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Mean Value Coordinates for Closed Triangular Meshes

Mean Value Coordinates for Closed Triangular Meshes. Tao Ju Scott Schaefer Joe Warren. Overview. About the authors About the work Previous works MVC for closed polygons MVC for closed meshes Algorithm Applications Conclusions and future work. About author.

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Mean Value Coordinates for Closed Triangular Meshes

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  1. Mean Value Coordinates for Closed Triangular Meshes Tao Ju Scott Schaefer Joe Warren

  2. Overview • About the authors • About the work • Previous works • MVC for closed polygons • MVC for closed meshes • Algorithm • Applications • Conclusions and future work

  3. About author • Tao Ju: M.S. and Ph.D Rice UniversityAdvisor, Dr. Joe Warren B.S. Tsinghua University Research interests lie in the field of computer graphics and its applications in bio-medical research. • Scott Schaefer: M.S. Computer Science Rice University 2003 Currently a Ph.D. B.S. Computer Science and Mathematics Trinity University 2000 Research interests lie in the field of computer graphics • Joe Warren: Department of Computer Science6100 South MainRice niversity Research interests are centered around the general problem of representing geometric shape.

  4. About the work Given a closed Triangular Mesh, construct a function that interpolates a set of values defined at vertices of the mesh. Parameterize the interior points of the mesh.

  5. Illustration

  6. Previous works Let be points in the plane with arranged in an anticlockwise ordering Around .The points form a star -shaped polygon with in its kernel. Our Aim is to study sets of weights such that

  7. Wachspress[75] Shortcoming:

  8. Mean value coordinates[03] Mean value theorem for harmonic functions Mean value coordinates

  9. Example Pole(divsions by 0) No Pole

  10. Mean value interpolation Discrete :

  11. Mean value interpolation Continuous:

  12. Important properties

  13. MVC for closed polygons

  14. MVC for closed polygons

  15. MVC for closed polygons

  16. MVC for closed meshes • Project surface onto sphere centered at v • m = mean vector (integral of unit normal over spherical triangle) • Symmetry:

  17. Given spherical triangle, compute mean vector (integral of unit normal) Build wedge with face normals Apply Stokes’Theorem, MVC for closed meshes

  18. Compute mean vector: Calculate weights By Sum over all triangles MVC for closed meshes

  19. Algorithm

  20. Robust algorithm

  21. Pseudo-code

  22. Applications • Boundary value interpolation • Volumetric textures • Surface Deformation

  23. Boundary value interpolation

  24. Volumetric textures

  25. Surface Deformation

  26. Conclusions and Future work • Mean value coordinates are a simple,but powerful method for creating functions that interpolate values assigned to vertices of a closed mesh. • One important generalization would be to derive mean value coordinates for piecewise linear mesh with arbitrary closed polygons as faces.

  27. Thanks all!

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