Efficient evaluation of subdivision schemes with polynomial reproduction property

• The method is applicable to all subdivision schemes with polynomial reproduction.
• It performs exact evaluation at rational parameters and approximate evaluation at other arbitrary parameters with tolerance control.
• It is efficient and robust for the presented schemes with corresponding coefficient matrix being strictly and diagonally dominant.
• It can also evaluate derivatives under the same framework.
• Extension of the method to surface cases is straightforward.

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam’s method with that of this paper.