Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)

Transcription

1 Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang 1 Guoping Wang 2 1 University of Science and Technology of China 2 Peking University, China SIAM Conference on Geometric and Physical Modeling

2 Doo-Sabin Surfaces Generalization of uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Doo and Sabin 1978]. Limit point rule: For an n-sided face, its centroid is the limit position of its associated extraordinary point. The extraordinary points are at the "centers" of n-sided faces. Convergence: The Doo-Sabin refinement is convergent for extraordinary points with arbitrary valence.

3 Quadratic NURSSes Generalization of non-uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Sederberg et al. 1998]. No closed form limit point rules. Converge for n-sided faces with n 12, but may diverge if n > 12 [Qin et al. 1998].

4 Doo-Sabin Subdivision n 1 P i = w ij P j, i = 0,..., n 1. j=0 Doo-Sabin version [Doo and Sabin 1978], extended to quadratic NURSS: w ij = { n+5 4n, i = j 3+2 cos(2π(i j)/n) 4n, i j Catmull-Clark variant [Catmull and Clark 1978]: n, i j = 0 1 w ij = n, i j = 1, i j > 1 1 4n

5 Catmull-Clark Variant of Doo-Sabin Subdivision Repeated averaging [Stam 2001, Zorin and Schröder 2001]: Linear subdivision: Dual averaging: E i = 1 2 (P i + P i+1 ), P i = 1 4 (P i + E i 1 + E i + F) F = 1 n n 1 P j j=0 = ( n )P i + ( n )(P i+1 + P i 1 ) + 1 4n P j. Non-Uniform Recursive Doo-Sabin Surfaces i j >1(NURDSes)

6 Non-uniform Quadratic B-spline Subdivision Non-uniform Quadratic B-spline Curves For a quadratic B-spline curve, a knot interval d i is assigned to each control point P i. A knot interval is the difference between two adjacent knots in the knot vector, i.e., the parameter length of a B-spline curve segment.

7 Non-uniform Quadratic B-spline Subdivision Non-uniform Quadratic B-spline Subdivision Refinement rules Repeated averaging: Non-uniform linear subdivision: Averaging: E i = d i+1p i + d i P i+1 d i + d i+1 Q 2i = 1 2 (P i + E i ) == (d i + 2d i+1 )P i + d i P i+1 2(d i + d i+1 ) Q 2i+1 = 1 2 (P i+1 + E i ) = d i+1p i + (2d i + d i+1 )P i+1. 2(d i + d i+1 )

8 Non-uniform Quadratic B-spline Subdivision Non-uniform Biquadratic B-spline Surfaces A horizonal knot interval d i and a vertical knot interval e j is assigned to each control point P i,j, as each control point corresponds to a biquadratic surface patch.

9 Non-uniform Quadratic B-spline Subdivision Non-uniform Biquadratic B-spline Subdivision Refinement rules Repeated averaging: Non-uniform linear subdivision: E 1 = d i+1p i,j + d i P i+1,j d i + d i+1, E 2 = e j+1p i,j + e j P i,j+1 e j + e j+1 F = e j+1(d i+1 P i,j + d i P i+1,j ) + e j (d i+1 P i,j+1 + d i P i+1,j+1 ) (d i + d i+1 )(e j + e j+1 ) Dual averaging: Q 2i,2j = 1 4 (P i,j + E 1 + E 2 + F)

10 Non-uniform Doo-Sabin Surfaces Each vertex is assigned a knot interval (possibly different) for each edge incident to it. After subdivision, new knot intervals d ij k can be specified as follows: d i,i+1 0 = d 1 i,i 1 = d0 i,i+1. d i,i 1 0 = d i,i+1 1 = di,i 1 0

11 Non-uniform Recursive Doo-Sabin Surfaces Refinement rules Repeated averaging: Non-uniform linear subdivision: p i = d 0 i,i+1, q i = d 0 i,i 1 E i = q i+1p i + p i P i+1 p i + q i+1 n 1 F = c j P j j=0 Dual averaging: P i = 1 4 (P i + E i 1 + E i + F)

