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 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 Symmetry:
Project surface onto sphere centered at v
m = mean vector (integral of unit normal over spherical triangle)
Symmetry:

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

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

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!