Binary univariate dual and primal subdivision schemes
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.
scientific article; zbMATH DE number 6388847
scientific article