Mean Value Coordinates for Closed Triangular Meshes

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Mean Value Coordinates for Closed Triangular
Meshes

• Tao Ju
• Scott Schaefer
• Joe Warren

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Boundary Value Interpolation

• Given , compute such that
• Given values at , construct a function
• Interpolates values at vertices
• Linear on boundary
• Smooth on interior

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Boundary Value Interpolation

• Given , compute such that
• Given values at , construct a function
• Interpolates values at vertices
• Linear on boundary
• Smooth on interior

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Previous Work
convex polygons Wachspress 1975
closed polygons Floater 2003, Hormann 2004
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Previous Work
convex polygons Wachspress 1975
closed polygons Floater 2003, Hormann 2004
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Previous Work
convex polygons Wachspress 1975
closed polygons Floater 2003, Hormann 2004
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Previous Work
convex polygons Wachspress 1975
closed polygons Floater 2003, Hormann 2004
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Previous Work
convex polygons Wachspress 1975
closed polygons Floater 2003, Hormann 2004
3D convex polyhedra Warren 1996 2004, Ju et al
2005
3D closed triangle meshes Floater et al 2005 Ju
2005
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ApplicationSurface Deformation
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ApplicationSurface Deformation
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ApplicationSurface Deformation
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ApplicationSurface Deformation
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3D Mean Value Coordinates
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3D Mean Value Coordinates
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3D Mean Value Coordinates

• Project surface onto sphere centered at

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3D Mean Value Coordinates

• Project surface onto sphere centered at
• mean vector (integral of unit normal
• over spherical triangle)

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3D Mean Value Coordinates

• Project surface onto sphere centered at
• mean vector (integral of unit normal
• over spherical triangle)
• Stokes Theorem

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3D Mean Value Coordinates

• Project surface onto sphere centered at
• mean vector (integral of unit normal
• over spherical triangle)
• Stokes Theorem

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3D Mean Value Coordinates

• Project surface onto sphere centered at
• mean vector (integral of unit normal
• over spherical triangle)
• Stokes Theorem

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3D Mean Value Coordinates

• Project surface onto sphere centered at
• mean vector (integral of unit normal
• over spherical triangle)
• Stokes Theorem

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Computing the Mean Vector

• Given spherical triangle, compute mean vector
  (integral of unit normal)

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Computing the Mean Vector

• Given spherical triangle, compute mean vector
  (integral of unit normal)

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Computing the Mean Vector

• Given spherical triangle, compute mean vector
  (integral of unit normal)
• Build wedge with face normals

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Computing the Mean Vector

• Given spherical triangle, compute mean vector
  (integral of unit normal)
• Build wedge with face normals
• Apply Stokes Theorem,

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Interpolant Computation

• Compute mean vector

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Interpolant Computation

• Compute mean vector
• Calculate weights

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Interpolant Computation

• Compute mean vector
• Calculate weights
• Sum over all triangles

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Implementation Considerations

• Special cases
• on boundary
• Numerical stability
• Small spherical triangles
• Large meshes
• Pseudo-code provided in paper

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ApplicationsBoundary Value Problems
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ApplicationsSolid Textures

• Extend texture to interior

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ApplicationsSurface Deformation
Control Mesh Surface Computing Weights Deformation
216 triangles 30,000 triangles 1.9 seconds 0.03 seconds
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ApplicationsSurface Deformation
Control Mesh Surface Computing Weights Deformation
98 triangles 96,966 triangles 3.3 seconds 0.09 seconds
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Summary

• Extend function from boundary to interior
• Closed-form solution for triangle meshes
• Numerically stable evaluation
• Extends to arbitrary smooth
• surfaces
• Useful for surface deformation

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Current Work

• General constructions for convex simplicial
  polytopes
• Polygons (2D), triangular meshes (3D),
• Include all barycentric coordiantes
• Cooridnates parameterized by a generating shape
• Wachspress (polar dual)
• Mean value (sphere)
• Discrete harmonics (original polytope)

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Current Work

• General constructions for continuous shapes in
  any dimensions
• Computes a family of barycentric coordinates
• Wachspress
• Mean value
• Discrete polyhedra are simply a special case with
  piecewise-linear functions

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Open Topics MVC and MLS

• Moving least squares
• Guarantees linear precision using linear
  functions
• Allows the control shape to be non-closed
  geometry, even points and edges
• Coordinates are expressed in integrals, but no
  explicit formulation exists

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Open Topics - Deformation

• Localized control with MVC
• Control shape has local effect on the model
• Save CPU time by using fewer weights
• Automatic skinning
• Generate control shapes from an arbitrary model
• Skeleton-driven deformation
• Deforming shape using curve skeletons
• Generalized cylinders as control shape