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!