Error bounds for a class of subdivision schemes based on the two-scale refinement equation

Abstract

Subdivision schemes are iterative procedures to construct curves and constitute fundamental tools in Computer Aided Design. Starting with an initial control polygon, a subdivision scheme refines the computed values at the previous step according to some basic rules. The scheme is said to be convergent if there exists a limit curve. The computed values define a control polygon in each step. This paper is devoted to estimate error bounds between the “ideal” limit curve and the control polygon defined after k subdivision stages. In particular, a stop criteria of convergence is obtained. The considered refinement rules in the paper are widely used in practice and are associated to the well known two-scale refinement equation including as particular examples Daubechies’ schemes. Companies such as Pixar have made subdivision schemes the basic tool for much of their computer graphicsmodelling software.