Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

1. Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds Pablo Diaz-Gutierrez M. Gopi Renato Pajarola University of California, Irvine

2. Introduction • Intersection of and relationship between three mesh problems: Simplification, stripification and connectivity compression • Each problem must be constrained • Imposed constraints allow further applicability http://graphics.ics.uci.edu

3. Introduction: Three problems • Simplification • Decimation • Vertex clustering • Edge collapsing • Compression • Valence-driven • Strip/edge-graph based • Stripification • Alternating linear strips • Generalized strip loops http://graphics.ics.uci.edu

4. Talk outline • Hierarchyless simplification • Simplification and stripification • Connectivity compression • Results • Conclusion http://graphics.ics.uci.edu

5. Mesh simplification • Popular approach: Edge collapse/vertex split • Problem: Dependencies between collapsed edges • Hierarchy of collapse/split operations Edge collapse Vertex split http://graphics.ics.uci.edu

6. Edge-collapse dependencies Edge-collapse A can’t be split before Edge-collapse B A B http://graphics.ics.uci.edu

7. Multi-edges Definition • Multi-edge: Edge representing multiple edges from the original mesh, after simplification. Edge collapse Vertex split http://graphics.ics.uci.edu

8. Edge-collapse dependencies A Multi-edges B Collapsing multi-edges produces dependencies http://graphics.ics.uci.edu

9. Multi-edges Avoiding dependencies • When one edge of a triangle is collapsed, the other two become multi-edges. • If we don’t collapse multi-edges: Only one edge per triangle is collapsed Edge collapse 6 5 6 5 1 Collapsing edge 1 9 9 7/8 7 8 4 4 2 2 3 3 http://graphics.ics.uci.edu

10. Hierarchyless simplification • Each triangle has at most one collapsible edge • One collapsible partner across that edge • Problem: Choosing one edge prevents others from collapsing • Optimize choice of collapsible edges http://graphics.ics.uci.edu

11. Hierarchyless simplification • Pose as graph problem in the dual graph of the triangle mesh • Choose collapsible edges • Graph matching • Maximal set of collapsible edges (triangle pairs) • Perfect graph matching • No multi-edges collapsed!! • (No collapsing dependencies) http://graphics.ics.uci.edu

12. Simplification example • Genus 0 manifold • 3 connected sets • Of collapsible edges http://graphics.ics.uci.edu

13. Simplification example Multi-edges http://graphics.ics.uci.edu

14. Simplification example http://graphics.ics.uci.edu

15. Simplification example http://graphics.ics.uci.edu

16. Simplification example http://graphics.ics.uci.edu

17. Simplification example http://graphics.ics.uci.edu

18. Simplification example Equivalent vertices http://graphics.ics.uci.edu

19. Simplification example http://graphics.ics.uci.edu

20. Simplification example Equivalent vertices http://graphics.ics.uci.edu

21. Simplification example http://graphics.ics.uci.edu

22. Simplification example • 3 connected components of collapsible edges • 3 vertices after complete simplification http://graphics.ics.uci.edu

23. Extremal simplification • All triangles collapsed • But not all vertices collapsed • Collapsible edges organized in connected components • Each connected component collapses to 1 vertex http://graphics.ics.uci.edu

24. Extremal simplification • Goal: Reduce number of vertices in final model • By reducing number of connected components of collapsible edges • Apply two operations: • Edge swap (next slide) • Matching reassignment • Minimum 1 or 2 connected components • In general, connected components are trees • Might have loops http://graphics.ics.uci.edu

25. Extremal simplificationConnecting collapsible edges Initially, 3 connected components of collapsible edges Choose an edge to swap Only two connected components now http://graphics.ics.uci.edu

26. Extremal simplification • 2 connected components of collapsible edges • 2 vertices after complete simplification http://graphics.ics.uci.edu

27. Talk outline • Hierarchyless simplification • Simplification and stripification • Connectivity compression • Results • Conclusion http://graphics.ics.uci.edu

28. Simplification and stripification • Each triangle has one collapsible edge • The other two connect it to a triangle strip loop • Removing collapsible edges creates disjoint triangle strip loops http://graphics.ics.uci.edu

