Consistent Parameterizations

1. Consistent Parameterizations Arul Asirvatham Committee Members Emil Praun Hugues Hoppe Peter Shirley

2. Parameterization • Mapping from a domain (plane, sphere, simplicial complex) to surface • Motivation: Texture mapping, surface reconstruction, remeshing …

3. Desirable Properties • One-to-one • Minimize some measure of distortion • Length preserving • Angle preserving • Area preserving • Stretch minimizing

4. Outline • Background • Commonly used Domains • Plane, Simplicial Complex, Sphere • Constrained Parameterizations • Consistent Parameterizations • Consistent Spherical Parameterizations • Inter-Surface Mapping • Summary and future work

5. Planar Parameterizations • Convex combination maps • p = i pi , i=1,…,n i =1 • [Tutte 63] • [Floater 97] • [Floater et al 03] • Conformal Maps • [Sheffer et al 01] • [Levy et al 02] • [Desbrun et al 02] • Stretch preserving maps • [Sander et al 01] g G

6. Simplicial Parameterizations • Planar parameterization techniques cut surface into disk like charts • Use domain of same topology • Work for arbitrary genus • Discontinuity along base domain edges [Eck et al 95, Lee et al 00, Guskov et al 00, Praun et al 01, Khodakovsky et al 03]

7. Spherical Parameterization • No cuts  less distortion • Restricted to genus zero meshes • [Shapiro et al 98] • [Alexa et al 00] • [Sheffer et al 00] • [Haker et al 00] • [Gu et al 03] • [Gotsman et al 03] • [Praun et al 03]

8. Constrained Parameterizations • Texture mapping [Levy et al 01, Eckstein et al 01, Kraevoy et al 03]

9. Consistent Parameterizations Base Domain DGP Applications • Motivation • Digital geometry processing • Morphing • Attribute transfer • Principal component analysis [Alexa 00, Levy et al 99, Praun et al 01] Input Meshes with Features Semi-Regular Meshes

10. Contributions • Consistent Spherical Parameterizations • Inter-surface maps

11. Consistent Spherical Parameterizations

12. Stretch Minimizing Spherical Parameterization [Praun & Hoppe 03] • Use multiresolution • Convert model to progressive mesh format • Map base tetrahedron to sphere • Add vertices one by one, maintaining valid embedding and minimizing stretch

13. Stretch Metric [Sander et al. 2001] linear map singular values: γ , Γ g G 2D texture domain surface in 3D

14. Conformal vs Stretch Conformal metric: can lead to undersampling Stretch Conformal Stretch metric encourages feature correspondence Conformal

15. Approach • Find “good” spherical locations • Use spherical parameterization of one model • Assymetric • Obtain spherical locations using all models • Constrained spherical parameterization • Create base mesh containing only feature vertices • Refine coarse-to-fine • Fix spherical locations of features

16. Finding spherical locations

17. Algorithm UCSP + step 1 CSP step 4 UCSP CSP step 5 step 2 step 3 step 6 • Find initial spherical locations using 1 model • Parameterize all models using those locations • Use spherical parameterizations to obtain remeshes • Concatenate to single mesh • Find good feature locations using all models • Compute final parameterizations using these locations

18. Constrained Spherical Parameterization

19. Approach

20. Consistent Partitioning • Compute shortest paths (possibly introducing Steiner vertices) • Add paths not violating legality conditions • Paths (and arcs) don’t intersect • Consistent neighbor ordering • Cycles don’t enclose unconnected vertices • First build spanning tree

21. Swirls • Unnecessarily long paths

22. Heuristics to avoid swirls • Insert paths in increasing order of length • Link extreme vertices first • Disallow spherical triangles with any angle < 10o • Sidedness test • Unswirl operator • Edge flips

23. Sidedness test E C E D B D B C A A

24. Morphing [Praun et al 03]

25. Morphing

26. Morphing

27. Attribute Transfer + Color Geometry

28. Attribute Transfer + Color Geometry

29. Face Database avg =

30. Timing • 2.4 GHz Pentinum 4 PC, 512 MB RAM

31. Inter Surface Maps

32. Introduction • No intermediate domain • Reduced distortion • Natural alignment of features

33. Comparison to CSP • No intermediate domain • Arbitrary genus • Limited to 2 models • Applications • Morphing • Digital geometry processing • Transfer of surface attributes • Deformation transfer

34. Contributions • Directly create inter-surface map • Symmetric coarse-to-fine optimization • Symmetric stretch metric  Automatic geometric feature alignment • Robust • Very little user input • Arbitrary genus • Hard constraints

35. Algorithm Overview • Consistent mesh partitioning • Constrained Simplification • Trivial map between base meshes • Coarse-to-fine optimization

36. Consistent Mesh Partitioning • Compute matching shortest paths (possibly introducing Steiner vertices) • Add paths not violating legality conditions

37. Legality Conditions • Paths don’t intersect • Consistent neighbor ordering • Cycles don’t enclose unconnected vertices • First build maximal graph without sep cycles • genus 0: spanning tree • genus > 0: spanning tree + 2g non-sep cycles

38. Separating/Non-separating cycles • Separating cycle breaks surface into 2 disjoint components Separating cycle Non separating cycle

39. Non-separating cycle test • Grow 2 fronts starting on both sides of AB • Non-separating if fronts meet B A

40. Tracing non separating cycle • Shortest path between AC is separating A C B

41. Tracing non separating cycle • Grow contour around AC • Contour wraps around and meets itself at O O A C B

42. Tracing non separating cycle • Trace paths from O to A and C O A C B

43. Automatic Insertion Of Feature Points Add features if not enough to resolve genus

44. Genus-0 example

45. Genus-1 example

46. Genus-2 example

47. Contributions • Consistent Spherical Parameterizations for several genus-zero surfaces • Robust method for Constrained Spherical Parameterization • Robust partitioning of two meshes of arbitrary genus • Methods to avoid swirls and to correct them when they arise

48. Future Work • Improve overall exectution time • Multiresolution path tracing algorithm • Linear stretch optimization • Construct maps between surfaces of different genus • Handle point cloud and volumetric data

49. Publications Consistent Spherical Parameterizations, Arul Asirvatham, Emil Praun, Hugues Hoppe, Computer Graphics and Geometric Modelling, 2005. Inter-Surface Mapping, John Schreiner, Arul Asirvatham, Emil Praun, Hugues Hoppe, ACM SIGGRAPH 2004.

50. Thank You