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

Subdivision schemes are iterative procedures for constructing curves and constitute fundamental tools in computer aided design. Starting with an initial control polygon, a subdivision scheme refines the values computed in 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 estimating error bounds between the limit curve and the control polygon defined after k subdivision stages. In particular, a stop criterion of convergence is obtained. The refinement rules considered in the paper are widely used in practice and are associated with the well known two-scale refinement equation including as particular examples the schemes based on Daubechies' filters. Our results generalize the previous analysis presented by Mustafa et al. in [G. Mustafa, F. Chen, J. Deng, Estimating error bounds for binary subdivision curves/surfaces, J. Comput. Appl. Math. 193 (2006) 596-613] and [G. Mustafa and M.S. Hashmi Subdivision depth computation for n-ary subdivision curves/surfaces, Vis. Comput. 26 (6-8) (2010) 841-851]. (C) 2011 Elsevier B.V. All rights reserved.