TY - JOUR

T1 - High order smoothness of non-linear Lane-Riesenfeld algorithms in the functional setting

AU - Dyn, Nira

AU - Goldman, Ron

AU - Levin, David

N1 - Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2019/5

Y1 - 2019/5

N2 - We investigate some variants of the linear Lane-Riesenfeld algorithm in the functional setting, generated by replacing the standard binary arithmetic averages between real numbers by non-linear, binary, symmetric averages between real numbers. For certain classes of non-linear averages we show that such generalized r-th order Lane-Riesenfeld algorithms generate limit functions with the same smoothness as the linear r-th order Lane-Riesenfeld algorithm. The analysis is based on the observation that a non-linear subdivision scheme can be regarded as a non-uniform linear scheme with mask coefficients depending on the initial data, and on ideas from a recent investigation of smoothly varying non-uniform linear subdivision schemes (Dyn et al., 2014). Our main result is motivated by an example in Duchamp et al. (2016). It extends a result in Duchamp et al. (2018), derived for subdivision schemes on smooth manifolds, and applied to the non-linear Lane-Riesenfeld algorithm in the functional setting, by allowing different non-linear averages at different locations in a smoothing step.

AB - We investigate some variants of the linear Lane-Riesenfeld algorithm in the functional setting, generated by replacing the standard binary arithmetic averages between real numbers by non-linear, binary, symmetric averages between real numbers. For certain classes of non-linear averages we show that such generalized r-th order Lane-Riesenfeld algorithms generate limit functions with the same smoothness as the linear r-th order Lane-Riesenfeld algorithm. The analysis is based on the observation that a non-linear subdivision scheme can be regarded as a non-uniform linear scheme with mask coefficients depending on the initial data, and on ideas from a recent investigation of smoothly varying non-uniform linear subdivision schemes (Dyn et al., 2014). Our main result is motivated by an example in Duchamp et al. (2016). It extends a result in Duchamp et al. (2018), derived for subdivision schemes on smooth manifolds, and applied to the non-linear Lane-Riesenfeld algorithm in the functional setting, by allowing different non-linear averages at different locations in a smoothing step.

KW - Analysis of non-uniform subdivision

KW - Laurent polynomials

KW - Non uniform Lane-Riesenfeld Algorithm

KW - Smoothly varying averages

UR - http://www.scopus.com/inward/record.url?scp=85064277211&partnerID=8YFLogxK

U2 - 10.1016/j.cagd.2019.03.003

DO - 10.1016/j.cagd.2019.03.003

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AN - SCOPUS:85064277211

SN - 0167-8396

VL - 71

SP - 119

EP - 129

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

ER -