1. Geometry Images Steven Gortler Harvard University Steven Gortler Harvard University Xianfeng Gu Harvard University Xianfeng Gu Harvard University Hugues Hoppe Microsoft Research Hugues Hoppe Microsoft Research

2. Irregular meshes Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 … Face 2 1 3 Face 4 2 3 …

3. Texture mapping Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 … s1 t1s1 t1s2 t2s2 t2s1 t1s1 t1s2 t2s2 t2 normal map s t Face 2 1 3 Face 4 2 3 …

4. Complicated rendering process Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 … random access! s1 t1s1 t1s2 t2s2 t2s1 t1s1 t1s2 t2s2 t2 Face 2 1 3 Face 4 2 3 … ~40M Δ/sec

5. Semi-regular representations irregular vertex indices only semi-regular [Eck et al 1995] [Lee et al 1998] [Khodakovsky 2000] [Guskov et al 2000] …

6. Geometry Image geometry image 257 x 257; 12 bits/channel 3D geometry completely regular sampling

7. Basic idea demo cut parametrize

8. Basic idea cut sample

9. cut [r,g,b] = [x,y,z] render store

10. How to cut ? sphere in 3D 2D surface disk

11. How to cut ? l Genus-0 surface  any tree of edges sphere in 3D 2D surface disk

12. How to cut ? l Genus-g surface  2g generator loops minimum torus (genus 1)

13. Surface cutting algorithm (1) Find topologically-sufficient cut: 2g loops [Dey and Schipper 1995] [Erickson and Har-Peled 2002] (2) Allow better parametrization: additional cut paths [Sheffer 2002] (1) Find topologically-sufficient cut: 2g loops [Dey and Schipper 1995] [Erickson and Har-Peled 2002] (2) Allow better parametrization: additional cut paths [Sheffer 2002]

14. Step 1: Find topologically-sufficient cut (a) retract 2-simplices (b) retract 1-simplices

15. Results of Step 1 genus 6 genus 0 genus 3

16. Step 2: Augment cut l Make the cut pass through “extrema” (note: not local phenomena). l Approach: parametrize and look for “bad” areas. l Make the cut pass through “extrema” (note: not local phenomena). l Approach: parametrize and look for “bad” areas.

17. Step 2: Augment cut …iterate while parametrization improves

18. Results of Steps 1 & 2 genus 1 genus 0

19. Parametrize boundary Constraints: n cut-path mates identical length n endpoints at grid points Constraints: n cut-path mates identical length n endpoints at grid points a a’ a a’  no cracks

20. Parametrize interior –optimizes point-sampled approx. [Sander et al 2002] n Geometric-stretch metric –minimizes undersampling [Sander et al 2001] n Geometric-stretch metric –minimizes undersampling [Sander et al 2001]

21. Previous metrics (Floater, harmonic, uniform, …) Stretch parametrization

22. SampleSample geometry image

23. RenderingRendering (65x65 geometry image)

24. rendering geometry image 257 2 x 12b/ch normal-map image 512 2 x 8b/ch Rendering with attributes

25. Advantages for hardware rendering l Regular sampling  no vertex indices. l Unified parametrization  no texture coordinates.  Raster-scan traversal of source data: geometry & attribute samples in lockstep. Summary: compact, regular, no indirection Summary: compact, regular, no indirection l Regular sampling  no vertex indices. l Unified parametrization  no texture coordinates.  Raster-scan traversal of source data: geometry & attribute samples in lockstep. Summary: compact, regular, no indirection Summary: compact, regular, no indirection

26. normal map 512x512; 8b/ch Normal-Mapped Demo geometry image 129x129; 12b/ch demo

27. color map 512x512; 8b/ch Pre-shaded Demo geometry image 129x129; 12b/ch

28. ResultsResults 257x257 normal-map 512x512

29. ResultsResults 257x257 color image 512x512

30. Mip-mappingMip-mapping 257x257129x12965x65 boundary constraints set for size 65x65

31. Hierarchical culling view-frustum culling backface culling geometry image normal-map image

32. CompressionCompression  1.5 KB + topological sideband (12 B) fused cut 295 KB Image wavelet-coder

33. Compression results 1.5 KB 3 KB 12 KB 49 KB 295 KB 

34. Rate distortion

35. Some artifacts aliasing anisotropic sampling

36. SummarySummary l Simple rendering: compact, no indirection, raster-scan stream. l Mipmapped geometry l Hierarchical culling l Compressible l Simple rendering: compact, no indirection, raster-scan stream. l Mipmapped geometry l Hierarchical culling l Compressible

37. Future work l Better cutting algorithms l Feature-sensitive remeshing l Tangent-frame compression l Bilinear and bicubic rendering l Build hardware l Better cutting algorithms l Feature-sensitive remeshing l Tangent-frame compression l Bilinear and bicubic rendering l Build hardware