Families of univariate and bivariate subdivision schemes originated from quartic B-spline
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Families of univariate and bivariate subdivision schemes originated from quartic B-spline Ghulam Mustafa1
· Rabia Hameed1
Received: 25 March 2016 / Accepted: 2 February 2017 © Springer Science+Business Media New York 2017
Abstract Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the LaneRiesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface. Keywords Approximating subdivision scheme · Non-tensor product scheme · Lane-Riesenfeld algorithm · Quartic B-spline · Polynomial generation and reproduction Mathematics Subject Classification (2010) 65D17 · 65D07 · 65U07 · 65D10
1 Introduction Subdivision schemes are powerful tools in the field of free form curve and surface modeling. Catmull and Clark [3] and Doo and Sabin [11] are the pioneers of the Communicated by: Tom Lyche Ghulam Mustafa
[email protected] Rabia Hameed [email protected] 1
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan
G. Mustafa and R. Hameed
surface modeling. They generalized the bi-quadratic and bi-cubic B-spline surfaces to quadrilateral meshes of arbitrary topology. Lane-Riesenfeld [19] introduced an algorithm for generating (n+1)-degree B-spline curve and tensor product surface. The Lane-Riesenfeld algorithm is a simplest form of the refine and smooth (RS) algorithm. Stam [22], Zorin and Schr¨oder [23] presented extension of the LaneRiesenfeld algorithm to arbitrary meshes. They showed that the Catmull and Clark’s [3] and Doo and Sabin’s [11] schemes are the special cases of their family of midpoint subdivision schemes. The Lane-Riesenfeld algorithm is actually based on the linear B-spline scheme. The symbol of the linear B-spline scheme is used as refine 2 operator r(z) = (1+z) and the symbol of its odd stencil is used as smoothing oper2 1+z ator s(z) = rodd (z) = 2 . So the n-th family member of Lane-Riesenfeld schemes has the symbol an (z) = (s(z))n r(z) =
(1 + z)n+2 , 2n+1
n ∈ N.
Cashman et al. [2] also proposed RS algorithm which is based on Dubuc-Deslauriers 4-point interpolatory scheme [10]. They used the symbol of 4-point interpolatory 4 2 (− z2 + 2z − 12 ) and the symbol of its odd scheme as refine operator r(z) = (1+z) 23 2 stencil as the smoothing operator s(z) = rodd (z) = 1+z (− z8 + 54 z − 18 ). The n-th 2 member of their family has the symbol an (z) = (s(z))n r(z) =
(1 + z)n+4 (−z2 + 1
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