Subdivision Curves and Surfaces
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1 Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration generates smoother + more refined mesh
2 What is wrong with NURBS? Very, very difficult to avoid seams Woody s hand from Pixar s Toy Story Subdivision no seams Geri s hand from Geri s Game Corner Cutting
3 Corner Cutting 3 : : 3 Corner Cutting
4 Corner Cutting Corner Cutting
5 Corner Cutting Corner Cutting
6 Corner Cutting Corner Cutting control point limit curve control polygon
7 The 4-point scheme The 4-point scheme
8 The 4-point scheme : : The 4-point scheme : 8
9 The 4-point scheme The 4-point scheme
10 The 4-point scheme The 4-point scheme
11 The 4-point scheme The 4-point scheme
12 The 4-point scheme The 4-point scheme
13 The 4-point scheme The 4-point scheme
14 The 4-point scheme The 4-point scheme control point limit curve control polygon
15 Subdivision curves Non interpolatory subdivision schemes Corner Cutting Interpolatory subdivision schemes The 4-point scheme Basic concepts of Subdivision Subdivision curve limit of recursive subdivision applied to given polygon Each iteration Increase number of vertices (approximately) * 2 Initial polygon - control polygon Central questions: Convergence: Given a subdivision operator and a control polygon, does the subdivision process converge? Smoothness: Does subdivision converge to smooth curve?
16 Subdivision schemes for surfaces Each iteration Subdivision refines control net (mesh) Increase number of vertices (approximately) * 4 Mesh vertices converge to limit surface Every subdivision method has: Method to generate net topology rules to calculate location of new vertices Triangular subdivision Defined for triangular meshes (control nets) Old vertices New vertices Every face replaced by 4 new triangular faces Two kinds of new vertices: Green vertices are associated with old edges Yellow vertices are associated with old vertices
17 Loop s scheme New vertex is weighted average of old vertices List of weights called subdivision mask or stencil Rule for new yellow vertices (n vertex valence) Rule for new green vertices w n = 40 w n 64n ( 3 + 2cos( 2π n) ) 2 n 3 3 The original control net
18 After st iteration After 2nd iteration
19 After 3rd iteration Loop Scheme 9
20 Loop Limit Surface Limit surfaces of Loop s subdivision is C 2 almost everywhere Butterfly Scheme Interpolatory scheme New yellow vertices inherit location of old vertices New green vertices calculated by following stencil:
21 The original control net After st iteration
22 After 2nd iteration After 3rd iteration
23 Butterfly Scheme Butterfly limit surface Limit surfaces of Butterfly subdivision are C, but do not have second derivative
24 Comparison Boundaries & Features: Loop Boundary vertices depend ONLY on other boundary vertices Corner vertices left in place Sometimes modified rules for corner neighbors Treat features (creases) like boundaries 6
25 Scheme Zoo More schemes: Proving scheme works: Catmul-Clark Kobbelt Duo-Sabin Convergence Degree of continuity Affine invariance Affine Invariance subdivide Transform T Transform T subdivide Coefficients of masks sum to weighted average 25
26 Affine Invariance Coefficients of masks must sum to n p = X a i p i displacement X ai (p i + t) = ³ X ai t + p Analysis of Subdivision Test: Convergence Smoothness Goals: Help to choose the rules Ensure that all surfaces have desired properties Plan Define subdivision surfaces Relate properties to coefficients
27 Subdivision Matrix Relate control on finer level to coarser level Useful for Analysis of properties smoothness affine invariance Formulas for normals Explicit evaluation of surfaces at arbitrary points of the domain Controls of K-Sided Patch Simplest case - consider only -ring p 3 p 4 p 2 p 5 c p p 6 p 0 p 2 p 3 p c p 0 p 4 p 6 p 5
28 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8
29 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8
30 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8
31 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8
32 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8 Subdivision Matrix 0 7=6 3=6 3=6 3= =8 3=8 =8 = =8 =8 3=8 = =8 =8 =8 3= =6 5=8 =6 =6 =6 0 0 =6 0 =6 =6 =6 5=8 =6 0 =6 0 =6 =6 0 =6 =6 =6 5=8 0 0 =6 0 =6 =6 =8 3=8 3= =8 0 0 B =8 0 3=8 3= =8 0 A =8 3=8 0 3= =8
33 Eigen Decomposition Diagonalize subdivision matrix eigenvectors eigenvalues p x i, i =0::N i, i =0::N : vector of points in a neighborhood p = X i a i x i (N+)-vector of 3D points 3D vector Eigenvectors Good case: λ & λ <, i =, K,n 0 = i x 0 = [; ;:::] S m p = a 0 x 0 + m a x + m2 a 2 x 2 + m3 a 3 x 3 + ::: limit position tangent vectors can make a 0 zero by moving control points (by affine invariance)
34 Subdominant Eigenvectors Next higher order terms assume = = 2 > j 3j a 0 =0 move control points so that m Sm p = (a x + a 2 x 2 ) + m µ 3 x 3 + ::: subdominant eigenvectors vanishes Example:Tangents Loop scheme Only one ring is necessary matrix of K+ rows Compute eigenvectors (left l & right x) Get coefficients from projection a i = ( l, P) i
35 Example: Limit Position p 3 p 4 p 2 p 5 Eigenvalue 0 c left and right vector are different l center 0 = d p p 6 p 0 p 2 p 3 p c p 0 p 4 p 6 p 5 l0 i = 8 d ; i =0:::K 3 a 0 = dc d X i d = p i 3 3+8k Subdivision Drawbacks Not always intuitive Can have artifacts Hard to control
36 Properties Works best on regular connectivity (valence 6) Easy to implement Efficiency is another matter Use Eigen-analysis for exact computation Local support operations depend on local data Order does not matter Data structure support Allow LOD View mesh at any level as control mesh Apply geometric modifications Continuous Scheme dependent Geri s Game
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