Recent Work on Laplacian Mesh Deformation

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Recent Work on Laplacian Mesh
Deformation

• Speaker Qianqian Hu
• Date 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
  in 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
  Siggraph06
• 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
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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Original
Edited
1) Isotropic scale 2) Rotation
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Definition of Ti

• A linear approximation to
• where is such that ?0, i.e.,

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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 3Build 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.

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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

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Weighting Scheme

• For mesh laplacian,
• For graph laplacian,

p1
p2
pi
Pj-1
Pj1
pj
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Local Transforms

• Propagating the local transforms over the whole
  mesh.

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Deformed neighbor points
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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

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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 Axb
Precomputing A-1 using LU decomposition
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Example
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Deformation from 2D curves
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Curve Editing
Cd
C
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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,
  Heung-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

are various deformation constraints
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Constraint Classification

• Soft constraints
• a nonlinear constraint which is quasi-linear.
• AXb(X)
• A a constant matrix,
• b(X) a vector function, JbltltA
• 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 XWP
  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 linear space spanned by the normals of the
  incident triangles

xi
Xi,j-1
Xi,j1
Xi,j
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Laplacian coordinate

• For the original mesh,
• In matrix form, di Ai µi, then µi Aidi
• For deformed mesh
• The differential coordinate

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Skeleton constraint

• For deforming articulated figures, some parts
  require unbendable constraint. Eg, humans 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,

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Volume constraint

• The total signed volume
• The volume constraint
• is the total volume of the original mesh

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Example

• Notice volume constraint can also be applied to
  local body parts

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Projection constraint

• Let pQpX, the projection constraint

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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!