Recent Work on Laplacian Mesh Deformation
Speaker: Qianqian HuDate: Nov. 8, 2006
Mesh Deformation
Producing visually pleasing results Preserving surface details
Approaches
Freeform deformation (FFD) Multi-resolution Gradient domain techniques
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)
Multi-resolution
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.
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.
Differential Coordinates
( )
( )
( ) ( ),
1.
i i ij i jj N i
ijj N i
L
δ v v v
Invariant only under translation!
Geometric meaning Approximating the local shape characteristics
The normal direction The mean curvature
Laplacian Matrix The transformation from absolute Cartesian
coordinates to differential coordinates
A sparse matrix
Energy function
The energy function with position constraints
The least squares
method
Characters
Advantages Detail preservation Linear system Sparse matrix
Disadvantages No rotation and scale invariants
Example
( )iT V
Original Edited
iδ ( )iL v
iT1) Isotropic scale
2) Rotation
Definition of Ti
A linear approximation to
where is such that γ=0, i.e.,exp( ) ( )Ts s T H I H h h
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
Be fit for large deformation No local self-intersection Visually-pleasing deformation
results
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
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
Volumetric Graph Step 2: Embed Min and M in a body-centered
cubic lattice. Remove lattice nodes outside Min.
Volumetric Graph Step 3:Build edge connections among M, Min,
and lattice nodes.
Edge connection
Volumetric Graph 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 The quadratic minimization problem
The deformed laplacian coordinates
Ti : a rotation and isotropic scale.
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
Weighting Scheme For mesh laplacian,
For graph laplacian,
i
j-1
j+1
j
βij αij
pi
p1 p2
Pj-1
pj
Pj+1
Local Transforms
Propagating the local transforms over the whole mesh.
Deformed neighbor points
C(u)
pup
t(u)
C’(u)
P ’Up
t’ (u)
Local Transformation
For each point on the control curve Rotation:
normal: linear combination of face normals tangent vector
Scale: s(up)
Propagation Scheme
The transform is propagated to all graph points via
a deformation strength field f(p) Constant Linear Gaussian
The shortest edge path
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:
Solution
By least square method
A sparse linear system: Ax=b
Precomputing A-1 using LU decomposition
Example
Deformation from 2D curves
2D
Projection
Back projection
3D
3D
Defo
rmatio
n
2D
Defo
rmatio
n
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
Example
Demo
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
Contributions
Linear and nonlinear constraints Volume constraint Skeleton constraint Projection constraint
Fit for non-manifold surface or objects with multiple disjoint components
Example
Deformation with nonlinear constraints
Example
Deformation of multi-component mesh
Laplacian Deformation The unconstrained energy
minimization problem
where 1
ˆ( ) ( ), ( ), 1if X LX X f X i
are various deformation constraints
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
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
Subspace Deformation
Build a coarse control mesh Control mesh is related to original
mesh X=WP using mean value interpolation
The energy minimization problem
Example
Constraints
Laplacian constraint Skeleton constraint Volume constraint Projection constraint
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
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
Skeleton constraint
For deforming articulated figures, some parts require unbendable constraint. Eg, human’s arm, leg.
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)
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
Volume constraint
The total signed volume:
The volume constraint
is the total volume of the original meshv̂
Example
Notice: volume constraint can also be applied to local body parts
Projection constraint
Let p=QpX, the projection constraint
p (ωx ,ωy )
Object space Eye space Projection plane
Projection constraint
The projection of p(=QpX)
In matrix form,
i.e.,
Example
Constrained Nonlinear Least Squares
The energy minimization problem
Iterative algorithm
Following the Gauss-Newton method, f(X) = LX-b(X) is linearized as
Iterative algorithm
At each iteration,
then When Xk =Xk-1 , stop
Stability Comparison
Example(Skeleton)
Example(Volume)
Example(non-manifold)
Demo
Thanks a lot!
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