Generalized spline subdivision

1
Generalized spline subdivision
Jorg Peters SurfLab (Purdue,UFL)

• Polynomial Heritage
• Computing Moments
• Shape and Eigenvalues

2
Polynomial heritageof generalized spline
subdivision

• Doo-Sabin Catmull-Clark
  

3
Polynomial heritageof generalized spline
subdivision

• Increasing regions are regular points and
  faces have standard valence

4
Polynomial heritageof generalized spline
subdivision

• Doo-Sabin bi-2 B-spline
• Catmull-Clark bi-3 B-spline
• Midedge Zwart-Powell
  C1 box-spline
• Loop C2 box-spline

box-spline generalization of B-spline to
shift-invariant partitions book de Boor,
Hollig, Riemenschneider 94
5
Polynomial heritageof generalized spline
subdivision

• Subdivision of the Zwart-Powell C1 quadratic
  box-spline

basis function
Subdivision
a
Subdivision Rule
c
d
Subdivision
6
Polynomial heritageof generalized spline
subdivision
Zwart-Powell subdivision 2 steps of Midedge
subdivision
Mid-edge Rule (simplest rule)
regular 4-valence, quadrilaterals
2 steps
4 steps
1
1
2
7
Polynomial heritageof generalized spline
subdivision

• Increasing regions are regular (polynomial)

• Union of surface layers at an extra-ordinary point

8
Polynomial heritageof generalized spline
subdivision

• Uses

Representation as Bezier patches
Evaluation at non-binary points
Fast moment computation
9
Generalized spline subdivision
Jorg Peters SurfLab (Purdue,UFL)

• Polynomial Heritage
• Computing Moments
• Shape and Eigenvalues

10
Moments of objects enclosed by generalized
subdivision surfaces
Inertia Frame

• Challenge Exponential increase in the number of
  facets!

Volume
11
Moments of objects enclosed by generalized
subdivision surfaces
Theory Gauss Divergence Theorem
ò

Ñ
f
dV
The integral of the divergence over the volume
V
equals the integral of the normal component over
the surface S
ò

dS
n
n/

f
S
ò
ò
ò





Ñ
f


dU
n
f

dS

n
n/

f

dV
V
S
U
12
Moments of objects enclosed by generalized
subdivision surfaces
Theory Change of variables
ò
dS
The area of the surface element S
S
equals the integral of the Jacobian n of the
surface parametrization (x,y,z) over the domain U
ò
dU
n

U
ò
ò




Ñ
f


dS

n
n/

f

dV
V
S
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Moments of objects enclosed by generalized
subdivision surfaces
For example, f0,0,z n f n
z is piecewise
polynomial
in regular regions
-
y
x
y
x
v
v
u
u
Volume
å
ò
-

p
p
p
p
p
dv
du

)
y
x
y
x
(
z
u
v
v
u
p
patch
Up
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Moments of objects enclosed by generalized
subdivision surfaces
ò
Volume patch p
-
p
p
p
p
p
dv
du

)
y
x
y
x
(
z
u
v
v
u
Up

Schema for bi-3 Bezier patch
15
Moments of objects enclosed by generalized
subdivision surfaces
ò
Volume patch p
-
p
p
p
p
p
dv
du

)
y
x
y
x
(
z
u
v
v
u
Up

16
Moments of objects enclosed by generalized
subdivision surfaces
ò
Volume patch p
-
p
p
p
p
p
dv
du

)
y
x
y
x
(
z
u
v
v
u
Up

17
Moments of objects enclosed by generalized
subdivision surfaces
ò
Volume patch p
-
p
p
p
p
p
dv
du

)
y
x
y
x
(
z
u
v
v
u
Up

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Moments of objects enclosed by generalized
subdivision surfaces
Work at each subdivision step linear for
each extraordinary point add volume
contribution of 3n patches
Doo-Sabin
19
Moments of objects enclosed by generalized
subdivision surfaces
å
ò
-
p
p
p
p
p

dv
du

)
y
x
y
x
(
z
V
u
v
v
u
i
i
layer
in

p
Up

m
å


W
V
Volume
V
m
V
i
i1
W

i
0
i
m
20
Moments of objects enclosed by generalized
subdivision surfaces

• Error estimation bounding boxes

21
Moments of objects enclosed by generalized
subdivision surfaces

• Geometric decay of error volume

1, 1/8, 1/64, ...
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Moments of objects enclosed by generalized
subdivision surfaces

• Computing geometry given a fixed volume

Bisection
23
Moments of objects enclosed by generalized
subdivision surfaces

• Higher moments and the inertia frame

center of mass
ò
ò
ò
dV

z
dV,
y
,
dV
x
V
V
V
ò
dV,...
xy
...,
inertia tensor
eigenvector frame
V
24
Moments of objects enclosed by generalized
subdivision surfaces

• Higher moments and the inertia frame

center of mass
25
Moments of objects enclosed by generalized
subdivision surfaces

• Physics-based animation

Center of mass support
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Moments of objects enclosed by generalized
subdivision surfaces

• Simple registration, comparison

matching frames computing a 3x3 matrix Q
IP Q IS
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Moments of objects enclosed by generalized
subdivision surfaces
Inertia Frame

• Solution Moments efficiently and exactly
  computed via Gauss theorem and polynomial
  heritage

Volume
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(No Transcript)
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Shape and eigenvalues

• Union of surface layers at an extra-ordinary
  point

• Control points transformed by the subdivision
  matrix

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Shape and eigenvalues
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Shape and eigenvalues
l

• If all lt 1, then collapse
• If some gt 1, then unbounded growth
• Good sequence 1, , , where
  lt 1
• Eigenvectors of determine the tangent plane

l
l
l
l
l
l
l
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Shape and eigenvalues

• Fast contraction of 3-sided facets
  l
  (1cos(2pi/ 3))/2 .25
• Slow contraction of large facets
  l (1cos(2pi/16))/2 .962...

midedge subdivision
33
Shape and eigenvalues

• adjust subdominant eigenvalues (modified
  midedge subdivision) ltgt l . 5

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Shape and eigenvalues
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Generalized spline subdivisionSummary

• Polynomial Heritage regular regions
• Computing Moments Gauss theorem
• Shape and Eigenvalues subdominant values