29. Extremal simplification andsingle-stripification • Connected components of collapsible edges are trees • Triangles around them form loops • Fewer collapsible edge components → fewer loops: • Reduce number of connected components of matched edges. • In manifolds, collapsible edges can be grouped in 1 or 2 connected components: • All triangulated manifolds can be made a single triangle strip loop. Schematic representation of triangle strips and medial axes. http://graphics.ics.uci.edu

30. Maintaining strips during simplification • Hierarchyless simplification automatically maintains triangle strips • Edge-collapses shorten strips and medial axes • But don’t change topology • Let’s see an example… http://graphics.ics.uci.edu

31. Maintaining strips during simplification http://graphics.ics.uci.edu

32. Maintaining strips during simplification http://graphics.ics.uci.edu

33. Maintaining strips during simplification http://graphics.ics.uci.edu

34. Maintaining strips during simplification http://graphics.ics.uci.edu

35. Maintaining strips during simplification http://graphics.ics.uci.edu

36. Quality considerations • Quality of simplification • Choose collapsible edges with the least quadric error • Quality of stripification • Application dependent (i.e. maximize strip locality) • Assign edge weights • Choose weight minimizingset of collapsible edges • Diaz-Gutierrez et al. "Constrained strip generation and management for efficient interactive 3D rendering“, CGI 2005 0 0 0 0 0 1 0 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 http://graphics.ics.uci.edu

37. Talk outline • Hierarchyless simplification • Simplification and stripification • Connectivity compression • Results • Conclusion http://graphics.ics.uci.edu

38. Collapsible edges & connectivity compression • Important: topological, not geometric compression. • Some existing techniques produce triangle strips as byproduct of compression. • A few compress connectivity along with strips. • We exploit duality of strips and medial axes. Images from http://www.gvu.gatech.edu/~jarek/ http://graphics.ics.uci.edu

39. 1 0 1 0 1 0 1 0 Encoding of a strip as a zipping of two trees “Hand and Glove” compression Genus-0 triangulated manifolds • Encode genus 0 mesh as: • Two vertex spanning trees of collapsible edges (“hand and glove” trees) • A bit string zips the trees together along the single strip loop • Guaranteed upper bound: 2 bits/face (i.e. 4 bits/vertex) http://graphics.ics.uci.edu

40. “Hand and Glove” compression Genus-0 triangulated manifolds • Predict direction of strip to improve compression • Slight modifications to handle: • Higher genus • Boundaries • Quadrilateral manifolds • Etc. • Very simple to code • One day for prototype program http://graphics.ics.uci.edu

41. Talk outline • Hierarchyless simplification • Simplification and stripification • Connectivity compression • Results • Conclusion http://graphics.ics.uci.edu

42. Results: Simplification Models with 1358, 454, 54 and 4 triangles. http://graphics.ics.uci.edu

43. Results: Simplification Models with 19778, 7238, 1500 and 778 triangles. Models with 16450, 6450, 2450 and 450triangles. http://graphics.ics.uci.edu

44. Results: Simplification Models with 101924, 33924, 9924 and 1924 triangles. http://graphics.ics.uci.edu

45. Results: View-dependent simplification Notice dramatic change in simplification http://graphics.ics.uci.edu

46. Results: Stripificationwith asymmetric simplification http://graphics.ics.uci.edu

47. Results: Compression bit ratios Bits per vertex obtained with Hand & Glove method. Comparison with Edgebreaker. The output of both methods is compressed with an arithmetic encoder. http://graphics.ics.uci.edu

48. Talk outline • Hierarchyless simplification • Simplification and stripification • Connectivity compression • Results • Conclusion http://graphics.ics.uci.edu

49. Summary and conclusion • This paper lays a theoretical foundation for combining three important areas of geometric computing. • By computing and appropriately managing sets of collapsible edges, we achieved: • Hierarchyless mesh simplification • Dynamic management of triangle strip loops • Efficient connectivity compression http://graphics.ics.uci.edu

50. Future work • Explore and improve Hand & Glove mesh compression. • Design a lighter data structure for computing errors in view-dependent simplification. • Extend current results on stripification (partially completed). http://graphics.ics.uci.edu