Square root 3 -Subdivision Schemes: Maximal Sum Rule Orders

Abstract

Abstract. Subdivision with finitely supported masks is an efficient method to create discrete multiscale representations of smooth surfaces for CAGD applications. Recently a new subdivision scheme for triangular meshes, called

$\sqrt 3$

-subdivision , has been studied. In comparison to dyadic subdivision, which is based on the dilation matrix 2I ,

$\sqrt 3$

-subdivision is based on a dilation M with det M=3 . This has certain advantages, for example, a slower growth for the number of control points.

This paper concerns the problem of achieving maximal sum rule orders for stationary

$\sqrt 3$

-subdivision schemes with given mask support, which is important because the sum rule order characterizes the order of the polynomial reproduction, and provides an upper bound on the Sobolev smoothness of the surface. We study both interpolating and approximating schemes for a natural family of symmetric mask support sets related to squares of sidelength 2n in Z ² , and obtain exact formulas for the maximal sum rule order for arbitrary n . For approximating schemes, the solution is simple, and schemes with maximal sum rule order are realized by an explicit family of schemes based on repeated averaging [15].

In the interpolating case, we use properties of multivariate Lagrange polynomial interpolation to prove the existence of interpolating schemes with maximal sum rule orders. These can be found by solving a linear system which can be reduced in size by using symmetries. From this, we construct some new examples of smooth (C ² ,C ³ ) interpolating

$\sqrt 3$

-subdivision schemes with maximal sum rule order and symmetric masks. The construction of associated dual schemes is also discussed.