12 Non-uniform Recursive Doo-Sabin Surfaces Coefficients c j Similarly to that in the Catmull-Clark variant of Doo-Sabin subdivision and non-uniform biquadratic B-spline subdivision, the face point F is the (weighted) centroid of the corresponding face and is the limit point corresponding to the center of the face. And, n 1 n 1 F = c j P j = c j P j. j=0 j=0 P i = 1 4 (1 + c i + q i+1 + p i 1 )P i + 1 p i + q i+1 p i 1 + q i (c i 1 + i j >1 c j P j q i )P i p i 1 + q i 4 (c p i i+1 + )P i+1. p i + q i+1

13 Non-uniform Recursive Doo-Sabin Surfaces Coefficients c j Combining the two equations, one obtains a system of linear equations with respect to c j, j = 0,..., n 1. Then we have where c j = α j n 1 k=0 α, k α j = 1 n 1 2 ( n 1 p j+k + q j k ) + k=0 k=0 Here, indices are taken modulo n. n 1 m ( m=1 k=1 n 1 q j+k k=m p j+k ).

14 Non-uniform Recursive Doo-Sabin Surfaces Quad case α 0 = p 0p 1 p 2 p 3 + q 0 q 1 q 2 q 3 2 α 1 = p 0p 1 p 2 p 3 + q 0 q 1 q 2 q 3 2 α 2 = p 0p 1 p 2 p 3 + q 0 q 1 q 2 q 3 2 α 3 = p 0p 1 p 2 p 3 + q 0 q 1 q 2 q q 1 p 1 p 2 p 3 + q 1 q 2 p 2 p 3 + q 1 q 2 q 3 p 3 + q 2 p 2 p 3 p 0 + q 2 q 3 p 3 p 0 + q 2 q 3 q 0 p 0 + q 3 p 3 p 0 p 1 + q 3 q 0 p 0 p 1 + q 3 q 0 q 1 p 1 + q 0 p 0 p 1 p 2 + q 0 q 1 p 1 p 2 + q 0 q 1 q 2 p 2

15 Non-uniform Recursive Doo-Sabin Surfaces A closer look at α j α j is the sum of some products of the n knot intervals, which correspond to the n vertices respectively. where α j = α j,j + k=0 n 1 l=0,l j α j,l = 1 n 1 2 ( n 1 p j+k + q j k ) + k=0 α j,j = 1 n 1 2 ( n 1 p j+k + q j k ) α j,l = m k=1 k=0 n 1 q j+k k=m k=0 n 1 m ( m=1 k=1 n 1 q j+k k=m p j+k, m = (l j) mod n p j+k ).

16 Non-uniform Recursive Doo-Sabin Surfaces I A closer look at α j α j,j = 1 2 ( n 1 k=0 p j+k + n 1 k=0 q j k) can be associated with P j as follows. There are two paths of length n from P j to itself. The clockwise path: P j P j+1 P n 1 P 0 P j 1 then the product of the associated knot intervals is n 1 n 1 p j+k = k=0 k=0 p k

17 Non-uniform Recursive Doo-Sabin Surfaces II A closer look at α j The counterclockwise path: P j P j 1 P 0 P n 1 P j+1 then the product of the associated knot intervals is n 1 n 1 q j k = k=0 k=0 α j,j is the average of the previous two products. q k

18 Non-uniform Recursive Doo-Sabin Surfaces I A closer look at α j For l j, α j,l = m k=1 q n 1 j+k k=m p j+k can be associated with P l in the following way. Here, m = (l j) mod n. There exist one path of length l j and one path of length n l j from P l to P j. If l > j, then we have The counterclockwise path of length m = l j: P l P l 1 P j+1 P j then the m associated knot intervals are q l, q l 1,..., q j+1, and their product is m k=1 q j+k.

19 Non-uniform Recursive Doo-Sabin Surfaces II A closer look at α j The clockwise path of length n m = n (l j): P l P l+1 P n 1 P 0 P j 1 then the n m associated knot intervals are p l, p l+1,..., p n 1, p 0,..., p j 1, and their product is n 1 k=m p j+k. Here, indices are taken modulo n. α j,l is the product of the above two products. For l < j, we have a similar explanation for α j,l.

