Geometry Image - PowerPoint PPT Presentation

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Geometry Image

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Geometry Image Xianfeng Gu, Steven Gortler, Hugues Hoppe SIGGRAPH 2002 Present by Pin Ren Feb 13, 2003 – PowerPoint PPT presentation

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Title: Geometry Image


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
s1 t1 s2 t2
Face 2 1 3 Face 4 2 3
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 Ds 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 0
genus 3
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
a
a
a
a
  • Constraints
  • cut-path mates identical length
  • endpoints at grid points

? no cracks
17
Parametrize interior
  • Geometric-stretch metric
  • minimizes undersampling Sander et al 2001
  • optimizes point-sampled approx. Sander et al
    2002

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
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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