Geometry Image

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Geometry Image

• Xianfeng Gu, Steven Gortler, Hugues Hoppe
• SIGGRAPH 2002

Present by Pin Ren Feb 13, 2003
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Irregular Triangle Meshes
Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2
Face 2 1 3 Face 4 2 3
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Texture mapping
Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2
s1 t1 s2 t2
Face 2 1 3 Face 4 2 3
random access!
t
random access!
s
normal map
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Irregular?Regular, How?

• Previous work

Eck et al 1995 Lee et al 1998 Khodakovsky
2000 Guskov et al 2000
Remesh into Semi-Regular Connectivity
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Geometry Image--completely regular sampling
geometry image257 x 257 12 bits/channel
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Basic idea
cut
parametrize
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Basic idea
cut
sample
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Basic idea
cut
store
render
r,g,b x,y,z
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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 Ds grid points
• Key Points
• Good Cut
• Good Parameterization
• Approach Combine those two goals together!

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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

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Step 1 Find topologically-sufficient cut
(a) retract 2-simplices
(b) retract 1-simplices
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Results of Step 1
genus 6
genus 0
genus 3
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Step 2 Augment cut

• Make the cut pass through extrema (note not
  local phenomena).
• Approach parametrize and look for bad areas.

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Step 2 Augment cut
iterate while parametrization improves
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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

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Parametrize boundary
a
a
a
a

• Constraints
• cut-path mates identical length
• endpoints at grid points

? no cracks
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Parametrize interior

• Geometric-stretch metric
• minimizes undersampling Sander et al 2001

• optimizes point-sampled approx. Sander et al
  2002

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Sampling
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Rendering
Span each quad of samples with two triangles.
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Rendering with Attributes
geometry image 2572 x 12b/ch
normal-map image 5122 x 8b/ch
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Mip-mapping
257x257
129x129
65x65
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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.

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

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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

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• All pictures credit to the original Siggraph02
  presentation slides

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More Pics1
257x257
normal-map 512x512
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More Pics2
257x257
color image 512x512
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More Pics3 artifacts
aliasing
anisotropic sampling
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Stretch parametrization
Previous metrics
(Floater, harmonic, uniform, )