jorg peters, surflab generalized spline subdivision polynomial heritage computing moments shape and...
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![Page 1: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/1.jpg)
Jorg Peters, SurfLab
Generalized spline subdivision
• Polynomial Heritage• Computing Moments• Shape and Eigenvalues
Jorg Peters SurfLab (Purdue,UFL)
![Page 2: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/2.jpg)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Doo-Sabin
Catmull-Clark
![Page 3: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/3.jpg)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Increasing regions are regular: points and faces have standard valence
![Page 4: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/4.jpg)
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]
![Page 5: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/5.jpg)
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
![Page 6: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/6.jpg)
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”)
![Page 7: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/7.jpg)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Increasing regions are regular (polynomial)
• Union of surface layers at an extra-ordinary point
![Page 8: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/8.jpg)
Jorg Peters, SurfLab
Polynomial heritageof generalized spline subdivision
• Uses:
Representation as Bezier patches
Evaluation at non-binary points
Fast moment computation
![Page 9: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/9.jpg)
Jorg Peters, SurfLab
Generalized spline subdivision
• Polynomial Heritage• Computing Moments• Shape and Eigenvalues
Jorg Peters SurfLab (Purdue,UFL)
![Page 10: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/10.jpg)
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
![Page 11: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/11.jpg)
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
![Page 12: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/12.jpg)
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
![Page 13: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/13.jpg)
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
![Page 14: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/14.jpg)
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
![Page 15: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/15.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
![Page 16: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/16.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
![Page 17: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/17.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
Volume patch p = Up
pu
pv
pv
pu
p dvdu )yxyx(z
![Page 18: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/18.jpg)
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
![Page 19: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/19.jpg)
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
![Page 20: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/20.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Error estimation: bounding boxes
![Page 21: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/21.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Geometric decay of error volume 1, 1/8, 1/64, ...
![Page 22: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/22.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Computing geometry given a fixed volume
Bisection
![Page 23: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/23.jpg)
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
![Page 24: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/24.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Higher moments and the inertia frame
center of mass
![Page 25: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/25.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Physics-based animation
Center of mass support
![Page 26: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/26.jpg)
Jorg Peters, SurfLab
Moments of objects enclosed by generalized subdivision surfaces
• Simple registration, comparison
matching frames = computing a 3x3 matrix Q:
IP Q = IS
![Page 27: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/27.jpg)
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
![Page 28: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/28.jpg)
Jorg Peters, SurfLab
![Page 29: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/29.jpg)
Jorg Peters, SurfLab
Shape and eigenvalues
• Union of surface layers at an extra-ordinary point
• Control points transformed by the subdivision matrix
![Page 30: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/30.jpg)
Jorg Peters, SurfLab
Shape and eigenvalues
i
eigenvector expansion
![Page 31: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/31.jpg)
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
![Page 32: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/32.jpg)
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
![Page 33: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/33.jpg)
Jorg Peters, SurfLab
Shape and eigenvalues
• adjust subdominant eigenvalues (modified midedge subdivision)
<=> =. 5
![Page 34: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/34.jpg)
Jorg Peters, SurfLab
Shape and eigenvalues
![Page 35: Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)](https://reader036.vdocument.in/reader036/viewer/2022081513/56649d645503460f94a46a9f/html5/thumbnails/35.jpg)
Jorg Peters, SurfLab
Generalized spline subdivisionSummary
• Polynomial Heritage regular regions
• Computing MomentsGauss’ theorem
• Shape and Eigenvaluessubdominant values