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

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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

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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.

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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.

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5 JZP Illustration

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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

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7 JZP Wachspress[75] Shortcoming:

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

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9 JZP Example Pole(divsions by 0) No Pole

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10 JZP Mean value interpolation Discrete :

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11 JZP Mean value interpolation Continuous:

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12 JZP Important properties

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13 JZP MVC for closed polygons

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14 JZP MVC for closed polygons

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15 JZP MVC for closed polygons

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

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

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

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19 JZP Algorithm

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20 JZP Robust algorithm

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21 JZP Pseudo-code

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22 JZP Applications Boundary value interpolation Volumetric textures Surface Deformation

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23 JZP Boundary value interpolation

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24 JZP Volumetric textures

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25 JZP Surface Deformation

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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.

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27 JZP Thanks all!

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