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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Noninterpolatory Hermite subdivision schemes


Authors: Bin Han, Thomas P.-Y. Yu and Yonggang Xue
Journal: Math. Comp. 74 (2005), 1345-1367
MSC (2000): Primary 41A05, 41A15, 41A63, 42C40, 65T60, 65F15
Published electronically: September 10, 2004
MathSciNet review: 2137006
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Abstract: Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.

A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.


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Additional Information

Bin Han
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: bhan@math.ualberta.ca

Thomas P.-Y. Yu
Affiliation: Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
Email: yut@rpi.edu

Yonggang Xue
Affiliation: Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
Email: xuey@rpi.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-04-01704-1
PII: S 0025-5718(04)01704-1
Keywords: Refinable function, vector refinability, subdivision scheme, shift invariant subspace, subdivision surface, spline
Received by editor(s): April 15, 2003
Received by editor(s) in revised form: December 10, 2003
Published electronically: September 10, 2004
Additional Notes: The first author’s research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under grant G121210654
The second author’s research was supported in part by an NSF CAREER Award (CCR 9984501)
Article copyright: © Copyright 2004 American Mathematical Society