3D Geometry Coding using Mixture Models and the Estimation Quantization Algorithm

1. 3D Geometry Coding usingMixture Models andthe Estimation Quantization Algorithm Sridhar Lavu Masters Defense Electrical & Computer Engineering DSP Group September 2002 Rice University

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 Original Coarse Multilevel 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 Non-normal vertices Wavelet coefficients

15. Related Work - Zerotree Zerotrees Mesh zerotree coding Mesh zerotree 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) •  Shape Fixed at each level •  Variance Local neighborhood •  Mean Zero

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 • Normal Higher • Tangential Lower • Precision • Normal Higher Lower l • Tangential Lower Higher 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. Performance Measure • Error metrics • MSE ? • Hausdorff distance • Min, max, mean, mean squared

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 andConnectivity • 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

49. Results – Rabbit PSNR