Noninterpolatory Hermite subdivision schemes |
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Authors: | Bin Han Thomas P.-Y. Yu Yonggang Xue. |
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Affiliation: | Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 ; Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 ; Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 |
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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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Keywords: | Refinable function vector refinability subdivision scheme shift invariant subspace subdivision surface spline |
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