Letter to the Editor
Surface subdivision schemes generated by refinable bivariate spline function vectors

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Abstract

The objective of this paper is to introduce a direct approach for generating local averaging rules for both the 3 and 1-to-4 vector subdivision schemes for computer-aided design of smooth surfaces. Our innovation is to directly construct refinable bivariate spline function vectors with minimum supports and highest approximation orders on the six-directional mesh, and to compute their refinement masks which give rise to the matrix-valued coefficient stencils for the surface subdivision schemes. Both the C1-quadratic and C2-cubic spaces are studied in some detail. In particular, we show that our C2-cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loop's surface subdivision scheme.

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The research was supported by NSF Grants CCR-9988289 and CCR-0098331; and ARO Grant DAAD 19-00-1-0512. Author is also with Department of Statistics, Stanford University, Stanford, CA 94305, USA.