Surface subdivision schemes generated by refinable bivariate spline function vectors

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 C 1-quadratic and C 2-cubic spaces are studied in some detail. In particular, we show that our C 2-cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loop's surface subdivision scheme.

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

“…For the 1-to-4 split rule, the dilation matrix to be selected is simply 2I 2 , both for triangular and quadrilateral meshes. Other topological rules of interest include the √ 3 [22,23,20,28,21,6] and the √ 2 split [40,41,12,14,24] rules, with dilation matrices given, for example, by…”

“…In our recent work [6,7], we introduced subdivision templates of matrices to gain certain desirable properties, such as shape control parameters, smaller template size, and interpolation. (See also [13] for the construction of Hermite interpolation schemes by the parametric approach.)…”

“…In our recent work [6,7], we have constructed refinable function vectors Φ with each component being a bivariate spline function (with small support) on the 6-directional mesh 3 . More precisely, 3 is obtained by triangulation of the x-y plane IR 2 with the grid lines x = i, y = j, x − y = k, x + y = , x + 2y = m, and 2x + y = n, where i, j, k, , m, n ∈ Z Z (see a truncated portion shown on the left of Fig.…”

“…3). We remark that the reason for the choice of 3 here, as opposed to [6,7], is to use the same domain of the characteristic map as that considered in [39] properties of order 1 and 2, respectively (see [6] and [7] for the corresponding templates for Hermite interpolating matrix-valued subdivision schemes).…”

“…In [6], we have also constructed a basis function φ b 1 ∈ S 2 3 ( 3 ) with (minimum) support shown on the right of Fig. 3, where its nonzero Bézier coefficients are displayed.…”