Dr. Scott Schaefer

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Surface Parameterization

• Dr. Scott Schaefer

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Parameterization and Texturing
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Mercator Projection
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Mercator Projection
Image taken from http//idvux.spaces.live.com
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Types of Distortion

• Isometric Mappings
• Preserve lengths
• Conformal Mappings
• Preserve angles
• Equiareal Mappings
• Preserve area

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Mathematics of Parameterization
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Mathematics of Parameterization
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Mathematics of Parameterization
singular values
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Mathematics of Parameterization
conformal
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Mathematics of Parameterization
equiareal
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Mathematics of Parameterization
isometric
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Conformal and Harmonic Mappings
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Conformal and Harmonic Mappings
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Conformal and Harmonic Mappings
Conformal gt Harmonic
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Harmonic Maps
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Harmonic Maps
ith row contains cotan weights for ith vertex
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Mean Value Map
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Mean Value Map
Always positive!
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Mean Value Map
ith row contains mean value weights for ith vertex
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Comparison
Original
Barycenter
Harmonic
Mean value
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
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Conformal Maps
Sum over all triangles Linear system of equations!
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Conformal Maps
Part of discrete Laplacian
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Iterative Optimization
Input 3D mesh
Repeat
Output 2D parameterization
Image taken from A Local/Global Approach to Mesh
Parameterization
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Conclusions

• Many, many, many methods exist for
  parameterization
• Not all guarantee 1-to-1 mappings (fold-overs)
• Most measure some degree of conformality
• Need a balance between
  distortions