Simple regularity criteria for subdivision schemes
SIAM Journal on Mathematical Analysis
Polynomial generation and quasi-interlpolation in stationary non-uniform subdivision
Computer Aided Geometric Design
On the support of recursive subdivision
ACM Transactions on Graphics (TOG)
Stationary subdivision schemes reproducing polynomials
Computer Aided Geometric Design
Artifacts in box-spline surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
A 4-point interpolatory subdivision scheme for curve design
Computer Aided Geometric Design
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Polynomial reproduction by symmetric subdivision schemes
Journal of Approximation Theory
An approximating C2 non-stationary subdivision scheme
Computer Aided Geometric Design
Integration of CAD and boundary element analysis through subdivision methods
Computers and Industrial Engineering
Incenter subdivision scheme for curve interpolation
Computer Aided Geometric Design
Variations on the four-point subdivision scheme
Computer Aided Geometric Design
Full length article: Polynomial reproduction for univariate subdivision schemes of any arity
Journal of Approximation Theory
A new four-point shape-preserving C3 subdivision scheme
Computer Aided Geometric Design
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The four-point subdivision scheme is well known as an interpolating subdivision scheme, but it has recently come to our notice that it is but the first scheme in a family all of whose members have the property that if all the control points lie equally spaced along the same cubic polynomial, the limit curve is exactly that polynomial. Other members of the family have higher smoothness. We study these schemes as functions, where the ordinate is given by the scheme, while the abscissae of the control points are equally spaced. Because all schemes include linear functions in their precision set, this may be regarded as a particular case of the parametric setting, rather than as a special case. This paper introduces the family and determines how the support, the Holder regularity, the precision set, the degree of polynomials spanned by the limit curves, and the artifact behavior vary with the integer parameter that identifies the members of the family. For the family members with an even parameter value, most of these properties have also been shown by Dong and Shen (Dong, B., Shen, Z., 2007. Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22 (1), 78-104), as they turn out to be a particular kind of pseudo-splines. But regarding the regularity exponent of the limit functions, we derive the exact values and thus improve the lower bounds given by Dong and Shen in that paper. Moreover, our analysis also covers the family members with an odd parameter value which do not seem to fit into the general framework of pseudo-splines. Just before this paper was submitted, (Choi, S.W., Lee, B.-G., Lee, Y.J., Yoon, J., 2006. Stationary subdivision schemes reproducing polynomials. Comput. Aided Geom. Design 23 (4), 351-360) appeared, which also discusses a family of subdivision schemes. The high order members of that family achieve higher degrees of polynomial reproduction, whereas ours aim only at cubic reproduction. This allows us to gain higher continuity for a given mask width.