Date post: | 21-Dec-2015 |
Category: |
Documents |
View: | 219 times |
Download: | 0 times |
Jorg Peters, SurfLab
Generalized spline subdivision
• Polynomial Heritage• Computing Moments• Shape and Eigenvalues
Jorg Peters SurfLab (Purdue,UFL)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Doo-Sabin
Catmull-Clark
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Increasing regions are regular: points and faces have standard valence
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Doo-Sabin bi-2 B-spline
• Catmull-Clark bi-3 B-spline
• Midedge Zwart-Powell C^1 box-spline
• Loop C^2 box-spline
box-spline = generalization of B-spline to shift-invariant partitions book: [de Boor, Hollig, Riemenschneider 94]
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Subdivision of the Zwart-Powell C^1 quadratic box-spline
a
c dSubdivision
Subdivision
basis function
Subdivision Rule
Subdivision
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
2 steps 4 steps
Zwart-Powell subdivision = 2 steps of Midedge subdivision
21
1
regular: 4-valence, quadrilaterals
Mid-edge Rule (“simplest rule”)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Increasing regions are regular (polynomial)
• Union of surface layers at an extra-ordinary point
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Uses:
Representation as Bezier patches
Evaluation at non-binary points
Fast moment computation
Jorg Peters, SurfLab
Generalized spline subdivision
• Polynomial Heritage• Computing Moments• Shape and Eigenvalues
Jorg Peters SurfLab (Purdue,UFL)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Challenge: Exponential increase in the number of facets!
Volume
Inertia Frame
Center of mass
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
V
f
V S U
dUn f dS nn/ f f dV
S
dSnn/ f
Theory: Gauss’ Divergence Theorem:
The integral of the divergence over the volume
equals the integral of the normal component over the surface S
dV
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
U
n
Theory: Change of variables
The area of the surface element S
equals the integral of the Jacobian |n| of the surface parametrization (x,y,z) over the domain U
S
dS
dU
V S
U
dUn f dS nn/ f f dV
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
V U
uvvu dvdu ]yx-y[x z dV1
For example, f=[0,0,z] n =
f n = z is piecewise polynomial in regular regions
Volume =
uvvu yxyx
uvvu yxyx
ppatch Up
pu
pv
pv
pu
p dvdu )yxyx(z
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
“Volume” patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
Schema for bi-3 Bezier patch
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
Jorg Peters, SurfLab
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
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Up
pu
pv
pv
pu
p dvdu )yxyx(z
m
m
ii
WV 0V iVi+1
i
iVlayer in p
Volume
mW
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Error estimation: bounding boxes
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Geometric decay of error volume 1, 1/8, 1/64, ...
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Computing geometry given a fixed volume
Bisection
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Higher moments and the inertia frame
center of mass:
V VV
dV zdV,y ,dVx
inertia tensor:
V
dV,...xy ..., eigenvector frame
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Higher moments and the inertia frame
center of mass
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Physics-based animation
Center of mass support
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Simple registration, comparison
matching frames = computing a 3x3 matrix Q:
IP Q = IS
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Solution: Moments efficiently and exactly computed via Gauss’ theorem and polynomial heritage
Volume
Inertia Frame
Center of mass
Jorg Peters, SurfLab
Jorg Peters, SurfLab
Shape and eigenvalues
• Union of surface layers at an extra-ordinary point
• Control points transformed by the subdivision matrix
Jorg Peters, SurfLab
Shape and eigenvalues
i
eigenvector expansion
Jorg Peters, SurfLab
Shape and eigenvalues
• If all < 1, then collapse• If some > 1, then unbounded growth• Good sequence: 1, , , … where | | < 1• Eigenvectors of determine the tangent plane
Jorg Peters, SurfLab
Shape and eigenvalues• Fast contraction of 3-sided facets
= (1+cos(2pi/ 3))/2 = .25
• Slow contraction of large facets
= (1+cos(2pi/16))/2 = .962...midedge subdivision
Jorg Peters, SurfLab
Shape and eigenvalues
• adjust subdominant eigenvalues (modified midedge subdivision)
<=> =. 5
Jorg Peters, SurfLab
Shape and eigenvalues
Jorg Peters, SurfLab
Generalized spline subdivisionSummary
• Polynomial Heritage regular regions
• Computing MomentsGauss’ theorem
• Shape and Eigenvaluessubdominant values