文献类型：期刊文献

英文题名：Smoothing nonlinear subdivision schemes by averaging

作者：Duchamp, Tom[1];Xie, Gang[2];Yu, Thomas[3]

机构：[1]Univ Washington, Dept Math, Box 354350, Seattle, WA 98195 USA;[2]East China Univ Sci & Technol, Dept Math, Shanghai 200237, Peoples R China;[3]Drexel Univ, Dept Math, Philadelphia, PA 19104 USA

年份：2018

卷号：77

期号：2

起止页码：361

外文期刊名：NUMERICAL ALGORITHMS

收录：;WOS:【SCI-EXPANDED(收录号:WOS:000423042200003)】；

基金：Gang Xie's research was supported by the Fundamental Research Funds for the Central Universities and the National, Natural Science Foundation of China (No.11101146).Thomas Yu's research was partially supported by the National Science Foundation grants DMS 1115915 and DMS 1522337.

语种：英文

外文关键词：Averaging; Differential proximity condition; Nonlinear subdivision; B-Spline; Lane-Riesenfeld algorithm; Manifold

摘要：In the theory of linear subdivision algorithms, it is well-known that the regularity of a linear subdivision scheme can be elevated by one order (say, from C (k) to C (k+1)) by composing it with an averaging step (equivalently, by multiplying to the subdivision mask a(z) a (1 + z) factor. In this paper, we show that the same can be done to nonlinear subdivision schemes: by composing with it any nonlinear, smooth, 2-point averaging step, the lifted nonlinear subdivision scheme has an extra order of regularity than the original scheme. A notable application of this result shows that the classical Lane-Riesenfeld algorithm for uniform B-Spline, when extended to Riemannian manifolds based on geodesic midpoint, produces curves with the same regularity as their linear counterparts. (In particular, curvature does not obstruct the nonlinear Lane-Riesenfeld algorithm to inherit regularity from the linear algorithm.) Our main result uses the recently developed technique of differential proximity conditions.