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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Overview

• About the authors
• About the work
• Previous works
• MVC for closed polygons
• MVC for closed meshes
• Algorithm
• Applications
• Conclusions and future work

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About author

• Tao Ju M.S. and Ph.D Rice
  UniversityAdvisor, Dr. Joe Warren B.S.
  Tsinghua University Research interests lie
  in the field of computer graphics and its
  applications in bio-medical research.

• Scott Schaefer M.S. Computer Science Rice
  University 2003 Currently a Ph.D. B.S.
  Computer Science and Mathematics Trinity
  University 2000 Research interests lie in
  the field of computer graphics

• Joe Warren Department of Computer
  Science6100 South MainRice niversity
  Research interests are centered around the
  general problem of representing geometric
  shape.

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About the work

• Given a closed Triangular Mesh,
• construct a function that interpolates a set of
• values defined at vertices of the mesh.
• Parameterize the interior points of the mesh.

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Illustration
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Previous works
Let be points in the plane
with arranged in an anticlockwise
ordering Around .The points
form a star -shaped polygon with in its
kernel. Our Aim is to study sets of weights such
that
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Wachspress75
Shortcoming
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Mean value coordinates03
Mean value theorem for harmonic functions
Mean value coordinates
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Example
Pole(divsions by 0)
No Pole
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Mean value interpolation
Discrete
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Mean value interpolation
Continuous
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Important properties
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MVC for closed polygons
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MVC for closed polygons
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MVC for closed polygons
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MVC for closed meshes

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

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MVC for closed meshes

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

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MVC for closed meshes

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

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Algorithm
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Robust algorithm
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Pseudo-code
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Applications

• Boundary value interpolation
• Volumetric textures
• Surface Deformation

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Boundary value interpolation
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Volumetric textures
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Surface Deformation
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Conclusions and Future work

• Mean value coordinates are a simple,but powerful
  method for creating functions that interpolate
  values assigned to vertices of a closed mesh.
• One important generalization would be to derive
  mean value coordinates for piecewise linear mesh
  with arbitrary closed polygons as faces.

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Thanks all!