Generalized spline subdivision

Transcription

1 Generalized sline subdivision Jorg Peters SurfLab (Purdue,UFL) Polynomial Heritage Comuting Moments Shae and Eigenvalues

2 Polynomial heritage of generalized sline subdivision Doo-Sabin Catmull-Clark

3 Polynomial heritage of generalized sline subdivision Increasing regions are regular: oints and faces have standard valence

4 Polynomial heritage of generalized sline subdivision Doo-Sabin bi-2 B-sline Catmull-Clark bi-3 B-sline Midedge Zwart-Powell C^1 box-sline Loo C^2 box-sline box-sline = generalization of B-sline to shift-invariant artitions book: [de Boor, Hollig, Riemenschneider 94]

5 Polynomial heritage of generalized sline subdivision Subdivision of the Zwart-Powell C^1 quadratic box-sline basis function Subdi vision a Subdivision Rule c d Subdi vision Subdi vision

6 Polynomial heritage of generalized sline subdivision Mid-edge Rule ( simlest rule ) Zwart-Powell subdivision = 2 stes of Midedge subdivision regular: 4-valence, quadrilaterals 2 stes 4 stes 1 2 1

7 Polynomial heritage of generalized sline subdivision Increasing regions are regular (olynomial) Union of surface layers at an extraordinary oint

8 Polynomial heritage of generalized sline subdivision Uses: Reresentation as Bezier atches Evaluation at non-binary oints Fast moment comutation

9 Generalized sline subdivision Jorg Peters SurfLab (Purdue,UFL) Polynomial Heritage Comuting Moments Shae and Eigenvalues

10 Inertia Frame Challenge: Exonential increase in the number of facets! Volume Center of mass

11 Theory: Gauss Divergence Theorem: The integral of the divergence over the volume equals the integral of the normal comonent over the surface S f S n/ n ds V f dv f dv = f n/ n ds = f V S U n du

12 Theory: Change of variables The area of the surface element S equals the integral of the Jacobian n of the surface arametrization (x,y,z) over the domain U S U ds n du f dv = f n/ n ds = V S f U n du

13 For examle, f=[0,0,z] n = x y u v x v y u f n = z x u y v x v y u is iecewise olynomial in regular regions Volume = 1 dv = V U = z [x u y v - x v y u ] du dv z (x u y v atch U x v y u ) du dv

14 Volume atch = z (x y U u v x v y u ) du dv Schema for bi-3 Bezier atch

15 Volume atch = z (x y U u v x v y u ) du dv

16 Volume atch = z (x y U u v x v y u ) du dv

17 Volume atch = z (x y U u v x v y u ) du dv

18 Work: at each subdivision ste linear for each extraordinary oint add volume contribution of 3n atches Doo-Sabin

19 V i = in layer i z (x y u v U x v y u ) du dv V i+1 W m m = Volume i V i i = 0 V +W m

20 Error estimation: bounding boxes

21 Geometric decay of error volume 1, 1/8, 1/64,...

22 Comuting geometry given a fixed volume Bisection

23 Higher moments and the inertia frame center of mass: x dv, y dv, V V V z dv inertia tensor:..., xy dv,... eigenvector frame V

24 Higher moments and the inertia frame center of mass

25 Physics-based animation Center of mass suort

26 Simle registration, comarison matching frames = comuting a 3x3 matrix Q: IP Q = IS

27 Inertia Frame Volume Center of mass Solution: Moments efficiently and exactly comuted via Gauss theorem and olynomial heritage

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29 Shae and eigenvalues Union of surface layers at an extraordinary oint Control oints transformed by the subdivision matrix

30 Shae and eigenvalues eigenvector exansion i

31 Shae and eigenvalues If all λ < 1, then collase If some λ > 1, then unbounded growth Good sequence: 1, ll,, where l < 1 Eigenvectors of l determine the tangent lane

32 Shae and eigenvalues Fast contraction of 3-sided facets l (1+cos(2i/ 3))/2 =.25 Slow contraction of large facets l (1+cos(2i/16))/2 = midedge subdivision

33 Shae and eigenvalues adjust subdominant eigenvalues (modified midedge subdivision) <=> l. 5

34 Shae and eigenvalues

35 Generalized sline subdivision Summary Polynomial Heritage regular regions Comuting Moments Gauss theorem Shae and Eigenvalues subdominant values