recent work on laplacian mesh deformation speaker: qianqian hu date: nov. 8, 2006
TRANSCRIPT
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Recent Work on Laplacian Mesh Deformation
Speaker: Qianqian HuDate: Nov. 8, 2006
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Mesh Deformation
Producing visually pleasing results Preserving surface details
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Approaches
Freeform deformation (FFD) Multi-resolution Gradient domain techniques
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FFD FFD is defined by uniformly spaced feature points i
n a parallelepiped lattice. Lattice-based (Sederberg et al, 1986) Curve-based (Singh et al, 1998) Point-based (Hsu et al, 1992)
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Multi-resolution
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Gradient domain Techniques Surface details: local differences or derivatives An energy minimization problem
Least squares method (Linear) Alexa 03; Lipman 04; Yu 04; Sorkine 04; Zhou 05; Lipman 05; Nealen 05. Iteration (Nonlinear) Huang 06.
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References 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.
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Differential Coordinates
( )
( )
( ) ( ),
1.
i i ij i jj N i
ijj N i
L
δ v v v
Invariant only under translation!
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Geometric meaning Approximating the local shape characteristics
The normal direction The mean curvature
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Laplacian Matrix The transformation from absolute Cartesian
coordinates to differential coordinates
A sparse matrix
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Energy function
The energy function with position constraints
The least squares
method
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Characters
Advantages Detail preservation Linear system Sparse matrix
Disadvantages No rotation and scale invariants
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Example
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( )iT V
Original Edited
iδ ( )iL v
iT1) Isotropic scale
2) Rotation
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Definition of Ti
A linear approximation to
where is such that γ=0, i.e.,exp( ) ( )Ts s T H I H h h
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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
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Comparison
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Contribution
Be fit for large deformation No local self-intersection Visually-pleasing deformation
results
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Outline
Construct VG (Volumetric Graph) Gin (avoid large volume changes) Gout (avoid local self-intersection)
Deform VG based on volumetric graph laplacian
Deform from 2D curves
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Volumetric Graph 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
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Volumetric Graph Step 2: Embed Min and M in a body-centered
cubic lattice. Remove lattice nodes outside Min.
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Volumetric Graph Step 3:Build edge connections among M, Min,
and lattice nodes.
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Edge connection
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Volumetric Graph Step 4: Simplify the graph using edge collapse
and smooth the graph.Simplification:
Smoothing:
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VG Example
Left: Gin (Red); Right: Gout (Green); Original Mesh (Blue)
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Laplacian Approximation The quadratic minimization problem
The deformed laplacian coordinates
Ti : a rotation and isotropic scale.
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Volumetric Graph LA The energy function is
Preserving surface details
Enforcing the user-specified deformation locations
Preserving volumetric details
i i iT i i iT
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Weighting Scheme For mesh laplacian,
For graph laplacian,
i
j-1
j+1
j
βij αij
pi
p1 p2
Pj-1
pj
Pj+1
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Local Transforms
Propagating the local transforms over the whole mesh.
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Deformed neighbor points
C(u)
pup
t(u)
C’(u)
P ’Up
t’ (u)
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Local Transformation
For each point on the control curve Rotation:
normal: linear combination of face normals tangent vector
Scale: s(up)
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Propagation Scheme
The transform is propagated to all graph points via
a deformation strength field f(p) Constant Linear Gaussian
The shortest edge path
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Propagation Scheme A smoother result: computing a weighted
average over all the vertices on the control curve.
Weight: The reciprocal of distance: A Gaussian function:
Transform matrix:
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Solution
By least square method
A sparse linear system: Ax=b
Precomputing A-1 using LU decomposition
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Example
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Deformation from 2D curves
2D
Projection
Back projection
3D
3D
Defo
rmatio
n
2D
Defo
rmatio
n
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Curve Editing
CLeast square fitting
3 bspline curve
Cb Cd
Editin
g
C ’bC ’d
A rotation and scale mapping Ti
discrete
C ’
2
1
min ( )i
N
i i ii
pL p Tδ
Laplacian deformation
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Example
Demo
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Subspace Gradient Domain Mesh Deformation
Jin Huang, Xiaohan Shi, Xinguo Liu, Kun Zhou, Liyi Wei, Shang-Hua Teng, Hujun Bao, Baining Guo, Heun
g-Yeung Shum
Microsoft Research Asia, Zhejiang University, Boston University
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Contributions
Linear and nonlinear constraints Volume constraint Skeleton constraint Projection constraint
Fit for non-manifold surface or objects with multiple disjoint components
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Example
Deformation with nonlinear constraints
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Example
Deformation of multi-component mesh
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Laplacian Deformation The unconstrained energy
minimization problem
where 1
ˆ( ) ( ), ( ), 1if X LX X f X i
are various deformation constraints
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Constraint Classification Soft constraints
a nonlinear constraint which is quasi-linear. AX=b(X)A: a constant matrix, b(X): a vector function, ||Jb||<<||A||
Hard constraints those with low-dimensional restriction and
nonlinear constraints that are not quasi-linear
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Deformation with constraints
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
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Subspace Deformation
Build a coarse control mesh Control mesh is related to original
mesh X=WP using mean value interpolation
The energy minimization problem
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Example
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Constraints
Laplacian constraint Skeleton constraint Volume constraint Projection constraint
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Laplacian constraint a) the Laplacian is a discrete approximation of the
curvature normal b) the cotangent form Laplacian lies exactly in the l
inear space spanned by the normals of the incident triangles
xi
Xi,j-1
Xi,j
Xi,j+1
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Laplacian coordinate For the original mesh,
In matrix form, δi = Ai μi, then μi = Ai+δi
For deformed mesh
The differential coordinateˆ ( ) ( )ii i
i
X d Xd
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Skeleton constraint
For deforming articulated figures, some parts require unbendable constraint. Eg, human’s arm, leg.
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Skeleton specificaation
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)
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Skeleton constraint Preserving both the straightness and the
length
In matrix form,
a bsi Si+1
1, 0
0
,
1( ) ( ) ( )
( )
i ij jj
ij ij i j rj j
j rj j
For each point s k x
k k k kr
k k
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Volume constraint
The total signed volume:
The volume constraint
is the total volume of the original meshv̂
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Example
Notice: volume constraint can also be applied to local body parts
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Projection constraint
Let p=QpX, the projection constraint
p (ωx ,ωy )
Object space Eye space Projection plane
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Projection constraint
The projection of p(=QpX)
In matrix form,
i.e.,
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Example
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Constrained Nonlinear Least Squares
The energy minimization problem
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Iterative algorithm
Following the Gauss-Newton method, f(X) = LX-b(X) is linearized as
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Iterative algorithm
At each iteration,
then When Xk =Xk-1 , stop
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Stability Comparison
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Example(Skeleton)
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Example(Volume)
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Example(non-manifold)
Demo
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Thanks a lot!