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Mesh Parameterization: Theory and Practice

Mesh Parameterization: Theory and Practice. Barycentric Mappings. Triangle Mesh Parameterization. triangle mesh vertices triangles parameter mesh parameter points parameter triangles parameterization piecewise linear map . The Spring Model. replace edges by springs

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Mesh Parameterization: Theory and Practice

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  1. Mesh Parameterization:Theory and Practice Barycentric Mappings

  2. Triangle Mesh Parameterization • triangle mesh • vertices • triangles • parameter mesh • parameter points • parameter triangles • parameterization • piecewise linear map Mesh Parameterization: Theory and PracticeBarycentric Mappings

  3. The Spring Model • replace edges by springs • fix boundary vertices • relaxation process • energy of spring between and : • spring constant • spring length • total energy Mesh Parameterization: Theory and PracticeBarycentric Mappings

  4. Energy Minimization • interior vertices • ’s neighbours • overall spring energy • partial derivative Mesh Parameterization: Theory and PracticeBarycentric Mappings

  5. Energy Minimization • minimum of spring energy for all interior points • is a convex combination of its neighbors with weights Mesh Parameterization: Theory and PracticeBarycentric Mappings

  6. The Linear System • separation of variables unknown parameter points fixed • linear system Mesh Parameterization: Theory and PracticeBarycentric Mappings

  7. The Linear System • solve system twice for and coordinates of interior parameter points • matrix is • sparse • diagonally dominant • nonsingular as long as all Mesh Parameterization: Theory and PracticeBarycentric Mappings

  8. Choice of Weights • uniform spring constants • , • chordal spring constants • , • no fold-overs for convex boundary • no linear reproduction • planar meshes are distorted Mesh Parameterization: Theory and PracticeBarycentric Mappings

  9. Choice of Weights • suppose is a planar mesh • specify weights such that • barycentric coordinates of • then solving reproduces Mesh Parameterization: Theory and PracticeBarycentric Mappings

  10. Barycentric Coordinates • Wachspress coordinates • discrete harmonic coordinates • mean value coordinates normalization Mesh Parameterization: Theory and PracticeBarycentric Mappings

  11. Example – Pyramid • fold-overs for negative coordinates • affine combinations , • numerically unstable if • mean value coordinates guaranteed to be positive Wachspress discrete harmonic mean value Mesh Parameterization: Theory and PracticeBarycentric Mappings

  12. The Boundary Mapping • chordal parameterization around convex shape • circle • rectangle • projection into least squares plane • may lead to fold-overs Mesh Parameterization: Theory and PracticeBarycentric Mappings

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