1. Max-Plank Institut für Informatik systematic error parallelogram rule polygonal rules exact prediction Geometry Prediction for High Degree Polygons Martin Isenburg Stefan Gumhold Ioannis Ivrissimtzis Hans-Peter Seidel INFORMATIK

2. Compression real stuff – sleeping bags – compressed air polygon meshes – faster downloads / less storage – collaborative CAD – distribution of simulation results – archival of spare parts / history

3. Movies “Rustboy” animated short by Brian Taylor

4. Engineering “Audi A8” created by Roland Wolf

5. Architectural Visualization “Atrium” created by Karol Myszkowski and Frederic Drago

6. Product Catalogues “Bedroom set-model Assisi” created by Stolid

7. Historical Study scanning of “Michelangelo’s David” courtesy of Marc Levoy

8. Computer Games screen shot of “The village of Gnisis”, The Elder Scrolls III

9. – Efficient Rendering – Progressive Transmission – Maximum Compression Connectivity Geometry Properties Mesh Compression Geometry Compression [ Deering, 95 ] storage / network main memory Maximum Compression

10. Most Popular Coder Triangle Mesh Compression [ Touma & Gotsman, 98 ]... 6444 M 54 S 66 6 connectivity with vertex degrees ( ) -3 -2 1 ( ) 7 4 -3 ( ) 2 0 -2... ( ) 1 -1 -1 ( ) -2 0 0 ( ) -4 7 -2 ( ) 1 -2 1 ( ) 2 4 -1 geometry with corrective vectors

11. Coding Connectivity

12. Predicting Geometry

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

14. Coding Polygon Connectivity Compressing Polygon Mesh Connectivity with Degree Duality … [ Isenburg, 02 ]  same compression in primal and dual !!

15. Predicting Polygon Geometry Compressing Polygon Mesh Geometry with Parallelogram … [ Isenburg & Alliez, 02 ]  but … does not work well in the dual !!

16. High Degree Polygons v2v2 v1v1 v0v0 v4v4 v2v2 v1v1 v0v0 v3v3 c 0 = 0.8090 c 1 =-0.3090 c 2 =-0.3090 c 3 =0.8090 c 0 = 0.8 c 1 =-0.6 c 2 =-0.4 c 3 =1.2 v3v3 v4v4 c 0 =0.9009 c 1 = -0.6234 c 2 = 0.2225 c 3 =0.2225 c 4 =-0.6234 c 0 =0.9009 v2v2 v1v1 v0v0 v3v3 v4v4 v5v5 v6v6 v2v2 v1v1 v3v3 v0v0 c 0 = 1.0 c 1 =-1.0 c 2 =1.0 polygonal rules: v p = c 0 v 0 + c 1 v 1 + … + c p-1 v p-1 v2v2 v1v1 v2v2 v1v1 v0v0 v3v3 v3v3 v0v0 v2v2 v1v1 v0v0 v3v3 v2v2 v1v1 v0v0 v3v3 parallelogram rule: v 3 = v 0 – v 1 + v 2

17. Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

18. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Geometry Compression [ Deering, 95 ] Geometric Compression through topological surgery [ Taubin & Rossignac, 98 ] Triangle Mesh Compression [ Touma & Gotsman, 98 ] Java3DMPEG - 4Virtue3D

19. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates

20. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Spectral Compression of Mesh Geometry [ Karni & Gotsman, 00 ] expensive numerical computations

21. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Progressive Geometry Compression [ Khodakovsky et al., 00 ] modifies mesh prior to compression

22. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Geometric Compression for interactive transmission [ Devillers & Gandoin, 00 ] poly-soups; complex geometric algorithms

23. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Vertex data compression for triangle meshes [ Lee & Ko, 00 ] local coord-system + vector-quantization

24. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Angle-Analyzer: A triangle- quad mesh codec [ Lee, Alliez & Desbrun, 02 ] dihedral + internal = heavy trigonometry

25. Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates High-Pass Quantization for Mesh Encoding [ Sorkine et al., 03 ] basis transformation with Laplacian matrix

26. Linear Prediction Schemes 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

27. 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

28. 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

29. 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

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

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

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

33.  “within”- predictions often find existing parallelograms ( i.e. quadrilateral faces ) “within” versus “across”  “within”- predictions avoid creases within-prediction across-prediction

34. Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

35. Discrete Fourier Transform ( 1 ) where. The Discrete Fourier Transform ( DFT ) of a complex vector is a basis transform that is described by the matrix:

36. Discrete Fourier Transform ( 2 ) Here is the Fourier Transform of.

37. Discrete Fourier Transform ( 3 ) Rewriting the equation makes the change of basis more obvious. This basis is called the Fourier Basis. basis vectors

38. Geometric Interpretation v2v2 v1v1 v0v0 v3v3 v4v4

39. The parallelogram rule predicts the highest frequency to be zero: Predict with Low Frequencies v2v2 v1v1 v0v0 v3v3 v3v3 v2v2 v1v1 v0v0 v3v3 v3v3

40. Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

41. Eliminate High Frequencies ( 1 ) v3v3 v2v2 v1v1 v4v4 v0v0 v3v3 v5v5 v3v3 v2v2 v1v1 v4v4 v0v0 v3v3 v5v5

42. Eliminate High Frequencies ( 2 ) v2v2 v3v3 v1v1 v0v0 v4v4 v1v1 v0v0 v4v4 v2v2 v3v3

43. Eliminate High Frequencies ( 3 ) v1v1 v0v0 v4v4 v3v3 v2v2 v1v1 v0v0 v4v4 v3v3 v2v2

44. Computing the Coefficients given k of n points are known: 1. write polygon in Fourier basis 2. put n-k highest frequencies to zero 3. invert known sub-matrix 4. calculate prediction coefficients known points missing points

45. Example: n = 5, k = 3 v0v0 v1v1 v4v4 v2v2 v3v3 missing points v0v0 v1v1 v4v4 v2v2 v3v3 known points

46. Example: n = 5, k = 3 v0v0 v1v1 v4v4 v2v2 v3v3 missing points known points

47. Polygonal Predictors v2v2 v1v1 v0v0 v3v3 v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 =-1.6180 c 2 =1.6180 c 0 = 1.0 c 1 =-2.0 c 2 =2.0 v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 =-2.2470 c 2 =2.2470 v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 =-2.4142 c 2 =2.4142 three vertices are known v2v2 v1v1 v0v0 v4v4 v3v3 c 0 = 0.8090 c 1 =-0.3090 c 2 =-0.3090 c 3 =0.8090 v3v3 v2v2 v0v0 v3v3 c 0 = 1.0 c 1 =-1.0 c 2 =1.0 c 3 =-1.0 c 4 =1.0 v4v4 v5v5 c 0 =0.9009 c 1 = -0.6234 c 2 = 0.2225 c 3 =0.2225 c 4 =-0.6234 c 5 =0.9009 v2v2 v1v1 v0v0 v3v3 v4v4 v5v5 v6v6 v2v2 v1v1 v0v0 v3v3 v4v4 c 0 =1.0 c 1 =-1.0 c 2 =1.0 c 3 =-1.0 c 4 =1.0 c 5 =-1.0 c 6 = 1.0 v7v7 v5v5 v6v6 v1v1 one vertex is missing

48. Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

49. Test Set of Dual Meshes

50. Parallelogram vs. Polygonal

51. Prediction Rule Histogram hexagon heptagon pentagon octagon

52. Dual vs. Primal Compression ( coordinates quantized at 14 bits of precision )

53. Average Prediction Error

54. Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

55. Validation of Predictors eliminating the highest frequency in a mesh element – + + parallelogram predictor [ Touma & Gotsman, 98 ] + – + – + – + Lorenzo predictor [ Ibarria et al, 03 ]

56. Exact barycentric prediction after dualization polygons of even order have a highest frequency of zero

57. Thank You INFORMATIK