Analysis of stationary subdivision schemes for curve design based on radial basis function interpolation

Abstract

This paper provides a large family of interpolatory stationary subdivision schemes based on radial basis functions (RBFs) which are positive definite or conditionally positive definite. A radial basis function considered in this study has a tension parameter λ > 0 such that it provides design flexibility. We prove that for a sufficiently large λ ≥ λ0, the proposed 2 L-point (L ∈ N) scheme has the same smoothness as the well-known 2 L-point Deslauriers-Dubuc scheme, which is based on 2 L - 1 degree polynomial interpolation. Some numerical examples are presented to illustrate the performance of the new schemes, adapting subdivision rules on bounded intervals in a way of keeping the same smoothness and accuracy of the pre-existing schemes on R. We observe that, with proper tension parameters, the new scheme can alleviate undesirable artifacts near boundaries, which usually appear to interpolatory schemes with irregularly distributed control points. © 2009 Elsevier Inc. All rights reserved.