Mesh Parameterization: Theory and Practice

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Mesh ParameterizationTheory and Practice

• Barycentric Mappings

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Triangle Mesh Parameterization

• triangle mesh
• vertices
• triangles
• parameter mesh
• parameter points
• parameter triangles
• parameterization
• piecewise linear map

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The Spring Model

• replace edges by springs
• fix boundary vertices
• relaxation process
• energy of spring between and
• spring constant
• spring length
• total energy

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Energy Minimization

• interior vertices
• s neighbours
• overall spring energy
• partial derivative

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Energy Minimization

• minimum of spring energy
• for all interior points
• is a convex combination of its neighbors
• with weights

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The Linear System

• separation of variables
• unknown parameter points fixed
• linear system

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The Linear System

• solve system twice
• for and coordinates of interior
  parameter points
• matrix is
• sparse
• diagonally dominant
• nonsingular
• as long as all

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Choice of Weights

• uniform spring constants
• ,
• chordal spring constants
• ,
• no fold-overs for convex boundary
• no linear reproduction
• planar meshes are distorted

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Choice of Weights

• suppose is a planar mesh
• specify weights such that
• barycentric coordinates of
• then solving
• reproduces

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Barycentric Coordinates

• Wachspress coordinates
• discrete harmonic coordinates
• mean value coordinates

normalization
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Example Pyramid
Wachspress
discrete harmonic
mean value

• fold-overs for negative coordinates
• affine combinations
  ,
• numerically unstable if
• mean value coordinates guaranteed to be positive

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The Boundary Mapping

• chordal parameterization around convex shape
• circle
• rectangle
• projection into least squares plane
• may lead to fold-overs