1. Martin Isenburg UC Berkeley Jack Snoeyink UNC Chapel Hill Early Split Coding of Triangle Mesh Connectivity

2. Geometry Connectivity – Efficient Rendering – Progressive Transmission – Compact Storage Mesh Compression “Geometry Compression” [ Deering, 95 ] Surface Meshes – triangular – polygonal Volume Meshes – tetrahedral, hexahedral storage / network main memory

3. Connectivity Triangle Mesh Connectivity v -9.7 1.6 -0.2 v -6.6 1.4 3.3 v -1.6 1.4 2.3 v -3.3 -1.4 3.0 v 5.6 1.7 4.5 v -6.6 1.4 6.1 v -3.3 –4.4 3.7 v -1.6 1.4 2.1 v -5.4 -1.6 6.4... f 1 2 3 f 3 2 4 f 2 1 5 f 2 5 4 f 6 4 5 f 6 5 7 f 5 1 8 f 4 8 7 f 7 8 9 f 9 8 10 f 10 11 7... Geometry each index uses log 2 ( v ) bits  6 log 2 ( v ) bpv  3.24 bpv enumeration of triangulated planar graphs [ Tutte 62 ]

4. “Out-of-Core Compression”[ Isenburg & Gumhold ‘03 ] “Delphi Coding”[ Coors and Rossignac ‘04 ] “FreeLence”[ Kälberer et al. ‘05 ] Compression Schemes “Cut-Border Machine”[ Gumhold & Strasser ‘98 ] “Triangle Mesh Compression”[ Touma & Gotsman ‘98 ] “Edgebreaker”[ Rossignac ‘99 ] “Face Fixer”[ Isenburg & Snoeyink ‘00 ] “Angle Analyzer”[ Lee, Alliez & Desbrun ‘02 ] “Degree Duality Coder”[ Isenburg ‘02 ] “Near-optimal Coding”[ Khodakovsky et al. ‘02 ] “Valence-Driven Encoding”[ Alliez & Desbrun ‘01 ] Computer Graphics “Succinct encodings of planar graphs”[ He, Yao, & Hu ‘99 ] “Short encodings of planar graphs”[ Keeler & Westbrook ‘95 ] “Succinct Representations of Graphs”[ Turan ‘84 ] “Optimal Coding”[ Poulahlon & Schaeffer ‘03 ] Planar Graph Coding

5. Connectivity Graph Traversal split offset

6. Connectivity Graph Traversal

7. CBM – Edgebreaker – TG coder label-based split offset “Cut-Border Machine”[ Gumhold & Strasser ‘98 ] “Edgebreaker”[ Rossignac ‘99 ] “Early Split Coding” [ this paper ] CRCRRS... RCRERC degree-based “Triangle Mesh Compression” [ Touma & Gotsman ‘98 ] 3... 64444544 S split offset

8. CBM and Edgebreaker

9. 5 processed region unprocessed region compression boundary

10. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C

11. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C C

12. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C C R

13. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C C C R C

14. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R C C R C R

15. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S C C R C R S 1 2 3 4 5 6 7 8 9 10 11 offset = 11

16. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C C R C R S L offset = 11

17. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C C C R C R S L C offset = 11

18. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C C R C R S L C R offset = 11

19. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C C C R C R S L C R C offset = 11

20. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C C R C R S L C R C R offset = 11

21. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C C C R C R S L C R C R C offset = 11

22. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R C C R C R S L C R C R C R offset = 11

23. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R C C R C R S L C R C R C R R offset = 11

24. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R C C R C R S L C R C R C R R R offset = 11

25. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C C R C R S L C R C R C R R R L offset = 11

26. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C C C R C R S L C R C R C R R R L C offset = 11

27. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R C C R C R S L C R C R C R R R L C R offset = 11

28. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R C C R C R S L C R C R C R R R L C R R offset = 11

29. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R C C R C R S L C R C R C R R R L C R R R offset = 11

30. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S C C R C R S L C R C R C R R R L C R R R S 1 2 offset = 3 offset = 11

31. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R C C R C R S L C R C R C R R R L C R R R S R S L C R C R C R R R L C R R R S R offset = 3 offset = 11

32. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E R C C R C R S L C R C R C R R R L C R R R S R E offset = 3 offset = 11

33. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E R E C C R C R S L C R C R C R R R L C R R R S R E E offset = 11 offset = 3

34. CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E R E C C R C R S L C R C R C R R R L C R R R S R E E C offset = 11 offset = 3 +1+3 +1+1 +1+3 = 4 = 12 C...

