1. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Direct Quad-Dominated Anisotropic Remeshing Martin Marinov and Leif Kobbelt

2. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt CAD meshes

3. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Meshes from scanned data

4. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing Goal: Produce CAD quality mesh elements

5. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work – Remeshing algorithms – Anisotropic alignment Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing Contributions and results

6. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Connectivity Regularization [Kobbelt et al.'99], [Surazhsky&Gotsman'03]

7. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Isotropic Remeshing [Alliez et al.'03], [Surazhsky et al.'03], [Botsch&Kobbelt'04]

8. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Normal Noise Reduction [Botsch&Kobbelt'01]

9. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Variational Shape Approximation [Cohen-Steiner et al.'04]

10. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Remeshing algorithms Anisotropic Remeshing [Alliez et al.'03]

11. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Anisotropic alignment Approximation properties: – Optimal L 2 [Nadler'86] and L p [Simpson'91] approximation (except for hyperbolic regions) – Optimal normal field approximation (L 2,1 metric)[Cohen-Steiner et al.'04]

12. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Anisotropic elements in CAD Designers and engineers compose models using combinations of anisotropic and isotropic objects Quad elements reflect the symmetry of the shape

13. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Lines of principal curvature Visual perception of the shape: Line-art drawing [Bradly et al.'85] [Elber'98] [Hertzmann&Zorin'00] [Rössl&Kobbelt'01]

14. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing Results

15. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature tensor field estimation Minimum principal curvature directionsMaximum principal curvature directions

16. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature lines sampling Minimum curvature linesMaximum curvature lines

17. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Meshing Minimum and maximum curvature lines intersectionAnisotropic remesh

18. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation – Local tensor flattening – Confidence estimation – Filtering Curvature lines integration Meshing Results

19. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature tensor field estimation Integrate edge tensors over a surface area B [Cohen-Steiner&Morvan'03]

20. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Local tensor flattening Flattening Barycentric interpolation

21. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Umbilic points Trisector Wedge Spherical or flat point on the tensor field '92 [Delmarcelle&Hesselink'92], [Tricoche'02]

22. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Tensor field singularities Where the estimated curvature directions are reliable?

23. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Confidence estimation High confidence Low confidence

24. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Tensor field filtering

25. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration – Local parameterization approach – Seeding and density estimation – Proximity queries – Line snapping Meshing Results

26. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Curvature line integration Current sample Integrate the next sample parameterization Next sample Current sample parameterization

27. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Local parameterization

28. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Seeding and density estimation – Placing seeds in the most confident regions – Minimum curvature lines samples seed maximum curvature lines and vice versa – Lines density depends on the local curvature estimation and the user-specified approx. tolerance

29. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Proximity queries Sampling density P-cell Lines' samples are associated with the original mesh faces

30. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt P-cell evolution P-cell contents maintenance: – Include adjacent faces which are inside the local sampling density r – Remove already included faces which are now outside r

31. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Isotropic regions Flat, spherical and transition regions Low confidence: Curvature directions not reliable Curvature line-based sampling senseless

32. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Line snapping

33. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing – Vertices, edges and faces construction Results

34. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Vertices construction Lines intersections constitute the new mesh vertices Use sample groups corresponding to the original mesh faces for effective computation

35. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Edges and faces construction Meshing directly in 3D: – Edges are defined between subsequent line's intersections – Use vertex normal information obtained on the original surface – Halfedge structure construction – Convex partitioning of concave faces

36. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Overview Previous work Anisotropic remeshing overview Curvature tensor field estimation Curvature lines integration Meshing Contributions and results

37. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Direction propagation into isotropic regions – Confidence estimation – Confidence-based filtering – Line-snapping

38. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Local parameterization approach – Arbitrary genus meshes handling – Reduces computational burden and simplifies implementation

39. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Contributions and results Proximity queries algorithm – Dramatically improves the scalability and the performance – Simple data structures

40. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Comparison to Alliez et al.'03 Alliez et al.'03 – Seeding at umbilic points – Point based sampling in isotropic regions and CDT meshing – Global parameterization – CDT proximity queries Marinov&Kobbelt'04 – Seeding in anisotropic regions – Geodesic line sampling in isotropic regions, quad- dominated meshing – Local parameterization – Fast proximity queries based on the original mesh connectivity

41. Informatik VIII Computer Graphics & Multimedia Martin Marinov and Leif Kobbelt Future work Improving the distribution of the integrated curvature lines – Global uniformity constraints – Multiresolution approach Improved approximation – Using asymptotic instead of principal directions in hyperbolic regions