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Geometry Image

Geometry Image. Xianfeng Gu, Steven Gortler, Hugues Hoppe SIGGRAPH 2002. Present by Pin Ren Feb 13, 2003. Irregular Triangle 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

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Geometry Image

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  1. Geometry Image Xianfeng Gu, Steven Gortler, Hugues Hoppe SIGGRAPH 2002 Present by Pin Ren Feb 13, 2003

  2. Irregular Triangle 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 random access! t random access! s normal map

  4. IrregularRegular, How? • Previous work: [Eck et al 1995] [Lee et al 1998] [Khodakovsky 2000] [Guskov et al 2000] … Remesh into Semi-Regular Connectivity

  5. Geometry Image--completely regular sampling geometry image257 x 257; 12 bits/channel

  6. Basic idea cut parametrize

  7. Basic idea cut sample

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

  9. Creation of Geometry Image • How can we get the Geometry Image? • Cut M into M’ which has the topology of a disk • Parameterize: piecewise linear map from domain unit square D to M’ • Resample it at D’s grid points • Key Points: • Good Cut • Good Parameterization • Approach: Combine those two goals together!

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

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

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

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

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

  15. Parameterize Methods • Boundary • To avoid Crack: constraints apply • To avoid degeneracy: more constraints • Minor adjustments for better result • Interior • Geometric-Stretch metric • Other metric: Floater …

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

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

  18. Sampling

  19. Rendering Span each quad of samples with two triangles.

  20. Rendering with Attributes geometry image 2572 x 12b/ch normal-map image 5122 x 8b/ch

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

  22. Advantages • Regular Sampling – no vertex indices • Unified Parameterization – no texture coord. • Directly Mip-mapping, • Rendering process is done in SCAN ORDER! • Much simpler than traditional rendering process • Inherently natural for hardware acceleration.

  23. Compression • Completely regular sample means: • Can take full advantages of off-the-shelf image compression codes. Image Wavelets Coder: 295KB1.5KB plus 12B sideband

  24. Compression Results 295KB 1.5KB 3KB 12KB 49KB

  25. Limitations • Higher genus can be problematic • Since it is based on sampling approach, • it does suffer from artifacts • Has difficulty to capture sharp surface features.

  26. Summary • Geometry Image is a novel method to represent geometries in a completely regular and simple way. • It has some very valuable advantages over traditional triangular meshes. • May Inspire new hardware rendering tech. • Based on sampling, may not be able to capture all the details

  27. All pictures credit to the original Siggraph02 presentation slides

  28. More Pics1 257x257 normal-map 512x512

  29. More Pics2 257x257 color image 512x512

  30. More Pics3 – artifacts aliasing anisotropic sampling

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

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