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Shape Deformation
Reporter: Zhang, Lei
5/30/2006
Stuff Vector Field Based Shape Deformatio
n (VFSD) Multigrid Alogrithm for Deformation Edit Deforming Surface Animation Subspace Gradient Domain Mesh Defo
rmationJ. Huang, X. H. Shi, X. G. Liu, K. Zhou, L. Y. Wei, S. H. Teng, H. J. Bao, B. G. Guo and H. Y. Shum.
Vector Field Based Shape Deformations
Wolfram von Funck, Holger Theisel, Hans-Peter Seidel
MPI Informatik
Basic Model
Moving vertex along the deformation orbit – defined by the path lines of a vector field v.
Path Line of Vector Field
X(t0)
X(t)
Given a time-dependent vector field V(X, t), a Path Line in space is X(t):
t0 t
0 0X( ) X( ), , X( ) Xd
t V t t tdt
OR0
0X( ) X (X( ), )t
tt V s s ds
Vector Field Selection
Deformation Request: No self-intersection Volume-preserving Details-preserving Smoothness of shape in deformation
Divergence-free Vector Field: V=(V1, V2, V3)
31 2 VV V0div
x y z
V
Construction of V
Divergence-free
p, q: two scalar field
2D space: ( , )
( , )( , )y
x
p x yx y
p x y
V
3D space: ( , , ) ( , , ) ( , , )x y z p x y z q x y z V
Vector Field for Special Deformation
Constant Vector Field V : translation
( ) , ( )T T
e f -x u x c x w x c
e f V = u w
0, 1 uw = u w
Deformation tX X+V
2( ) , ( )T T
e f -x a x c x a x c
Vector Field for Special Deformation
Linear Vector Field V : rotation
Deformation
, 2( ) 2e f - - a x c x c a a
a
c
xftX X+V
e f V =
Piecewise Field for Deformation
Deformation for a selected region Define piecewise continuous field
Inner region: V
Outer region: zero
Intermediate region: blending
Region specified by an implicit function
And thresholds
( )r cx
i or r
( ) ir rx
( )i or r r x
( )or r x
Piecewise Field for Deformation
Inner region
Intermediate region
Outer region
( , , ) ( , , ) ( , , )x y z p x y z q x y z V
Piecewise Field for Deformation
( )
( ) 1 ( ) 0
0
e
p b e b
x
x x
( )
( ) 1 ( ) 0
0
f
q b f b
x
x x
4
4
0
( ) ii i
i o i
r rb b r w B
r r
x
if
if
if
if
if
if
( ) ir rx
( )i or r r x
( )or r x
( ) ir rx
( )i or r r x
( )or r x
Deformation Tools
Translation: constant vector field
, ( )
1 ( ) 0 1 ( ) 0 , ( )
0, ( )
i
i o
o
r r
b e b b f b r r r
r r
v x
V x x x
x
e f v = ( ) , ( )T T
e f -x u x c x w x c
Deformation Tool
Rotation: linear vector field
, ( )
1 ( ) 0 1 ( ) 0 , ( )
0, ( )
i
i o
o
r r
b e b b f b r r r
r r
v x
V x x x
x
e f v = , 2( ) 2e f - - a x c x c a a
Path Line Computation
0 0X( ) X( ), , X( ) Xd
t V t t tdt
Runge-Kutta Integration 1 1,i it c
1 1 1 1, ,i i i i i i i it t t t V c V c c c
11 1
, 1 i ii i
i i i i
t t t tt
t t t t
V x V V
,i itc
For each vertex v(x, ti), integrating vector field above to v(x’, ti+1)
Remeshing
Edge Split
Examples
Demo
Examples
Performance
Benchmark Test
AMD 2.6GHz
2 GB RAM
GeForce 6800 GT GPU
Conclusion Embeded in Vector Field
FFD Parallel processing Salient Strength
No self-intersection Volume-preserving Details-preserving Smoothness of shape in deformation
