1. 3D Geometry Coding using Mixture Models and the Estimation Quantization Algorithm Sridhar Lavu Masters Defense Electrical & Computer Engineering DSP GroupRice UniversitySeptember 2002

2. 3D Surfaces Video games Animations - Bug’s Life, Toy Story 2 3D object modeling - CAD e-commerce

3. 3D Surfaces Geometry, color, texture 3D scanning Polygon meshes Problem - large data sets Geometry compression 100,000 triangles

4. Contribution 3D geometry coder Multilevel representation –Normal meshes EQ algorithm –Estimation-Quantization (EQ) –Local context information RD optimization

5. Related Work Zerotree coder for the wavelet coefficients of normal meshes RD optimization based quantization algorithm for the wavelet coefficients of meshes

6. Outline 3D surface data Multilevel representation Normal meshes Wavelet transform EQ algorithm Error metrics Results

7. 3D geometry data Geometry Polygon meshes Geometry & connectivity Geometry 0.0 0.0 0.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 0.0 0.5 0.5 1.0 Connectivity 0 1 2 2 3 1 0 1 4 1 2 4 2 3 4 3 0 4

8. Multilevel Representations OriginalCoarseMultilevel triangular meshes Original  Normal meshes

9. Normal meshes Multilevel representation Base mesh Successively refine the mesh –Subdivision

10. Subdivision Linear subdivision Butterfly subdivision Loop subdivision

11. Butterfly Subdivision

12. Normal Meshes Predict b and n Find intersection Store offset 1 number per vertex

13. Wavelet Transforms Irregular data Lifting scheme – predict and update Subdivision – predict step Wavelet transforms –Butterfly wavelet transform –Loop wavelet transform

14. Wavelet Transforms and Normal Meshes Wavelet coefficients Non-normal vertices

15. Related Work - Zerotree Zerotrees Zerotree coding Mesh zerotree Mesh zerotree coding EQ coding

16. Review Multilevel representations for meshes Normal meshes Wavelet transforms –Subdivision –Lifting Related work - ZT based algorithm Contribution – EQ based algorithm

17. 3D EQ Coder Local context information Model for wavelet coefficients –Generalized Gaussian distribution EQ Algorithm –Estimate Step –Quantize Step –RD Optimization

18. Wavelet Coefficient Model Generalized Gaussian distribution

19. Wavelet Coefficient Model Generalized Gaussian (GGD) – ShapeFixed at each level –  VarianceLocal neighborhood –  MeanZero

20. EQ Algorithm Scan the vertices –Estimate, quantize, encode Estimate step - variance –Local neighborhood –Causal neighborhood –Quantized neighbors Quantize step –Deadzone quantizer –RD optimization

21. EQ Algorithm (cont.) RD optimization –Rate = -log(probability) –Distortion = MSE of coefficients Entropy coding –Arithmetic coder

22. Normal vs. Tangential Smooth surfaces Global error contribution –NormalHigher –TangentialLower Precision –NormalHigherLower l –Tangential LowerHigher l Most tangential components are zero –Single quantizer per level

23. Neighborhood

24. Ordering - Base Triangles

25. Ordering - Vertices

26. Summary of EQ Algorithm Pick l Determine ordering –Ordering of base triangles –Ordering inside each base triangle Local causal neighborhood Estimate s Quantize using RD optimization Normal vs. tangential

27. Error metrics MSE ? Hausdorff distance Min, max, mean, mean squared Performance Measure

28. Results Metric - PSNR Bits-per-vertex (bpv) Reconstructed mesh vs. original mesh Metro and MeshDev software tools

29. Results - EQ vs. ZT

30. Results EQ vs. ZT (Lifted Butterfly)

31. Results - EQ vs. ZT (Loop Wavelets)

32. Results (Bounds) Upper bound –Complete context Lower bound –No context

33. Summary Multilevel representations Normal meshes Wavelet transforms GGD model Local context based coder EQ vs. ZT

34. Conclusion & Future Work Conclusions –GGD model + EQ algorithm –0.5 – 1 dB gain Future work –Vertex based error for RD optimization New algorithms –Space-Frequency quantization (SFQ)

36. Scaling Coefficients and Connectivity Scaling coefficients –Vertices of base mesh –Uniform quantization Connectivity –Semi-regular connectivity –Base mesh connectivity –TG Coder (lossless)

37. Lifting (Predict, Update) Forward Inverse

38. Lifting - Haar Split Predict Update

39. Loop Wavelet Transform

40. Causal Neighborhoods

41. EQ – Unpredictable sets Empty causal neighborhood Zero s estimate Classify as unpredictable (U) set Model U set as zero-mean GGD Use a single s and n for U set

42. EQ – Threshold step Iteration of E and Q steps First iteration Threshold coefficients Partition U and P sets Estimate s and n Use estimates in next iteration

43. Normal Predictable Set

44. Normal Unpredictable Set

45. Tangential Set

46. Hausdorff Distance

47. Mesh Zerotree Coding

48. Results – Venus PSNR BPV0.250.51.02.03.04.0 EQ lifted BW63.768.674.279.281.783.2 ZT lifted BW63.068.273.778.981.781.9 EQ unlifted BW63.568.674.178.981.483.0 ZT unlifted BW62.467.873.078.481.281.5 EQ Loop Wavelet60.065.371.376.479.481.4 ZT Loop Wavelet60.966.171.877.179.7

49. Results – Rabbit PSNR BPV0.250.51.02.02.53.0 EQ lifted BW70.375.780.984.285.185.6 ZT lifted BW69.375.180.984.084.1 EQ unlifted BW70.075.380.684.085.085.5 EQ unlifted BW68.774.780.483.6