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Geometry Images

Geometry Images. Xianfeng Gu Harvard University. Steven Gortler Harvard University. Hugues Hoppe Microsoft Research. 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 …. Texture mapping. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 ….

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Geometry Images

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

  2. Irregular meshes Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2 … Face 2 1 3 Face 4 2 3 …

  3. Texture mapping Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2 … Face 2 1 3 Face 4 2 3 … s1 t1 s2 t2 t normal map s

  4. Complicated rendering process Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2 … Face 2 1 3 Face 4 2 3 … s1 t1 s2 t2 random access! random access! ~40M Δ/sec

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

  6. Geometry Image completely regular sampling 3D geometry geometry image257 x 257; 12 bits/channel

  7. Basic idea cut parametrize demo

  8. Basic idea cut sample

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

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

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

  12. How to cut ? • 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]

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

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

  16. Step 2: Augment cut • Make the cut pass through “extrema” (note: not local phenomena). • 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: • cut-path mates identical length • endpoints at grid points a a’ a’ a  no cracks

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

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

  22. Sample geometry image

  23. Rendering (65x65 geometry image)

  24. Rendering with attributes geometry image 2572 x 12b/ch normal-map image 5122 x 8b/ch rendering

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

  26. Normal-Mapped Demo geometry image129x129; 12b/ch normal map512x512; 8b/ch demo

  27. Pre-shaded Demo geometry image129x129; 12b/ch color map512x512; 8b/ch demo

  28. Results 257x257 normal-map 512x512

  29. Results 257x257 color image 512x512

  30. boundary constraintsset for size 65x65 Mip-mapping 257x257 129x129 65x65

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

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

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

  34. Rate distortion

  35. Some artifacts aliasing anisotropic sampling

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

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

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