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

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Presentation on theme: "Mean Value Coordinates for Closed Triangular Meshes Tao Ju Scott Schaefer Joe Warren."— Presentation transcript:

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

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

3 3 JZP 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 4 JZP 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 5 JZP Illustration

6 6 JZP 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 7 JZP Wachspress[75] Shortcoming:

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

9 9 JZP Example Pole(divsions by 0) No Pole

10 10 JZP Mean value interpolation Discrete :

11 11 JZP Mean value interpolation Continuous:

12 12 JZP Important properties

13 13 JZP MVC for closed polygons

14 14 JZP MVC for closed polygons

15 15 JZP MVC for closed polygons

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

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

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

19 19 JZP Algorithm

20 20 JZP Robust algorithm

21 21 JZP Pseudo-code

22 22 JZP Applications Boundary value interpolation Volumetric textures Surface Deformation

23 23 JZP Boundary value interpolation

24 24 JZP Volumetric textures

25 25 JZP Surface Deformation

26 26 JZP 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 27 JZP Thanks all!


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