35. Subsequences of Labels are self-contained Encodings

36. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained reversible ……

37. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible ……

38. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained reversible E E ……

39. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E ……

40. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E ……

41. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E ……

42. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E ……

43. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E ……

44. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S ……

45. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R ……

46. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R ……

47. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R ……

48. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C ……

49. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L ……

50. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R ……

51. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R R ……

52. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR ……

53. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C ……

54. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C R ……

55. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C R C ……

56. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C R C R ……

57. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C R C R …… C

58. S L C R C R C R R R L C R R R S R E E Subsequences are self-contained E reversible R E S R R R C L R RR C R C R …… C L we did not need split offsets !!!

59. TG coder

60. 5 processed region unprocessed region compression boundary slot counts

61. TG coder 5 processed region unprocessed region compression boundary 7 7

62. TG coder 5 processed region unprocessed region compression boundary 7 6 7 6 zero slot

63. TG coder 5 processed region unprocessed region compression boundary 7 6 7 6

64. TG coder 5 processed region unprocessed region compression boundary 7 6 5 7 6 5 zero slot

65. TG coder 5 processed region unprocessed region compression boundary 7 6 5 7 6 5

66. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 2 4 6 7 8 11 12 13 15 16 18 offset = 18 7 6 5 S

67. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 S offset = 18

68. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 7 6 5 S 7 offset = 18

69. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 7 6 5 S 7 offset = 18 zero slot

70. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 7 6 5 S 7 offset = 18 zero slot

71. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 7 6 5 S 7 offset = 18

72. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 7 6 5 S 7 6 offset = 18

73. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 7 6 5 S 7 6 offset = 18

74. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 7 6 5 S 7 6 5 offset = 18

75. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 7 6 5 S 7 6 5 offset = 18

76. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 7 6 5 S 7 6 5 offset = 18

77. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 7 6 5 S 7 6 5 offset = 18

78. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 7 6 5 S 7 6 5 offset = 18

79. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

80. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

81. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

82. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

83. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

84. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 offset = 18

85. TG coder 5 processed region unprocessed region compression boundary 7 6 5 S 7 6 5 6 7 6 5 S 7 6 5 6 7 offset = 18 7 … E

86. Subsequences of Degrees are not self-contained Encodings

87. Subsequences are not self-contained … S 7 6 5 6 E … 5 S

88. 5 S Subsequences are not self-contained

89. … S 7 6 5 6 E … 5 S Subsequences are not self-contained

90. … S 7 6 5 6 E … 5 S Subsequences are not self-contained

91. … S 7 6 5 6 E … 5 S Subsequences are not self-contained

92. … S 7 6 5 6 E … 5 S Subsequences are not self-contained

93. Not unique without split offsets … S 4 3 E 4 3 E 4 3 3 4 4 3 4 3 6 6 5 6 6 5 6 …5 7 5 6 6 6 5 …

94. Early Split Coder

95. encode with labels or degrees traverse triangles in same order – new label W – earlier split operations one-to-one mapping  insights: – how to turn degree subsequences into self-contained encodings – decoding strategy decides success

96. R C R C C R The new label W

97. R C C R C C R

98. R C C R C C R C

99. R C C R C R C C R wart R C R C C R

100. The new label W R C R C C R C R C C R C R C C R wart zero slot

101. The new label W R C R C C R C W R C C R C R C C R wart

102. The new label W R C R C C R C W C R C C R C R C C R wart

103. The new label W R C R C C R C W C R R C C R C R C C R wart

104. The “early split” operation R R C R C

105. S R R C R C

106. S R R C R C R

107. S R R C R C R L

108. S R R C R C R L R

109. S R R C R C R L R L zero slot

110. The “early split” operation S R R C R C R L R L

111. S R R C R C R L R L

112. S R R C R C R L R L

113. S R R C R C R L R L

114. S R R C R C R L R L R R C R C C

115. S R R C R C R L R L R R C R C C S zero slot

116. The “early split” operation S R R C R C R L R L R R C R C C S W zero slot

117. The “early split” operation S R R C R C R L R L R R C R C C W S W

118. S R R C R C R L R L R R C R C C W S W C

119. S R R C R C R L R L R R C R C C W S W C R

120. S R R C R C R L R L R R C R C C W S W C R R zero slot

121. The “early split” operation S R R C R C R L R L R R C R C C W S W C R R W

122. Complete Example

123. One Scheme & Four Encodings label-based degree-based

124. Forward Decoding with offsets

125. V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

126. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

127. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

128. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

129. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

130. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

131. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

132. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

133. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

134. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

135. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

136. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

137. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

138. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

139. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

140. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

141. Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

142. Reverse Decoding without offsets

143. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

144. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

145. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

146. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

147. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

148. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

149. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

150. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

151. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

152. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

153. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

154. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

155. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

156. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

157. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

158. Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 E R C E R R C S C R C W R C R C R C C C C

159. Results

160. Compression Results

161. Improving the TG coder 5 “Valence-Driven Encoding”[ Alliez & Desbrun ‘01 ] degree of vertex must be at least 5 ( at least 6 for early-split ) degree of vertex must be at least 4 ( at least 5 for early-split )

162. Thank You