Title: Geometry Image
1Geometry Image
- Xianfeng Gu, Steven Gortler, Hugues Hoppe
- SIGGRAPH 2002
Present by Pin Ren Feb 13, 2003
2Irregular Triangle Meshes
Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2
Face 2 1 3 Face 4 2 3
3Texture 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
4Irregular?Regular, How?
Eck et al 1995 Lee et al 1998 Khodakovsky
2000 Guskov et al 2000
Remesh into Semi-Regular Connectivity
5Geometry Image--completely regular sampling
geometry image257 x 257 12 bits/channel
6Basic idea
cut
parametrize
7Basic idea
cut
sample
8Basic idea
cut
store
render
r,g,b x,y,z
9Creation 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!
10Surface 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
11Step 1 Find topologically-sufficient cut
(a) retract 2-simplices
(b) retract 1-simplices
12Results of Step 1
genus 6
genus 0
genus 3
13Step 2 Augment cut
- Make the cut pass through extrema (note not
local phenomena). - Approach parametrize and look for bad areas.
14Step 2 Augment cut
iterate while parametrization improves
15Parameterize Methods
- Boundary
- To avoid Crack constraints apply
- To avoid degeneracy more constraints
- Minor adjustments for better result
- Interior
- Geometric-Stretch metric
- Other metric Floater
16Parametrize boundary
a
a
a
a
- Constraints
- cut-path mates identical length
- endpoints at grid points
? no cracks
17Parametrize interior
- Geometric-stretch metric
- minimizes undersampling Sander et al 2001
- optimizes point-sampled approx. Sander et al
2002
18Sampling
19Rendering
Span each quad of samples with two triangles.
20Rendering with Attributes
geometry image 2572 x 12b/ch
normal-map image 5122 x 8b/ch
21Mip-mapping
257x257
129x129
65x65
22Advantages
- 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.
23Compression
- Completely regular sample means
- Can take full advantages of off-the-shelf image
compression codes. -
Image Wavelets Coder 295KB?1.5KB plus 12B
sideband
24Compression Results
295KB
1.5KB
3KB
12KB
49KB
25Limitations
- Higher genus can be problematic
- Since it is based on sampling approach,
- it does suffer from artifacts
- Has difficulty to capture sharp surface features.
26Summary
- 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
28More Pics1
257x257
normal-map 512x512
29More Pics2
257x257
color image 512x512
30More Pics3 artifacts
aliasing
anisotropic sampling
31Stretch parametrization
Previous metrics
(Floater, harmonic, uniform, )