Subdivision Methods for Geometric Design: A Constructive by Joe Warren, Henrik Weimer

By Joe Warren, Henrik Weimer

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Returning to our initial example, the ref inement coeff icients for n1 [x] def ine the subdivision mask s 1 [x] = 1 + x . 1, which follows. 6 The B-spline basis function of order one as the sum of two translates of its dilates. 1 s m[x] = Proof 1 (1 + x)s m−1 [x]. 6) The proof is inductive. Assume that the subdivision mask s m−1 [x] encodes the coeff icients of the ref inement relation for n m−1 [x]. Our task is to show that the mask 12 (1 + x) s m−1 [x] encodes the coeff icients of the ref inement relation for n m[x].

The key idea is to def ine a set of direction vectors of the form {{a i , b i } ∈ Z2 | i = 1, . . , m} and then consider the cross-sectional volume of the form m volm−2 {t 1 , . . , t m} ∈ Hm {a i , b i }t i == {x, y} . 14) i =1 If this function is normalized to have unit integral, the resulting function n [x, y] is the box-spline scaling function associated with the direction vectors . Note that instead of computing the (m − 1)-dimensional volume of the intersection of Hm and an (m−1)-dimensional hyperplane this def inition computes the (m − 2)-dimensional volume of the intersection of Hm and two (m − 1)-dimensional hyperplanes of the form im=1 ai t i == x and im=1 b i t i == y.

Just as in the linear case, this subdivision matrix S is a bi-inf inite matrix whose columns are two-shifts of the subdivision mask s m. This two-shifting arises from the fact that translating a ref ineable scaling function nm[x] of the form s m[[ i ]] nm[2x − i ] by j units on Z induces a shift of 2 j units in the translates of the i dilated scaling function nm[2x] on 12 Z; that is, nm[x − j ] == s m[[ i ]] nm[2x − i − 2 j ]. 8) 36 CHAPTER 2 An Integral Approach to Uniform Subdivision Based on this formula, we observe that the i j th entry of the matrix S is s m[[ i − 2 j ]].

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