G 2 Tensor Product Splines over Extraordinary Vertices Charles - PowerPoint PPT Presentation

G 2 Tensor Product Splines over Extraordinary Vertices Charles Loop Scott Schaefer Microsoft Research Texas A&M University

Spline Rings

Spline Rings

Spline Rings

Problems with Catmull-Clark Subdivision • Composed of an infinite number of patches – Hard to evaluate/analyze/process – Not well suited to GPU pipeline • Not C 2 at extraordinary vertices – Not “Class A” surface – Limits use to entertainment scenarios

Problem Statement Fill the hole in an n-valent Catmull-Clark spline ring with n tensor product patches that join each other and the spline ring with second order smoothness.

Geometric Continuity k k G C • Definition      P P P P [DeRose ’85]   i i 1 i i 1 • Correspondence Map   2 2 ฀ ฀ : – Assumed to be identity on edge • Matrix Equation   D D  P i+1 i i 1 P i  • Chain Rule Matrix 

Cocycle Condition • For cyclic collection of patches incident on a common vertex I           D D [Hahn ’89]   0 0 n 1 n 2 1 0  1 P 0 ,…,P n-1  0  n -1

Correspondence Maps 1

Correspondence Maps 1 2

Correspondence Maps 3 1 2

Correspondence Maps 1 2 interior

Correspondence Maps 3 2 exterior

Interior Correspondence Maps   2 2 ฀ ฀ : n      2 0 cos               1 1 u v , b u n b v n x ,     1 1      2 0 sin               n 1 1 u v , b u b v   n y ,    0 tan       n

Interior Correspondence Maps Interior Correspondence Map

Interior Correspondence Maps Cocycle Condition:   n        1 1 R n n n at type 1 vertex

Interior Correspondence Maps   1 n        1 R n n n

Interior Correspondence Maps      1 n n

Exterior Correspondence Maps 3 2

Exterior Correspondence Maps       q u v , u , v      s u v , v u , Cocycle Condition at type 2 vertex:       1 1                    1 1 1 1 Q S R Q S R n n n n n m m m m m

Exterior Correspondence Maps             1 1                    1 1 1 1 Q S R Q S R n n n n n m m m m m

Exterior Correspondence Maps    1           1 1 Q S R n n n n n

Exterior Correspondence Maps   2 2 ฀ ฀ :     n  s u v , v u , Cocycle Condition at type 3 vertex:                                  1     1     1     1 S S S S S S S S k k l l m m n n

Exterior Correspondence Maps    n u v , 4 5 8 ∞ n = 3

Boundary Data    10 11 21 a a a    i 2 i 1 i 1   10 00 10 20 a a a a          T    3 i 1 i i 1 i 1 3 H u v , B u B v   i 11 10 11 12 a a a a    i 1 i i i   12 20 21 22   a a a a  i 1 i i i

Boundary Data    10 11 21 a a a    i 2 i 1 i 1   10 00 10 20 a a a a          T    3 i 1 i i 1 i 1 3 H u v , B u B v   i 11 10 11 12 a a a a    i 1 i i i   12 20 21 22   a a a a  i 1 i i i

Patch Smoothness Constraints • External Constraints            j j  1, 1, , 0,1,2 P t H t j  j i  j i n u u • Internal Constraints                   j k j k 1 1    0, ,0 , 0,1,2 P t P r t j k   j k i  j k i 1 n n n u v u v • Constraint System      55 64 7 64 n n n n ฀ ฀ C , W Cp Wa

Bicubic Energy 1 1         2 2       4 4 energy P u v , P u v , du dv  i  i 4 4 u v 0 0

Bicubic Energy 1 1         2 2       4 4 energy P u v , P u v , du dv  i  i 4 4 u v 0 0   T p E p i i

Constrained Minimization       T p 0 E C        a  W  C 0           E 0 0    ˆ      ˆ  H p 0 E 0 E c     j        j  E    , j 0, n 1  ˆ   ˆ  ˆ 0     w c 0         j j j    0 0 E

Basis Functions

Support Constraints

Support Constraints

Basis Function Plots

Results Catmull-Clark This Scheme Loop ‘04

Results Catmull-Clark This Scheme

Results Catmull-Clark This Scheme

Boundary Basis Functions

Results Catmull-Clark This Scheme

Results Catmull-Clark This Scheme

Conclusions • G 2 piecewise polynomial surface – Bidegree 7 – Finitely many pieces – Handle meshes with boundary • Negative weights – Convex minimum? • Valence 3 has high curvature hot spots – Solve as special case?

Thank you for your attention