Geometry Images

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Geometry Images
Steven Gortler Harvard University
Xianfeng Gu Harvard University
Hugues Hoppe Microsoft Research
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Irregular 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
t
normal map
s
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Complicated rendering process
Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2
s1 t1 s2 t2
Face 2 1 3 Face 4 2 3
random access!
random access!
40M ?/sec
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Semi-regular representations
Eck et al 1995 Lee et al 1998 Khodakovsky
2000 Guskov et al 2000
only semi-regular
irregular vertex indices
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Geometry Image
3D geometry
completely regular sampling
geometry image257 x 257 12 bits/channel
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Basic idea
cut
parametrize
demo
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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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How to cut ?
2D surface disk
sphere in 3D
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How to cut ?
2D surface disk
sphere in 3D

• Genus-0 surface ? any tree of edges

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How to cut ?
torus (genus 1)

• Genus-g surface ? 2g generator loops minimum

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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

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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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Results of Steps 1 2
genus 1
genus 0
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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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Stretch parametrization
Previous metrics
(Floater, harmonic, uniform, )
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Sample
geometry image
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Rendering
(65x65 geometry image)
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Rendering with attributes
geometry image 2572 x 12b/ch
normal-map image 5122 x 8b/ch
rendering
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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

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Normal-Mapped Demo
geometry image129x129 12b/ch
normal map512x512 8b/ch
demo
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Results
257x257
normal-map 512x512
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Results
257x257
color image 512x512
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Mip-mapping
257x257
129x129
65x65
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Hierarchical culling
view-frustum culling
geometry image
backface culling
normal-map image
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Compression
Image wavelet-coder
? 1.5 KB
295 KB
fused cut
topological sideband (12 B)
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Compression results
295 KB ?
1.5 KB
3 KB
12 KB
49 KB
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Rate distortion
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Some artifacts
aliasing
anisotropic sampling
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Summary

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

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Future work

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

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(No Transcript)
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Pre-shaded Demo
geometry image129x129 12b/ch
color map512x512 8b/ch
demo