1. Compressing Texture Coordinates Martin IsenburgJack Snoeyink University of North Carolina at Chapel Hill with h Selective Linear Predictions

2. Take this home: “We predict texture coordinates with four different rules – taking into account the presence of mapping discontinuities.” “We compress the corresponding corrective vectors with different arithmetic contexts.”

3. Overview Background Compressing Vertex Positions Linear Prediction Schemes Texture Coordinate Mappings Compressing Texture Coordinates Alternative Approaches Summary

4. Background

5. Meshes - Basic Ingredients connectivity geometry face 1 1 2 3 4 face 2 3 4 3 face 3 5 2 1 3 face f 725 6291 6293 vertex 1 ( x 1 y 1 z 1 ) vertex 2 ( x 2 y 2 z 2 ) vertex 3 ( x 3 y 3 z 3 ) vertex v ( x v y v z v )

6. Meshes - Optional Properties mapping values face 1 1 2 3 4 face 2 3 4 5 face 3 5 6 1 7 face f 9152 123 271 texcoord 1 ( u 1 v 1 ) texcoord 2 ( u 2 v 2 ) texcoord 3 ( u 3 v 3 ) texcoord t ( u t v t )

7. – Fast Rendering – Progressive Transmission – Maximum Compression Geometry Compression [ Deering, 95 ] Mesh Compression Maximum Compression

8. Geometry Compression [ Deering, 95 ] – Fast Rendering – Progressive Transmission – Maximum Compression Connectivity Geometry Properties Geometry Compression [ Deering, 95 ] Mesh Compression Properties Maximum Compression

9. Geometry Compression [ Deering, 95 ] – Fast Rendering – Progressive Transmission – Maximum Compression Connectivity Geometry Properties – Mapping – Values Geometry Compression [ Deering, 95 ] Mesh Compression Properties Values Maximum Compression Normals Colors Material Attributes Texture Coordinates Texture Coordinates

10. Triangle Mesh Compression Texture Coordinates ??? Triangle Mesh Compression [ Touma & Gotsman, Graphics Interface 98 ] “… treat like vertex positions …” Yes! But … “parallelogram prediction” - most popular! Geometry “coding with vertex degrees” - optimal? Connectivity

11. Generalization of TG coder Connectivity Compressing Polygon Mesh Connectivity with Degree Duality Prediction [ Isenburg, 02 ] Near-optimal connectivity coding of Polygonal Meshes [ Khodakovsky et al, 02 ] Compressing Polygon Mesh Geometry with Parallelogram Prediction [ Isenburg & Alliez, 02 ] Geometry

12. Compressing Vertex Positions

13. Classic approaches [ 95 – 98 ]: – linear prediction Geometry Compression [ Deering, 95 ] Java3D Geometric Compression through topological surgery [ Taubin & Rossignac, 98 ] MPEG - 4 Triangle Mesh Compression [ Touma & Gotsman, 98 ] Virtue3D

14. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based

15. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based Spectral Compression of Mesh Geometry [ Karni & Gotsman, 00 ] expensive numerical computations

16. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based Progressive Geometry Compression [ Khodakovsky et al., 00 ] modifies mesh prior to compression

17. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based Geometric Compression for interactive transmission [ Devillers & Gandoin, 00 ] poly-soups; complex geometric algorithms

18. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based Vertex data compression for triangle meshes [ Lee & Ko, 00 ] local coord-system + vector-quantization

19. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based certain 3D models + expensive matching Compression of engineering models by repeated feature [ Shikhare et al., 01 ] discovery

20. Compressing Vertex Positions Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 02 ]: – spectral – re-meshing – space-dividing – vector-quantization – feature discovery – angle-based Angle-Analyzer: A triangle- quad mesh codec [ Lee, Alliez & Desbrun, 02 ] dihedral + internal = heavy trigonometry

21. Linear Prediction Schemes

22. 1.quantize positions with b bits 2.traverse positions 3.linear prediction from neighbors 4.store corrective vector ( 1.2045, -0.2045, 0.7045 ) ( 1008, 68, 718 ) floating point integer

23. Linear Prediction Schemes 1.quantize positions with b bits 2.traverse positions 3.linear prediction from neighbors 4.store corrective vector use traversal order implied by the connectivity coder

24. Linear Prediction Schemes 1.quantize positions with b bits 2.traverse positions 3.linear prediction from neighbors 4.store corrective vector ( 1004, 71, 723 ) apply prediction rule prediction

25. Linear Prediction Schemes 1.quantize positions with b bits 2.traverse positions 3.linear prediction from neighbors 4.store corrective vector 0 10 20 30 40 50 60 70 position distribution 0 500 1000 1500 2000 2500 3000 3500 corrector distribution ( 1004, 71, 723 )( 1008, 68, 718 ) position ( 4, -3, -5 ) correctorprediction

