Transcript Slide 1

Recent Work on Laplacian
Mesh Deformation
Speaker: Qianqian Hu
Date: Nov. 8, 2006
Mesh Deformation
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Producing visually pleasing results
Preserving surface details
Approaches
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Freeform deformation (FFD)
Multi-resolution
Gradient domain techniques
FFD
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FFD is defined by uniformly spaced feature points in
a parallelepiped lattice.
 Lattice-based (Sederberg et al, 1986)
 Curve-based (Singh et al, 1998)
 Point-based (Hsu et al, 1992)
Multi-resolution
Gradient domain Techniques
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Surface details:
local differences or derivatives
An energy minimization problem
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Least squares method (Linear)
Alexa 03; Lipman 04; Yu 04; Sorkine 04;
Zhou 05; Lipman 05; Nealen 05.
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Iteration (Nonlinear)
Huang 06.
References
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Zhou, K, Huang, J., Snyder, J., Liu, X., Bao, H., and Shum, H.Y.
2005. Large Mesh Deformation Using the Volumetric Graph
Laplacian. ACM Trans. Graph. 24, 3, 496-503.
Huang, J., Shi, X., Liu, X., Zhou, K., Wei, L., Teng, S.H., Bao, H.,
G, B., Shum, H.Y. 2006. Subspace Gradient Domain Mesh
Deformation. In Siggraph’06
Sorkine, O., Lipman, Y., Cohen-or,D., Alexa, M., Rossl, C., Seidel,
H.P. 2004. Laplacian surface editing. In Symposium on
Geometry Processing, ACM SIGGRAPH/Eurographics, 179-188.
Differential Coordinates
δi  L(vi ) 

jN ( i )
ij

jN ( i )
ij
(vi  v j ),
 1.
Invariant only under translation!
Geometric meaning
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Approximating the local shape characteristics
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The normal direction
The mean curvature
Laplacian Matrix
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The transformation from absolute Cartesian
coordinates to differential coordinates
A sparse matrix
Energy function
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The energy function with position
constraints
The least squares method
Characters
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Advantages
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Detail preservation
Linear system
Sparse matrix
Disadvantages
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No rotation and scale invariants
Example
Original
δi
Ti
1) Isotropic scale
2) Rotation
Edited
Ti (V )
L(vi)
Definition of Ti
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A linear approximation to
T  s exp( H )  s( I   H   hT h)
where
is such that γ=0, i.e.,
Large Mesh Deformation Using
the Volumetric Graph Laplacian
Kun Zhou, Jin Huang, John Snyder, Xinguo Liu, Hujun
Bao, Baining Guo, Heung-Yeung Shum
Microsoft Research Asia,
Zhejiang University, Microsoft Research
Comparison
Contribution
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Be fit for large deformation
No local self-intersection
Visually-pleasing deformation results
Outline
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Construct VG (Volumetric Graph)
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Gin (avoid large volume changes)
Gout (avoid local self-intersection)
Deform VG based on volumetric graph
laplacian
Deform from 2D curves
Volumetric Graph
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Step 1: Construct an inner shell Min for the mesh by
offsetting each vertex a distance opposite its normal.
An iterative method
based on simplification envelopes
Volumetric Graph
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Step 2: Embed Min and M in a body-centered cubic
lattice. Remove lattice nodes outside Min.
Volumetric Graph
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Step 3:Build edge connections among M, Min, and
lattice nodes.
Edge connection
Volumetric Graph
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Step 4: Simplify the graph using edge collapse and
smooth the graph.
Simplification:
Smoothing:
VG Example
Left: Gin (Red); Right: Gout (Green); Original Mesh (Blue)
Laplacian Approximation
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The quadratic minimization problem
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The deformed laplacian coordinates
Ti : a rotation and isotropic scale.
Volumetric Graph LA
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The energy function is
Preserving
surface
details
 i  Ti i
Enforcing the userspecified deformation
locations
Preserving
volumetric
details
 i  Ti i
Weighting Scheme
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i
For mesh laplacian,
j+1 βij
αij
j-1
j
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For graph laplacian,
p1
p2
pi
Pj-1
Pj+1
pj
Local Transforms
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Propagating the local transforms over
the whole mesh.
Deformed neighbor points
up
p
t(u)
C(u)
Up
P’
t’ (u)
C’(u)
Local Transformation
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For each point on the control curve
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Rotation:
normal: linear combination of face normals
tangent vector
Scale: s(up)
Propagation Scheme
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The transform is propagated to all
graph points via
a deformation strength field f(p)
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Constant
Linear
Gaussian
The shortest
edge path
Propagation Scheme
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A smoother result: computing a weighted average
over all the vertices on the control curve.
Weight:
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The reciprocal of distance:
A Gaussian function:
Transform matrix:
Solution
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By least square method
A sparse linear system: Ax=b
Precomputing A-1 using LU decomposition
Example
Deformation from 2D curves
3D
2D
Projection
Deformation
Deformation
Back projection
3D
2D
Curve Editing
Cb
C
Least square
fitting
Cd
discrete
3 bspline curve
Editing
Laplacian
deformation
A rotation
and scale
mapping Ti
C’
 N
2
min   L( pi)  Ti δi 
pi
 i 1

