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

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

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

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

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

6. Mesh Parameterization: Theory and Practice Non-Planar Domains 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 distortion with transition functions [Khodakovsky et al. 2003]

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

8. Mesh Parameterization: Theory and Practice Non-Planar Domains 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]

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

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

11. Mesh Parameterization: Theory and Practice Non-Planar Domains Polycubes Po·ly·cube: n. (Geom.) A solid composed of multiple unit cubes attached face to face 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]

12. Mesh Parameterization: Theory and Practice Non-Planar Domains 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 projectionMIPS

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

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