20 Non-uniform Recursive Doo-Sabin Surfaces Convergence Theorem (Convergence) The NURDS scheme is convergent at extraordinary points of arbitrary valence. Corollary (Limit point rule) For an n-sided face, its face point (i.e. the weighted centroid) is the limit point of its associated extraordinary point.

21 Non-uniform Recursive Doo-Sabin Surfaces Continuity NURDSes have stationary subdivision rules. We prove that the NURDS scheme is G 1 at vertices of valence 3 or 4. Because the subdivision matrix has no obvious symmetries, it is difficult to perform an eigenanalysis for extraordinary points with valence n 5. Numerical experiments and examples show that limit surfaces are G 1 at these points.

22 Non-uniform Recursive Doo-Sabin Surfaces Examples (a) (b) (c) (d) (a) initial control mesh, (b) uniform biquadratic B-spline surface, (c) biquadratic NURBS surface with a crease, (d) NURDS with a dart.

23 NURDSes vs Quadratic NURSSes Consider the configuration surrounding a type F face of valence n, and assume that p 0 = 1000 and all other knot intervals equal 1. For valence 3 n 30, we construct subdivision matrices for quadratic NURSSes and NURDSes respectively, and then investigate eigenstructure and continuity.

24 NURDSes vs Quadratic NURSSes Quadratic NURSSes Concerning spectrum and continuity, we have the following results. For 3 n 30, λ 0, λ 1 and λ 2 may be negative, while λ 3 is always positive. For n 15, λ 0 > 1, the subdivision process is divergent. For 3 n 14, λ 0 = 1 > λ 1, the subdivision process is convergent. For 3 n 10 and n = 12, λ 0 = 1 > λ 1 = λ 2 > λ 3, quadratic NURSSes are G 1 continuous at the extraordinary vertices. For n = 13 and 14, λ 0 = 1 > λ 1 > λ 2 > λ 3, quadratic NURSSes are G 1 continuous at the extraordinary vertices. For n = 11, λ 0 = 1 > λ 1 > λ 2 = λ 3, quadratic NURSSes are only G 0 continuous at the extraordinary vertices.

25 NURDSes vs Quadratic NURSSes Quadratic NURSSes Figure: Absolute values of the first four eigenvalues for quadratic NURSSes for 3 n 30.

26 NURDSes vs Quadratic NURSSes NURDSes Regarding spectrum and continuity, it follows that For 3 n 30, λ 0, λ 1, λ 2 and λ 3 are all positive. For 3 n 30, λ 0 = 1 > λ 1, the subdivision process is convergent. For 3 n 30, λ 0 = 1 > λ 1 = λ 2 > λ 3, NURDSes are G 1 continuous at the extraordinary vertices.

27 NURDSes vs Quadratic NURSSes NURDSes Figure: Values of the first four eigenvalues for NURDSes for 3 n 30.

28 NURDSes vs Quadratic NURSSes Both NURDSes and quadratic NURSSes are the subdivision surfaces that generalize non-uniform biquadratic B-spline surfaces to control grids of arbitrary topology. Differences: NURDSes reduce to Catmull-Clark-variant Doo-Sabin surfaces whereas quadratic NURSSes degenerate to original Doo-Sabin surfaces. NURDSes are convergent at extraordinary points of arbitrary valence while quadratic NURSSes may diverge for valences larger than 12. NURDSes have closed form limit point rules whereas quadratic NURSSes do not.

29 Summary Summary NURDSes are a generalization of Catmull-Clark-variant Doo-Sabin surfaces and biquadratic NURBS surfaces. NURDS refinement can be factored into non-uniform linear subdivision followed by dual averaging. NURDSes are convergent for arbitrary n-sided faces. NURDSes have closed form limit point rules. Future work: Rigorous analysis for G 1 continuity for valence n > 5. Boundary rules for open meshes. Generalization of repeated averaging to higher degree cases, such as bicubic NURBS.

30 Thanks Thanks!