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Refinable bivariate quartic $C^2$-splines for multi-level data representation and surface display

Authors: Charles K. Chui and Qingtang Jiang
Journal: Math. Comp. 74 (2005), 1369-1390
MSC (2000): Primary 65D07, 65D18; Secondary 41A15
Published electronically: July 28, 2004
MathSciNet review: 2137007
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Abstract: In this paper, a second-order Hermite basis of the space of $C^2$-quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of $C^2$-quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.

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Additional Information

Charles K. Chui
Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121 and Department of Statistics, Stanford University, Stanford, California 94305

Qingtang Jiang
Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121

Keywords: Multi-level data representation, Hermite interpolation, refinable quartic $C^2$-splines, vector subdivision, $\sqrt 3$ topological rule, $2$-point coefficient stencils
Received by editor(s): July 8, 2003
Received by editor(s) in revised form: January 2, 2004
Published electronically: July 28, 2004
Additional Notes: The first author was supported in part by NSF Grants #CCR-9988289 and #CCR-0098331, and ARO Grant #DAAD 19-00-1-0512.
The second author was supported in part by University of Missouri–St. Louis Research Award 03
Article copyright: © Copyright 2004 American Mathematical Society