A family of subdivision schemes with cubic precision

Abstract

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 Hölder 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.