C ’b
C ’d
Example
Demo
Subspace Gradient Domain Mesh
Deformation
Jin Huang, Xiaohan Shi, Xinguo Liu, Kun Zhou, Liyi Wei,
Shang-Hua Teng, Hujun Bao, Baining Guo, HeungYeung Shum
Microsoft Research Asia,
Zhejiang University, Boston University
Contributions
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Linear and nonlinear constraints
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Volume constraint
Skeleton constraint
Projection constraint
Fit for non-manifold surface or objects
with multiple disjoint components
Example
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Deformation with nonlinear constraints
Example
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Deformation of multi-component mesh
Laplacian Deformation
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The unconstrained energy minimization
problem
where
f1 ( X )  LX  ˆ( X ),
fi ( X ), i  1
are various deformation constraints
Constraint Classification
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Soft constraints
a nonlinear constraint which is quasi-linear.
AX=b(X)
A: a constant matrix,
b(X): a vector function, ||Jb||<<||A||
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Hard constraints
those with low-dimensional restriction and
nonlinear constraints that are not quasi-linear
Deformation with constraints
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The energy minimization problem
where L is a constant matrix and g(X)
= 0 represents all hard constraints.
Soft constraints: laplacian, skeleton, position constraints
Hard constraints: volume, projection constraints
Subspace Deformation
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Build a coarse control mesh
Control mesh is related to original mesh
X=WP using mean value interpolation
The energy minimization problem
Example
Constraints
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Laplacian constraint
Skeleton constraint
Volume constraint
Projection constraint
Laplacian constraint
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a) the Laplacian is a discrete approximation of the
curvature normal
b) the cotangent form Laplacian lies exactly in the
linear space spanned by the normals of the incident
triangles
xi
Xi,j-1
Xi,j+1
Xi,j
Laplacian coordinate
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For the original mesh,
In matrix form, δi = Ai μi, then μi = Ai+δi
 For deformed mesh
The differential coordinate
i
ˆ
i ( X ) 
di ( X )
di
Skeleton constraint
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For deforming articulated figures, some
parts require unbendable constraint. Eg,
human’s arm, leg.
Skeleton specificaation
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A closed mesh: two virtual vertices(c1,c2), the
centroids of the boundary curve of the open ends:
Line segment ab: approximating the middle of the
front and back intersections(blue)
Skeleton constraint
Preserving both the straightness and the length
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a
si
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Si+1
b
In matrix form,
For each point , si   j kij x j
1
()ij  (kij  ki 1, j )  (krj  k0 j )
r
() j  krj  k0 j
Volume constraint
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The total signed volume:
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The volume constraint
vˆ is the total volume of the original mesh
Example
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Notice: volume constraint can also be
applied to local body parts
Projection constraint
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Let p=QpX, the projection constraint
p
Object space
(ωx ,ωy )
Eye space
Projection plane
Projection constraint
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The projection of p(=QpX)
In matrix form,
i.e.,
Example
Constrained Nonlinear Least
Squares
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The energy minimization problem
Iterative algorithm
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Following the Gauss-Newton method,
f(X) = LX-b(X) is linearized as
Iterative algorithm
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At each iteration,
then
 When Xk =Xk-1 , stop
Stability Comparison
Example(Skeleton)
Example(Volume)
Example(non-manifold)
Demo
Thanks a lot!