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Geometry Images - PowerPoint PPT Presentation


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Geometry Images. Xianfeng Gu Harvard University. Steven Gortler Harvard University. Hugues Hoppe Microsoft Research. Irregular meshes. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 …. Face 2 1 3 Face 4 2 3 …. Texture mapping. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 ….

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geometry images

Geometry Images

Xianfeng Gu

Harvard University

Steven Gortler

Harvard University

Hugues Hoppe

Microsoft Research

irregular meshes
Irregular meshes

Vertex 1 x1 y1 z1

Vertex 2 x2 y2 z2

Face 2 1 3

Face 4 2 3

texture mapping
Texture mapping

Vertex 1 x1 y1 z1

Vertex 2 x2 y2 z2

Face 2 1 3

Face 4 2 3

s1 t1

s2 t2

t

normal map

s

complicated rendering process
Complicated rendering process

Vertex 1 x1 y1 z1

Vertex 2 x2 y2 z2

Face 2 1 3

Face 4 2 3

s1 t1

s2 t2

random access!

random access!

~40M Δ/sec

semi regular representations
Semi-regular representations

[Eck et al 1995]

[Lee et al 1998]

[Khodakovsky 2000]

[Guskov et al 2000]

only semi-regular

irregular vertex indices

geometry image
Geometry Image

completely regular sampling

3D geometry

geometry image257 x 257; 12 bits/channel

basic idea
Basic idea

cut

parametrize

demo

basic idea8
Basic idea

cut

sample

basic idea9
Basic idea

cut

store

render

[r,g,b] = [x,y,z]

how to cut
How to cut ?

2D surface disk

sphere in 3D

how to cut11
How to cut ?
  • Genus-0 surface  any tree of edges

2D surface disk

sphere in 3D

how to cut12
How to cut ?
  • Genus-g surface  2g generator loops minimum

torus (genus 1)

surface cutting algorithm
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]

step 1 find topologically sufficient cut
Step 1: Find topologically-sufficient cut

(a) retract 2-simplices

(b) retract 1-simplices

results of step 1
Results of Step 1

genus 6

genus 3

genus 0

step 2 augment cut
Step 2: Augment cut
  • Make the cut pass through “extrema” (note: not local phenomena).
  • Approach: parametrize and look for “bad” areas.
step 2 augment cut17
Step 2: Augment cut

…iterate while parametrization improves

parametrize boundary
Parametrize boundary

Constraints:

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

a

a’

a’

a

 no cracks

parametrize interior
Parametrize interior
  • optimizes point-sampled approx. [Sander et al 2002]
  • Geometric-stretch metric
    • minimizes undersampling [Sander et al 2001]
slide21

Stretch parametrization

Previous metrics

(Floater, harmonic, uniform, …)

sample
Sample

geometry image

rendering
Rendering

(65x65 geometry image)

rendering with attributes
Rendering with attributes

geometry image 2572 x 12b/ch

normal-map image 5122 x 8b/ch

rendering

advantages for hardware rendering
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

normal mapped demo
Normal-Mapped Demo

geometry image129x129; 12b/ch

normal map512x512; 8b/ch

demo

results
Results

257x257

normal-map 512x512

results28
Results

257x257

color image 512x512

hierarchical culling
Hierarchical culling

view-frustum culling

geometry image

backface culling

normal-map image

compression
Compression

Image wavelet-coder

295 KB

 1.5 KB

fused cut

+ topological sideband (12 B)

compression results
Compression results

295 KB 

1.5 KB

3 KB

12 KB

49 KB

some artifacts
Some artifacts

aliasing

anisotropic sampling

summary
Summary
  • Simple rendering: compact, no indirection, raster-scan stream.
  • Mipmapped geometry
  • Hierarchical culling
  • Compressible
future work
Future work
  • Better cutting algorithms
  • Feature-sensitive remeshing
  • Tangent-frame compression
  • Bilinear and bicubic rendering
  • Build hardware
pre shaded demo
Pre-shaded Demo

geometry image129x129; 12b/ch

color map512x512; 8b/ch

demo