Wavelet Subdivision Methods : GEMS for Rendering Curves and by Charles Chui, Johan de Villiers

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 and Read more...


targeting subdivision curves and surfaces, this booklet info wavelet equipment for geometric layout, modifying, and set of rules improvement. It offers a examine of the idea, tools, and construction Read more...

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

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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.

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