Geometry Images - PowerPoint PPT Presentation

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Geometry Images

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Hierarchical culling. view-frustum culling. backface culling. geometry image ... Hierarchical culling. Compressible. Future work. Better cutting algorithms ... – PowerPoint PPT presentation

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Title: Geometry Images


1
Geometry Images
Steven Gortler Harvard University
Xianfeng Gu 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
s1 t1 s2 t2
Face 2 1 3 Face 4 2 3
t
normal map
s
4
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
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
3D geometry
completely regular sampling
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 ?
2D surface disk
sphere in 3D
  • Genus-0 surface ? any tree of edges

12
How to cut ?
torus (genus 1)
  • Genus-g surface ? 2g generator loops minimum

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 0
genus 3
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
a
a
a
a
  • Constraints
  • cut-path mates identical length
  • endpoints at grid points

? no cracks
20
Parametrize interior
  • Geometric-stretch metric
  • minimizes undersampling Sander et al 2001
  • optimizes point-sampled approx. Sander et al
    2002

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
Mip-mapping
257x257
129x129
65x65
31
Hierarchical culling
view-frustum culling
geometry image
backface culling
normal-map image
32
Compression
Image wavelet-coder
? 1.5 KB
295 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

38
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