Intrinsic Parameterization for Surface Meshes

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Intrinsic Parameterization for Surface Meshes

• Mathieu Desbrun, Mark Meyer, Pierre Alliez
• CS598MJG Presented by Wei-Wen Feng
• 2004/10/5

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Whats Parameterization?

• Find a mapping between original surface and a
  target domain ( Planar in general )

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What does it do?

• Most significant Texture Mapping
• Other applications include remeshing, morphing,
  etc.

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Two Directions in Research

• Define metric (energy) measuring distortion
• Minimize the energy to find mapping
• This papers main contribution

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Two Directions in Research

• Using the metric, and make it work on mesh
• Cut mesh into patches
• Considering arbitrary genus

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Outline

• Previous Work
• Intrinsic Properties
• DCP DAP
• Boundary Control
• Future Work

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Previous Work

• Discrete Harmonic Map (Eck. 95)
• Minimize Eharmh ½ SKi,j h(i) h(j)2
• K Spring constant
• The same as minimize Dirichlet energy

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Previous Work

• Shape Preserving Param. (Floater. 97)
• Represent vertex as convex combination of
  neigobors
• Trivial choice barycenter of neighbors
• Ensure valid embedding

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Previous Work

• Most Isometric Param. (MIPS) (K. Hormann . 99)
• Doesnt need to fix boundary
• Conformal but need to minimize non-linear energy

MIPS
Harmonic Map
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Previous Work

• Signal Specialized Param. (Sander. 02)
• Minimize signal stretch on the surface when
  reconstruct from parametrization

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Intrinsic Parameterization

• Motivation
• Find good distortion measure only depending on
  the intrinsic properties of mesh
• Develop good tools for fast parameterization
  design

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Intrinsic Properties

• Defined at discrete suraface, restricted at
  1-ring
• Notion
• F Return the score of surface patch M
• E(M,U) Distortion between mapping
• Intrinsic Properties
• Rotation Translation Invariance
• Continuity Converge to continuous surface
• Additivity f (A) f (B) f (A?B) f (A?B)

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Intrinsic Properties

• Minkowski Functional
• fA Area
• fc Euler characteristic
• fP Perimeter
• From Hadwiger, the only admissible intrinsic
  functional is
• f a fA b fc c fP

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Discrete Conformal Param.

• Measure of Area (Dirichlet Energy)
• Conformality is attained when Dirichlet energy is
  minimum
• When fixed boundary, it is in fact discrete
  harmonic map

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Discrete Authalic Param.

• Measure of Euler characteristic (Angle)
• Integral of Gaussian curvature
• Derived as Chi Energy

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Comparing DCP DAP

• DCP (Dirichlet Energy)
• Measure area extension
• Minimized when angles preserved
• DAP (Chi Energy)
• Measure angle excess
• Minimized when area preserved

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Solving Parametrization

• General distortion measure
• Fix the boundary, minimized the energy
• Very sparse linear systems ? Conjugate gradient

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Natural Boundary

• Instead fixed the boundary, solve for optimal
  conformal mapping which yields best boundary.
• For interior points
• For boundary points
• Constrain two points to avoid degeneracy.

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Compare with LSCM

• Least Square Conformal Map (Levy. 02)
• Start from Cauchy-Riemann Equation
• Theoretically equivalent to Natural Boundary Map
• Minimize conformal energy
• Natural Conformal Map
• Imposing boundary constraint for boundary points

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Extend to non-linear func.

• All parametrization could be expressed as
• U l UA (1-l) Uc
• Substitute U in a non-linear function reduces the
  problem into solving l
• Ex
• Could be reduced into root finding

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Boundary Control

• Precompute the impulse response
  parameterization for each boundary points
• New parameterization could be obtained by
  projecting boundary parameter onto its impulse
  response parameterization

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Boundary Optimization

• Minimized arbitrary energy with respect to
  boundary parameterization
• Using precomputed gradient to accelerate
  optimization

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Summary of Contributions

• A linear system solution for Natural Conformal
  Map
• A new geometric metric for parameterization (DAP)
• Real-time boundary control for better
  parameterization design

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Whats Next ?

• Mean Value Coordinate (Floater. 03)
• The same property of convex combination
• Approximating Harmonic Map but ensure a valid
  embedding

Tutte
Harmonic
Shape Preserving
Mean Value
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Whats Next ?

• Spherical Parameterization (Praun. 03)
• Smooth parameterization for genus-0 model
• Using existing metric

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Conclusion

• There seems to be less paper directly about
  finding metrics (or find a better way to model
  them) for parameterization.
• Now more efforts in finding globally smooth
  parameterization on arbitrary meshes

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• Thank You

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References

• (Eck. 95) Multiresolution Analysis of Arbitrary
  Meshes. Proceedings of SIGGRAPH 95\
• (Floater. 97) Parametrization and Smooth
  Approximationof Surface Triangulations. Computer
  Aided Geometric Design 14, 3 (1997)
• (K. Hormann . 99) MIPS An Efficient Global
  Parametrization Method. In Curve and Surface
  Design Saint-Malo 1999 (2000)
• (Sander. 02) Signal-Specialized
  Parameterization. In Eurographics Workshop on
  Rendering, 2002.

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References

• (Floater, Hormann 03) Surface Parameterization
  A Tutorial and Survey
• (Levy. 02) Least Squares Conformal Maps for
  Automatic Texture Atlas Generation. ACM SIGGRAPH
  Proceedings
• (Floater. 03) Mean Value Coordinates. Computer
  Aided Geometric Design 20, 2003