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Geometry Image

Learn about the innovative Geometry Image method for simplified 3D geometries, its advantages over traditional meshes, and limitations in capturing fine details. Explore the concept of irregular to regular mesh transformation, sampling techniques, and compression results. This revolutionary approach may inspire new hardware rendering technologies. (445 characters)

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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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