Transcript Mean Value Coordinates for Closed Triangular Meshes

Mean Value Coordinates for
Closed Triangular Meshes
Tao Ju
Scott Schaefer
Joe Warren
Overview
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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.
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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
anticlockwise ordering Around
.The points
-shaped polygon with in its kernel.
Our Aim is to study sets of weights such that
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arranged in an
form a star
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Wachspress[75]
Shortcoming:
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Mean value coordinates[03]
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 :
wp

v
w
i
i
i
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i
i
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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
wp

v
w
i
i
i
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 w (p
i
i
i
 v)  0
i
Project surface onto sphere
centered at v
m = mean vector (integral of unit
normal over spherical triangle)
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m   wk ( pk  v)
k 1
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
i
Symmetry:

j
mj  0
 w ( p  v)  0
i
i
i
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MVC for closed meshes
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Given spherical triangle, compute mean
vector m (integral of unit normal)
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Build wedge with face normals
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Apply Stokes’ Theorem,
nk
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1
 k nk  m  0

k 1 2
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MVC for closed meshes
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Compute mean vector:
Calculate weights
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By m 
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w (p
k 1
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k
k
1
k nk  m  0

k 1 2
 v)
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
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Boundary value interpolation
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Volumetric textures
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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
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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!
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