the schemes \(H_0\), \(H_1\) and \(H_2\) and next compare them with each other and with the magnitude of artifact of the first parent scheme.
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 6388847
zbMATH Keywords: subdivision curve
zbMATH Keywords: primal scheme
zbMATH Keywords: dual scheme
zbMATH Keywords: subdivision scheme
zbMATH Keywords: Hölder continuity
reviewed by: Krzysztof Gdawiec
MaRDI profile type: Publication
full work available at URL: s40324-014-0017-6
OpenAlex ID: W1997741487
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DOI: 10.1007/S40324-014-0017-6
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description: scientific article; zbMATH DE number 6388847