26. Deering, 95 Prediction: Delta-Coding A processed region unprocessed region P P = A

27. Taubin & Rossignac, 98 Prediction: Spanning Tree A B C D E processed region unprocessed region P P = α A + βB + γC + δD + εE + …

28. Touma & Gotsman, 98 Prediction: Parallelogram Rule processed region unprocessed region P P = A – B + C A B C

29. Parallelogram Rule good prediction bad prediction “non-convex” bad prediction “non-planar”

30. Not Triangles … Polygons! Face Fixer [ Isenburg & Snoeyink, 00 ]

31. Non-triangular Faces Question:Why would a mesh have a non-triangular face?

32. Non-triangular Faces Question:Why would a mesh have a non-triangular face? Answer:Because there was no reason to triangulate it! This face was “convex” and “planar”.  use this info for better predictions

33.  within-predictions often find existing parallelograms (  quadrilaterals ) “within” versus “across”  within-predictions avoid creases within-prediction across-prediction

34. Texture Coordinate Mappings

35. Texture Mapping (1) “the process of applying a texture image to a mesh” mesh textured mesh texture image

36. Texture Mapping (2) “putting every 3D polygon of the mesh in correspondence with a 2D polygon in the image”

37. “some vertices of the mesh have multiple corresponding locations in texture space” Discontinuities in Mapping (1) 3 5 5 5 3 3 5 5 5 3

38. 13 9 7 11 Discontinuities in Mapping (2) “vertices may have multiple associated texture coordinates” 3 3 5 5 9 11 7 13

39. crease corner 0 smooth corner 0 Encoding the Mapping (1) 1. distinguish vertices 2. distinguish corners smooth vertex crease vertex corner vertex 100 0 1

40. Encoding the Mapping (2) 27 23 24 25 26 1

41. 0 1 1 0 0 39 Encoding the Mapping (3) 40 37 38 0

42. Why Discontinuities ? cuts required to flatten mesh – no boundary – non-zero genus piece-wise texture mapping – author / artist decision – easier for automated techniques additional cuts to meet objectives – angle/area preserving parameterization – minimizing texture stretch

43. Piece-wise Mappings (1)

44. Piece-wise Mappings (2)

45. Compressing Texture Coordinates

46. Selective Linear Prediction analyze neighborhood use most promising predictor  “within” ( for polygon meshes only )  “across”  “nearby”  “center” avoid unreasonable predictions compress with different contexts

47. Unreasonable Predictions

48. Example Scenarios (1) processed vertex ring 1 17 16 11 13 within-prediction T 16 – T 11 + T 13

49. Example Scenarios (2) 0 1 1 0 0 29 30 21 19 20 22 25 26 smooth edge within-prediction T 20 – T 19 + T 21 across-prediction T 22 – T 25 + T 26

50. Example Scenarios (3) processed vertex ring 1 57 crease edge 50 nearby-prediction T 50 53 54 52 47 44

51. Example Scenarios (4) 0 1 1 0 0 0 82 83 81 80 74 within-prediction T 81 – T 74 + T 80 center-prediction BB mid

52. Example Bit-Rates ( 10 bit ) within by prediction type model acrossnearbycenter -.-7.59.818.9 -.-8.310.319.0 -.-7.310.718.3 “1398” “1412” “1510” 9.5 10.4 9.9 lion wolf raptor 5.68.59.720.2 5.99.111.120.2 5.58.48.919.5 total 6.3 6.6 6.3

53. Selection Histogram within frequency of prediction type model acrossnearbycenter -.-55 %35 %10 % -.-49 %39 %12 % -.-53 %36 %11 % “1398” “1412” “1510” lion wolf raptor 82 %14 %3 %1 % 84 %13 %2 %1 % 78 %18 %5 %1 %

54. Alternative Approaches

55. “make texture coordinates implicit ” change the mesh (re-meshing) Geometry Images [ Gu et al, 02 ] Bounded Distortion piece-wise Mesh Parameterization [ Sorkine et al, 02 ] Space-Optimized Texture Maps [ Balmelli et al, 02 ] change the image (re-texturing)

56. Summary & Acknowledgments

57. Summary (1) use the parallelogram predictor for texture coordinates avoid unreasonable predictions across discontinuities (“seams”) switch to less-promising predictor compress resulting corrective vectors with separate arithmetic contexts

58. Summary (2) compression software as benchmark for future research is available on the Web http://www.cs.unc.edu/~isenburg/pmc/ local link

59. Acknowledgments Archaeology Technology Labs, North Dakota State University Curious Labs, California

60. Thank You! http://www.cs.unc.edu/~isenburg/pmc/