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Mesh Parameterization: Theory and Practice

Learn the theory and practice of mesh parameterization including barycentric mappings, triangle mesh, vertices, triangles, piecewise linear map, energy minimization, spring model, linear system, weights choice, linear reproduction, and example applications. Dive into techniques like Wachspress, discrete harmonic, mean value coordinates, and boundary mapping. Understand how to solve linear systems, choose appropriate weights, and avoid distortions in planar meshes.

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Mesh Parameterization: Theory and Practice

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  1. Siggraph CourseMesh Parameterization: Theory and Practice Barycentric Mappings

  2. Triangle Mesh Parameterization • triangle mesh • vertices • triangles • parameter mesh • parameter points • parameter triangles • parameterization • piecewise linear map

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

  4. Energy Minimization • interior vertices • ’s neighbours • overall spring energy • partial derivative

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

  6. The Linear System • separation of variables unknown parameter points fixed • linear system

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

  8. Choice of Weights • uniform spring constants • , • chordal spring constants • , • no fold-overs for convex boundary • no linear reproduction • planar meshes are distorted

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

  10. Barycentric Coordinates • Wachspress coordinates • discrete harmonic coordinates • mean value coordinates normalization

  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

  12. The Boundary Mapping • chordal parameterization around convex shape • circle • rectangle • projection into least squares plane • may lead to fold-overs

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