Mesh Parameterization: Theory and Practice

1. Mesh Parameterization:Theory and Practice Non-Planar Domains

2. Limitations of planar domains • so far … parameter domain = topological disk • one connected component • one boundary • parameterization bijective ⇒ surface = topologicaldisk • what about other surfaces? Mesh Parameterization: Theory and PracticeNon-Planar Domains

3. Texture atlases: distortion or seams? seams distortion Mesh Parameterization: Theory and PracticeNon-Planar Domains

4. Beyond planar domains • alternative: adapt the parameter domain • same topology as the mesh • base complexes • simplified triangle mesh • spherical domains • limited to genus-zero meshes • polycubes • quadrilateral domain elements Mesh Parameterization: Theory and PracticeNon-Planar Domains

5. Generating base complexes • surface triangulation of seed points [Eck et al. 1995] • successive simplification[Lee et al. 1998] Mesh Parameterization: Theory and PracticeNon-Planar Domains

6. Computing the parameterization • initial parameterization • parameter points for mesh vertices • inherit correspondences during simplification • piecewise linear map per mesh triangle • optimization • Loop smoothing • global minimization of distortionwith transition functions[Khodakovsky et al. 2003] Mesh Parameterization: Theory and PracticeNon-Planar Domains

7. Applications and limitations • applications • remeshing • compression • surface fitting • morphing • limitations • not good for texture mapping • where to store the color data? Mesh Parameterization: Theory and PracticeNon-Planar Domains

8. Spherical parameterizations • projected Gauss-Seidel iterations[Kobbelt et al. 1999] • project all points onto sphere • compute barycentric average • reproject onto sphere • problems • does not guarantee bijectivity • diverges close to solution [Saba et al. 2005] • solution • spherical barycentric coordinates [Gotsman et al. 2003] Mesh Parameterization: Theory and PracticeNon-Planar Domains

9. Alternatives • successive simplification [Shapiro & Tal 1998] [Praun & Hoppe 2003] Mesh Parameterization: Theory and PracticeNon-Planar Domains

10. Applications and limitations • applications • remeshing [Praun & Hoppe 2003] • compression, morphing, … • cube maps • texture mapping • limitations • only spherical meshes Mesh Parameterization: Theory and PracticeNon-Planar Domains

11. Polycubes • polycubes as parameter domains [Tarini et al. 2004] • square domain elements • matching topology • similar coarse shape • not too many elements • construction • interactively [Tarini et al. 2004] • automatic [Lin et al. 2008] Po·ly·cube: n. (Geom.) A solid composed of multiple unit cubes attached face to face Mesh Parameterization: Theory and PracticeNon-Planar Domains

12. projection MIPS Area-MIPS Polycube-maps • computing the parameterization • initial projection onto the polycube • global optimization (Gauss-Seidel iterations) • applications • quadrilateral remeshing • texture mapping • shading textures • level-of-detail rendering Mesh Parameterization: Theory and PracticeNon-Planar Domains

13. not necessarily on the polycube surface: project a fragment with interpolated texture coord final texel value for the fragment map to 2D plus a tiny structure to store polycube layout Texture mapping with Polycube-maps object space texture space (3D!) stored in texture RAM w v u a packed texture image mesh polycube Mesh Parameterization: Theory and PracticeNon-Planar Domains

14. Summary • non-planar domains • base complexes • spherical domains • polycubes • applications • remeshing • texture mapping • morphing Mesh Parameterization: Theory and PracticeNon-Planar Domains