Martin Isenburg Jack Snoeyink University of North Carolina at Chapel Hill

2. SPIRALE REVERSI Reverse Decoding of theEdgebreaker Encoding Martin Isenburg Jack Snoeyink University of North Carolina at Chapel Hill

3. Introduction • Mesh compression is hip • Edgebreaker encodes mesh connectivity • Original decoding algorithm • worst case time complexity: O(n2) • Wrap & Zip decoding algorithm • worst case time complexity: O(n) Spirale Reversi decoding algorithm 

4. Overview • Connectivity Compression • Edgebreaker • Four Examples • Edgebreaker Encoding • Edgebreaker Decoding • Wrap & Zip Decoding • Spirale Reversi Decoding • Holes and Handles

5. Connectivity Compression

6. connectivity geometry n = 10,000 100 KB 60 KB n = 100,000 1245 KB 600 KB n = 1,000,000 15 MB 6 MB Standard Representation triangle1 1 2 3 triangle2 3 4 3 triangle3 5 2 1 trianglem vertex1 (x,y,z) vertex2 (x,y,z) vertex3 (x,y,z) vertexn

7. Compression Schemes • Geometry Compression, Deering, 1995 • Short Encodings of Planar Graphs and Maps, Keeler and Westbrook, 1995 • Geometric Compression through Topological Surgery, Taubin and Rossignac, 1996 • Good orders for incremental (re)construction, Snoeyink and vanKrefeld, 1997 • Encoding a triangulation as a permutation of its point set, Denny and Sohler, 1997 • Triangle Mesh Compression, Touma and Gotsman, 1998 • Real time compression of triangle mesh connectivity, Gumhold and Strasser, 1998 • Mesh Connectivity Coding by Dual Graph Approach, Li and Kuo, 1998 • Edgebreaker: Connectivity Compression for Triangle Meshes, Rossignac, 1998 • Mesh Collapse Compression, Isenburg and Snoeyink, 1999 • Single Resolution Compression of Arbitrary Triangular Meshes with Properties, Bajaj, Pascucci, and Zhuang, 1999 • Triangle Strip Compression, Isenburg, 2000 • Face Fixer: Compressing Polygon Meshes with Properties, Isenburg and Snoeyink, 2000 • Efficient Coding of non-triangular Meshes, Kronrod and Gotsman

8. Compression Schemes time • Geometry Compression, Deering, 1995 • Short Encodings of Planar Westbrook, 1995 • Geometric Compression ubin and Rossignac, 1996 • Good orders for increm nd vanKrefeld, 1997 • Encoding a triangulation y and Sohler, 1997 • Triangle Mesh Compre • Real time compression d and Strasser, 1998 • Mesh Connectivity Co o, 1998 • Edgebreaker: Connectivity ossignac, 1998 • Mesh Collapse Compressi • Single Resolution Compres hes with Properties, Bajaj, Pascucci, and Zhuang, 1999 • Triangle Strip Compression, Isenburg, 2000 • Face Fixer: Compressing Polygon Meshes with Properties, Isenburg and Snoeyink, 2000 • Efficient Coding of non-triangular Meshes, Kronrod and Gotsman esh Connec Edgebreaker esh Collaps

9. Edgebreaker

10. Encoding • Encodes meshconnectivity asa sequence oflabels • Best worst casebound for meshconnectivity • Less than 4 bits per vertex for meshes without holes and handles

11. Decoding • Original decoding replays encoding traversal • but O(n2) worst case time complexity • Wrap & Zip decoding “sort of” replays encoding traveral • only O(n) time complexity • but multiple traversals of labels (triangles) for mesh with boundary (handles) • “zipping” is un-necessary overhead

12. Edgebreaker Encoding

13. C

14. C R

15. C R R

16. C R R R

17. C L R R R

18. C S L R R R

19. C S L L R R R

20. C E S L L R R R

21. C C E S L L R R R

22. C C E S L L R R R R

23. C C E S L L R R R R R

24. C C E E S L L R R R R R

25. C C E E S L L R R R R R

26. EdgebreakerDecoding

27. C C E E S L L R R R R R Changes in boundary length Computing initial boundary length 0

28. C C C E E S L L R R R R R Changes in boundary length Computing initial boundary length +1 0 +1

29. C C E E S L L R R R R R R Changes in boundary length Computing initial boundary length +1 -1 0 +1 0

30. C C E E S L L R R R R R R Changes in boundary length Computing initial boundary length +1 -1 -1 0 +1 0 -1

31. C C E E S L L R R R R R R Changes in boundary length Computing initial boundary length +1 -1 -1 -1 0 +1 0 -1 -2

32. C C E E S L L L R R R R R Changes in boundary length Computing initial boundary length +1 -1 -1 -1 -1 0 +1 0 -1 -2 -3

33. C C E E S S L L R R R R R Changes in boundary length Computing initial boundary length +1 -1 -1 -1 -1 +1 0 +1 0 -1 -2 -3 -2

34. C C E E S S L L R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 0 +1 0 -1 -2 -3 -2 0

35. C C E E S S L L L R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 0 +1 0 -1 -2 -3 -2 -3 0 -1

36. C C E E E S S L L R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 -3 0 +1 0 -1 -2 -3 -2 -3 -6 0 -1 -4

37. C C C E E S S L L R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 -3 +1 0 +1 0 -1 -2 -3 -2 -3 -6 -5 0 -1 -4

38. C C E E S S L L R R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 -3 +1 -1 0 +1 0 -1 -2 -3 -2 -3 -6 -5 -6 0 -1 -4

39. C C E E S S L L R R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 -3 +1 -1 -1 0 +1 0 -1 -2 -3 -2 -3 -6 -5 -6 -7 0 -1 -4

40. C C E E E S S L L R R R R R Changes in boundary length Computing initial boundary length Computing offset for label +1 -1 -1 -1 -1 +1 -1 -3 +1 -1 -1 -3 0 +1 0 -1 -2 -3 -2 -3 -6 -5 -6 -7 -10 0 -1 -4

41. C C E E S L L R R R R R

42. C C C C E E S L L R R R R R

43. C C E E S L L R R R R R R R

44. C C E E S L L R R R R R R R

45. C C E E S L L R R R R R R R

46. C C E E S L L L L R R R R R

47. C C E E S S S L L R R R R R use pre-computed offset

48. C C E E S L L L L R R R R R

49. C C E E E E S L L R R R R R

50. C C C C E E S L L R R R R R