A Fast Multigrid Algorithm for Mesh Deformation
Lin Shi, Yizhou Yu, Nathan Bell, Wei-Wen Feng
University of Illinois at Urbana-Champaign
Basic Model
Two-pass pipeline Local Frame Update
Vertex Position Update
Multigrid Computation Method
R. Zayer, C. Rossl, Z. Karni and H. P. Seidel. Harmonic Guidance for Surface Deformation. EG2005.
Y. Lipman, O. Sorkine, D. Levin and D. Cohen-Or. Linear rotation-invariant coordinates for meshes. Siggraph2005.
Discrete Form (SIG’05)
ix
ikx
1ikx
ik
1 2 1i ik k x x
3
31 11 2 1 2
2 21 1 2 2, , 1 1, 1
( ) ,
,
2
i
i i i ik k k k
i i i
k k k k k k
I
g g g
x x x x
3, ,
i i ik kk kg x x
3
1, 1 ,i i i
k kk kg x x
1: det , ,i ii ik kkO sign x x N
First Discrete Form
Discrete Form (SIG’05)
ix
ikx
1ikx
ik
1 2 1i ik k x x
3 311 2
11 2
( ) , ,i ii i ik k
i ik k
I
L L
x N x N
3
,ii i
k kL x N
Second Discrete Form
Local Frame (SIG’05)
Discrete Frame at each vertex
iTM
iN
1ib
2ib
1
ix
1 2, ,i i ib b N forms a right-hand orthonormal basis.
ix
1 1 2 2, ,
j i i i ikk
i i ii i i i ikk k
L
L
N
b b b b N
x x x
x x
jx
1/ 21 1 1,1
i iix gb
First Pass (EG’05)
Harmonic guidance for local frame
2 0 0h Lh Boundary conditions:
1: edited vertex
0: fixed vertex
Scaling
RotationT ( ) ( ) 1 ( )sT h T h I x x x
1
0
Second Pass (SIG’05)
Solving vertex position
iTM
iN
1ib
2ib
1
ix
ix
jx
1 2 31 2j j j
i j ji ji jic c c x x b b N
1 2 31 2i i i
j i ij ij ijc c c x x b b N
1 2 3 1 2 31 2 1 2
1
2
i j ji
j j j i i iji ji ji ji ij ij ijc c c c c c
x x d
d b b N b b N
Second Pass
Solving vertex position
1 2 3 1 2 31 2 1 2
1
2
i j ji
j j j i i iji ji ji ji ij ij ijc c c c c c
x x d
d b b N b b N
( ) ( )
( ) ( )
ji j ji jj N i j N ii
ji jij N i j N i
w w
w w
x dx
( ) ( )
( )ji j i ji jij N i j N i
w w
x x d
“Normal Equation”
1
ix
ix
jx
Some Results
Computation
First Pass
Second Pass
2 0 0h Lh
( ) ( )
( )ji j i ji jij N i j N i
w w
x x d
Multigrid Method
Multigrid Method
=n n nA u b
nu
,= ;
nn n n
n n n
b b A u
A u b
2
2
2 2 2
,
,
= ;
n n
n n
n n n
b R b
A RA P
A u b
( 1)
( 1)
,
,
= .
pn p n
pn p n
pn pn pn
b R b
A RA P
A u bpnu
defect equation
coarsest level
Performance
Performance
Conclusion
Computation Method for large mesh
Editing Arbitrary Deforming Surface Animations
S. Kircher, M. Garland
University of Illinois at Urbana-Champaign
Problem
Deforming Surface
Editing Surface
Pyramid Scheme
Quadric Error Metric
, , ,1T
x y zv v v v =
4 4,TQ Q v v v
0x y z
M. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. SIGGRAPH’97.
Pyramid Scheme
1 0nM M M
Coarse Fine
1: k kr M M
1: k kr M M
2nd-order divided difference 1kM
kM
1kM
Detail vector
Construct by and adding detail vectors for level k.
kM 1kM
Sig’99
Adaptive Transform
is generated from by improving its error with respect to
Adaptive Transform
Multilevel Meshes (Sig’05)
0 1 2, , ,S S S S
0 1 2, , ,H H H H
iH1iH
1iS
Reclustering
, ,v a b
Swap
0H H Swap
Basis Smoothing Blockification Vertex Teleportation
PRE-processing: Time-varying multiresolution transform for a given animation sequence.
Editing Tool
Direct Manipulation
level k
level 0
0H
1H
Editing Tool
Direct Manipulation
Editing Tool
Direct Manipulation
Multiresolution Embossing
Multiresolution set of Edit
Conclusion
Multiresolution Edit
The End
