Refinable bivariate quartic -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

DOI:
https://doi.org/10.1090/S0025-5718-04-01702-8

Published electronically:
July 28, 2004

MathSciNet review:
2137007

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, a second-order Hermite basis of the space of -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 -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.

**1.**P. Alfeld, L. L. Schumaker,*Smooth macro-elements based on Powell-Sabin triangle splits*, Adv. Comput. Math.**16**(2002), 29-46. MR**2003a:65097****2.**C. K. Chui,*Multivariate Splines*, NSF-CBMS Series, vol. 54, SIAM Publ., Philadelphia, 1988. MR**92e:41009****3.**C. K. Chui,*Vertex splines and their applications to interpolation of discrete data*, In*Computation of Curves and Surfaces*, 137-181, W. Dahmen, M. Gasca and C.A. Micchelli (eds.), Kluwer Academic, 1990. MR**91f:65021****4.**C. K. Chui, H. C. Chui, T. X. He,*Shape-preserving interpolation by bivariate quadratic splines*, In*Workshop on Computational Geometry*, 21-75, A. Conte, V. Demichelis, F. Fontanella, and I. Galligani (eds.), World Sci. Publ. Co., Singapore, 1992. MR**96d:65033****5.**C. K. Chui, Q. T. Jiang,*Surface subdivision schemes generated by refinable bivariate spline function vectors*, Appl. Comput. Harmon. Anal.**15**(2003), 147-162. MR**2004h:65015****6.**N. Dyn, D. Levin, J. A. Gregory,*A butterfly subdivision scheme for surface interpolation with tension control*, ACM Trans. Graphics**2**(1990), 160-169.**7.**R. Q. Jia,*Shift-invariant spaces and linear operator equations*, Israel J. Math.**103**(1998), 259-288. MR**99d:41016****8.**R. Q. Jia, Q. T. Jiang,*Approximation power of refinable vectors of functions*, In*Wavelet analysis and applications*, 155-178, Studies Adv. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2002. MR**2003e:41030****9.**R. Q. Jia, Q. T. Jiang,*Spectral analysis of transition operators and its applications to smoothness analysis of wavelets*, SIAM J. Matrix Anal. Appl.**24**(2003), 1071-1109. MR**2004h:42043****10.**R. Q. Jia, C. A. Micchelli,*On linear independence of integer translates of a finite number of functions*, Proc. Edinburgh Math. Soc.**36**(1992), 69-85. MR**94e:41044****11.**L. Kobbelt,*-subdivision*, In*Computer Graphics Proceedings*, Annual Conference Series, 2000, pp. 103-112.**12.**U. Labsik, G. Greiner,*Interpolatory -subdivision*,*Proceedings of Eurographics 2000, Computer Graphics Forum*, vol. 19, 2000, pp. 131-138.**13.**C. Loop,*Smooth subdivision surfaces based on triangles*, Master's thesis, University of Utah, Department of Mathematics, Salt Lake City, 1987.**14.**G. Nürnberger, F. Zeilfelder,*Developments in bivariate spline interpolation*, J. Comput. Appl. Math.**121**(2000), 125-152. MR**2001e:41042****15.**M. J. D. Powell, M. A. Sabin,*Piecewise quadratic approximations on triangles*, ACM Trans. Math. Software**3**(1977), 316-325. MR**58:3319****16.**P. Sablonniére,*Error bounds for Hermite interpolation by quadratic splines on an -triangulation*, IMA J. Numer. Anal.**7**(1987), 495-508.MR**90a:65029**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65D07,
65D18,
41A15

Retrieve articles in all journals with MSC (2000): 65D07, 65D18, 41A15

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

Email:
ckchui@stanford.edu

**Qingtang Jiang**

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

Email:
jiang@math.umsl.edu

DOI:
https://doi.org/10.1090/S0025-5718-04-01702-8

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