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Mean Value Coordinates for Closed Triangular Meshes Scott Schaefer, Tao Ju, Joe Warren (Rice University) Presented in SIGGRAPH2005 Outline Abstract Preliminaries Previous work Mean value Interpolation 3D Mean value

  1. “Mean Value Coordinates for Closed Triangular Meshes” Scott Schaefer, Tao Ju, Joe Warren (Rice University) Presented in SIGGRAPH’2005

  2. Outline � Abstract � Preliminaries � Previous work � Mean value Interpolation � 3D Mean value coordinates for closed triangular meshes � Applications � Questions

  3. Abstract � Search for a function that can interpolate a set of values at the vertices of a mesh smoothly into its interior � Mean value coordinates have been used as an interpolant for closed 2D polygons.

  4. Abstract � This paper generalizes the mean value coordinates to closed triangular meshes � Interesting applications to surface deformation and volumetric textures

  5. Mean Value Theorem � Wikipedia :

  6. Harmonic functions and Mean Value property � A harmonic function is twice continuously differentiable function f: U->R which satisfies the laplace’s equation

  7. Harmonic functions and Mean Value property � They attain there maxima/minima only at the boundaries. � Let B(x,r) be a ball with center x and radius r, contained totally in U,

  8. Harmonic functions and Mean Value property � Then,

  9. Barycentric Coordinates (Mobius, 1827) � Given find weights such that

  10. Boundary Value Interpolation

  11. Boundary Value Interpolation

  12. Boundary Value Interpolation

  13. Previous work: Wachpress’s solution (1975)

  14. Barycentric coordinates for arbitrary polygons in the plane

  15. Floater : Mean Value Coordinates � These weights were derived by application of mean value theorem for harmonic functions. � They depend smoothly on the vertices

  16. Previous Work

  17. Previous Work

  18. Previous Work

  19. Previous Work

  21. Continuous Barycentric Coordinates

  22. Continuous Barycentric Coordinates

  23. Mean Value Interpolation � Continuous form of mean value coordinates � Consider evaluation of the numerator

  24. Mean Value Interpolation

  25. Mean Value Interpolation

  26. Mean Value Interpolation � Project the function f[x] onto the boundary of this circle

  27. Mean Value Interpolation � Integrate the projected function divided by (p[x]-v) over the circle S v and then normalize.

  28. Mean Value Interpolation

  29. Mean Value Interpolation

  30. Mean Value Interpolation

  31. Relation to Discrete Coordinates

  32. Relation to Discrete Coordinates

  33. 3D Mean Value Coordinates � Find weigths w i which allow us to represent any v as a weighted combination of the vertices of a closed triangular mesh and satisfy mean value interpolation

  34. 3D Mean Value Coordinates � Given a triangular mesh and a vertex v in its interior � Consider a unit sphere centered at vertex v

  35. 3D Mean Value Coordinates � Project the mesh onto the surface of the sphere � Planar triangles -> spherical triangles

  36. 3D Mean Value Coordinates � Define m as the mean vector = integral of unit normal over spherical triangle

  37. 3D Mean Value Coordinates � Given m, represent it as a weighted combination of the vertex v to the vertices p k of the triangle

  38. 3D Mean Value Coordinates

  39. Computing The Mean Vector

  40. Computing The Mean Vector

  41. Computing The Mean Vector

  42. Computing The Mean Vector

  43. Interpolant Computation

  44. Interpolant Computation

  45. Interpolant Computation

  46. Implementation Considerations

  47. Application: Surface Deformation

  48. Application: Surface Deformation

  49. Application: Surface Deformation

  50. Application: Surface Deformation

  51. Applications Boundary Value Problems

  52. Applications Solid Textures

  53. Applications Surface Deformation

  54. Applications Surface Deformation

  55. Applications Surface Deformation

  56. Summary

  57. Thank You � Questions?

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