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G2 Tensor Product Splines
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G 2 Tensor Product Splines over Extraordinary Vertices Charles - - PowerPoint PPT Presentation
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
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Spline Rings
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Spline Rings
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Problems with Catmull-Clark Subdivision
- Composed of an infinite number of patches
– Hard to evaluate/analyze/process – Not well suited to GPU pipeline
- Not C2 at extraordinary vertices
– Not “Class A” surface – Limits use to entertainment scenarios
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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.
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Geometric Continuity
- Definition
- Correspondence Map
– Assumed to be identity on edge
- Matrix Equation
- Chain Rule Matrix
1 1
k k
G C i i i i
P P P P
2 2
:
1 i i
D D
[DeRose ’85] Pi Pi+1
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Cocycle Condition
- For cyclic collection of patches incident on a common vertex
1 2 1
I
n n
D D
[Hahn ’89] 0 n-1 1 P0 ,…,Pn-1
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Correspondence Maps
1
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Correspondence Maps
1 2
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Correspondence Maps
1 2 3
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Correspondence Maps
interior
1 2
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Correspondence Maps
exterior
2 3
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Interior Correspondence Maps
2 1 1 , 2 1 1 ,
cos , b b 1 1 sin , b b tan
n n x n n y n
u v u v u v u v
2 2
:
n
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Interior Correspondence Maps
Interior Correspondence Map
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Interior Correspondence Maps
1 1 n n n n
R
Cocycle Condition: at type 1 vertex
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Interior Correspondence Maps
1 1 n n n n
R
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Interior Correspondence Maps
1 n n
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Exterior Correspondence Maps
2 3
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Exterior Correspondence Maps
1 1 1 1 1 1 n n n n n m m m m m
Q S R Q S R
, , , , q u v u v s u v v u
Cocycle Condition at type 2 vertex:
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Exterior Correspondence Maps
1 1 1 1 1 1 n n n n n m m m m m
Q S R Q S R
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Exterior Correspondence Maps
1 1 1 n n n n n
Q S R
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Exterior Correspondence Maps
1 1 1 1 k k l l m m n n
S S S S S S S S
Cocycle Condition at type 3 vertex:
2 2
:
n
, , s u v v u
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Exterior Correspondence Maps
3 4 5 8 ∞ n =
,
n u v
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Boundary Data
10 11 21 2 1 1 10 00 10 20 3 3 1 1 1 11 10 11 12 1 12 20 21 22 1
, B B
i i i T i i i i i i i i i i i i i
a a a a a a a H u v u v a a a a a a a a
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Boundary Data
10 11 21 2 1 1 10 00 10 20 3 3 1 1 1 11 10 11 12 1 12 20 21 22 1
, B B
i i i T i i i i i i i i i i i i i
a a a a a a a H u v u v a a a a a a a a
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Patch Smoothness Constraints
- External Constraints
- Internal Constraints
- Constraint System
1, 1, , 0,1,2
j j j j
i i n u u
P t H t j
1 1 1
0, ,0 , 0,1,2
j k j k j k j k
i i n n n u v u v
P t P r t j k
Cp Wa
55 64 7 64
,
n n n n
C W
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Bicubic Energy
4 4 4 4
1 1 2 2 0 0
, ,
i i u v
energy P u v P u v du dv
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Bicubic Energy
4 4 4 4
1 1 2 2 0 0
, ,
i i u v
energy P u v P u v du dv
T i i
p E p
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Constrained Minimization
T
p E C a W C
E E E E
ˆ ˆ , 0, 1 ˆ ˆ ˆ
H j j j j j
p E j n w c c
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Basis Functions
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Support Constraints
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Support Constraints
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Basis Function Plots
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Results
Catmull-Clark This Scheme Loop ‘04
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Results
Catmull-Clark This Scheme
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Results
Catmull-Clark This Scheme
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Boundary Basis Functions
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Results
Catmull-Clark This Scheme
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Results
Catmull-Clark This Scheme
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Conclusions
- G2 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?
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