By Charles Chui, Johan de Villiers

Evaluation Curve illustration and drawing Free-form parametric curves From subdivision to foundation services Wavelet subdivision and modifying floor subdivision foundation features FOR CURVE illustration Refinability and scaling features new release of tender foundation services Cardinal B-splines solid bases for integer-shift areas Splines and polynomial replica CURVE SUBDIVISION SCHEMES Subdivision matrices andRead more...

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**Read or Download Wavelet Subdivision Methods : GEMS for Rendering Curves and Surfaces PDF**

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**Extra resources for Wavelet Subdivision Methods : GEMS for Rendering Curves and Surfaces**

**Sample text**

For our current discussion, the symmetric sequence {pj } is said to be centered, if μ+ν = 0 or 1, so that μ ≤ −1 < 1 ≤ ν. If μ + ν = 0, then {pj } is said to be centered at p0 ; while if μ + ν = 1, then {pj } is said to have two centers p0 and p1 . In the remaining discussion throughout this section, we will always assume that the subdivision sequence p = {pj } is symmetric and centered. Next, let us discuss how the basis function φp could be constructed for a CHAPTER 1: OVERVIEW 15 convergent subdivision scheme deﬁned by the subdivision sequence p = {pj }.

3) can be applied to investigate the existence of some ﬁnite sequence {pj }, j ∈ Z2 , for which the box spline B(· |Dn ) satisﬁes the identity B(x |Dn ) = pj B(Ax − j |Dn ), x ∈ R2 . 5) is called the reﬁnement relation. 4 and studied in depth in Chapter 8. 6) B(w |Dn ) = ⎝ | det A| 2 j∈Z it follows immediately that a necessary and suﬃcient condition for B(· |Dn ) to CHAPTER 1: OVERVIEW 31 be reﬁnable, with respect to the dilation matrix A and with ﬁnite reﬁnement sequence {pj }, is that 1 | det A| pj zj = j∈Z2 B(w |Dn ) B(A−T w) −T is a Laurent polynomial in z := e−iA w (by using the notation zj := −T e−ij·A w ).

3) j where z is a complex variable, and the distinction of the notion of a two-scale symbol from that of a symbol is simply the need to multiply by 1/2. Observe that since both {crj } and {pj } are ﬁnite sequences, Cr (z) and P (z) are Laurent polynomials (that is, linear combinations of integer powers of z, where both positive and negative integers are allowed). 3) that Cr+1 (z) = 2P (z)Cr (z 2 ). 4) iteratively. In view of the deﬁnition of the sequences {p[r] }, r = 1, 2, . . 9). 4) can be formulated in terms of the two-scale symbol of the reﬁnement sequence, namely [r] pj = 2r 2π π r−1 k P ei2 −π θ e−ijθ dθ, j ∈ Z.