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Mean Value Coordinates for Closed Triangular Meshes

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Presentation on theme: "Mean Value Coordinates for Closed Triangular Meshes"— Presentation transcript:

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

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 weights
By 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!

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