Journal article
Smoothing nonlinear subdivision schemes by averaging
Numerical algorithms, v 77(2), pp 361-379
01 Feb 2018
Abstract
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.
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Details
- Title
- Smoothing nonlinear subdivision schemes by averaging
- Creators
- Tom Duchamp - University of WashingtonGang Xie - East China University of Science and TechnologyThomas Yu - Drexel University
- Publication Details
- Numerical algorithms, v 77(2), pp 361-379
- Publisher
- Springer Nature
- Number of pages
- 19
- Grant note
- Fundamental Research Funds for the Central Universities DMS 1115915; DMS 1522337 / National Science Foundation; National Science Foundation (NSF) 11101146 / National, Natural Science Foundation of China; National Natural Science Foundation of China (NSFC)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000423042200003
- Scopus ID
- 2-s2.0-85017129955
- Other Identifier
- 991021878015304721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied