TY - CHAP

T1 - Three Families of Nonlinear Subdivision Schemes

AU - Dyn, Nira

PY - 2006

Y1 - 2006

N2 - Three families of nonlinear subdivision schemes, derived from linear schemes, are reviewed. The nonlinearity is introduced into the linear schemes by adapting the schemes to the data. The first family, derived from the four-point interpolatory linear subdivision scheme, consists of geometrically controlled schemes, which are either shape preserving or artifact-free. The second family of schemes is designed for the functional setting, to be used in constructions of multiscale representations of piecewise smooth functions. The schemes are extensions of the Dubuc-Deslauriers 2N-point interpolatory schemes, with the classical local interpolation replaced by ENO or WENO local interpolation. The third family consists of subdivision schemes on smooth manifolds. These schemes are derived from converging linear schemes, represented in terms of repeated binary averages. The analysis of the nonlinear schemes is done either by proximity to the linear schemes from which they are derived, or by methods adapted from methods for linear schemes.

AB - Three families of nonlinear subdivision schemes, derived from linear schemes, are reviewed. The nonlinearity is introduced into the linear schemes by adapting the schemes to the data. The first family, derived from the four-point interpolatory linear subdivision scheme, consists of geometrically controlled schemes, which are either shape preserving or artifact-free. The second family of schemes is designed for the functional setting, to be used in constructions of multiscale representations of piecewise smooth functions. The schemes are extensions of the Dubuc-Deslauriers 2N-point interpolatory schemes, with the classical local interpolation replaced by ENO or WENO local interpolation. The third family consists of subdivision schemes on smooth manifolds. These schemes are derived from converging linear schemes, represented in terms of repeated binary averages. The analysis of the nonlinear schemes is done either by proximity to the linear schemes from which they are derived, or by methods adapted from methods for linear schemes.

KW - 2000 MSC 65D05

KW - 65D07

KW - 65D10

KW - 65D15

KW - 65D17

KW - 65U05

KW - 65U07

KW - ENO interpolation

KW - adaptive tension parameter

KW - convexity preserving scheme

KW - data dependent scheme

KW - geodesics

KW - linear and nonlinear subdivision scheme

KW - projection onto a manifold

KW - refinement on a manifold

KW - repeated binary averages

UR - http://www.scopus.com/inward/record.url?scp=77956696070&partnerID=8YFLogxK

U2 - 10.1016/S1570-579X(06)80003-0

DO - 10.1016/S1570-579X(06)80003-0

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AN - SCOPUS:77956696070

T3 - Studies in Computational Mathematics

SP - 23

EP - 38

BT - Studies in Computational Mathematics

PB - Elsevier

ER -