1. Shape Compression using Spherical Geometry Images
Surface parametrizations
2001/05/18
Shape Compression using Spherical Geometry Images
Hugues Hoppe, Microsoft Research
Emil Praun, University of Utah
IMA Workshop on Computer Graphics

2. Mesh representation
irregular
semi-regular
completely regular

3. What if images were represented with irregular meshes?
Drawbacks:
storage of connectivity
no random lookup
rendering
compositing
filtering
compression
demo

4. Simple 2D grid Advantages: implicit connectivity 2D lookup raster-scan
alpha blending
DSP
JPEG 2000

5. Representations for media
Audio: uniform 1D grid
Images: uniform 2D grid
Video: uniform 3D grid
Geometry: irregular mesh
historical artifact?

6. Surface parametrizations
2001/05/18
Geometry image
2D grid sampling
3D geometry
Our approach is to represent an arbitrary surface using a completely regular grid of samples on a square domain,
which we call a geometry image.
<click> Here is an example.
As you can see, it looks just like an ordinary image. It has 257-by-257 pixels. The only difference is that it has 12 bits/channel instead of the usual 8.
Why do we call this a geometry image?
Because its RGB colors encode XYZ positions, and in fact
<click> entirely describe this 3D shape.
As we will see later in the talk, the regularity of this representation can permit image processing techniques, such as compression, to be applied to geometry directly.
geometry image 257 x 257; 12 bits/channel
IMA Workshop on Computer Graphics

7. Surface parametrizations
2001/05/18
Geometry image
Here is an outline of the basic steps involved in creating a geometry image.
First, we slice open the mesh along an appropriate set of cut paths, creating a surface with the topology of a disk.
<click> Second, we parametrize this cut surface onto the square domain of the geometry image.
To illustrate this parametrization, we’ll show a demo.
<demo> Here we see a morph between the cut bunny and its embedding in the square.
You can see how the cut in blue is being morphed into the boundary of the square.
Because this is so much fun, I’ll show the morph one more time.
IMA Workshop on Computer Graphics

8. Surface parametrizations
2001/05/18
Geometry image
Here is an outline of the basic steps involved in creating a geometry image.
First, we slice open the mesh along an appropriate set of cut paths, creating a surface with the topology of a disk.
<click> Second, we parametrize this cut surface onto the square domain of the geometry image.
To illustrate this parametrization, we’ll show a demo.
<demo> Here we see a morph between the cut bunny and its embedding in the square.
You can see how the cut in blue is being morphed into the boundary of the square.
Because this is so much fun, I’ll show the morph one more time.
render
[r,g,b] = [x,y,z]
IMA Workshop on Computer Graphics

9. Advantages for hardware rendering
Surface parametrizations
2001/05/18
Advantages for hardware rendering
Regular sampling  no vertex indices.
Sequential traversal of source data
Unified parametrization  no texture coordinates.
IMA Workshop on Computer Graphics

10. Main questions
cut?
parametrize?

11. Construction approaches
General cut
Spherical
Multi-chart
[Gu et al. SIGGRAPH 2002]
[Praun & Hoppe. SIGGRAPH 2003]
[Sander et al. SGP 2003]
arbitrary surface
genus-zero surface
 cut symmetries
>1 chart
 zippering

12. Construction approaches
General cut
[Gu et al. SIGGRAPH 2002]
arbitrary surface
genus 6

13. Construction approaches
General cut
Spherical
Multi-chart
[Gu et al. SIGGRAPH 2002]
[Praun & Hoppe. SIGGRAPH 2003]
[Sander et al. SGP 2003]
arbitrary surface
genus-zero surface
 cut symmetries
>1 chart
 zippering
400x160
piecewise regular

14. Construction approaches
General cut
Spherical
Multi-chart
[Gu et al. SIGGRAPH 2002]
[Praun & Hoppe. SIGGRAPH 2003]
[Sander et al. SGP 2003]
arbitrary surface
genus-zero surface
 cut symmetries
>1 chart
 zippering

15. Spherical parameterization and remeshing
Surface parametrizations
2001/05/18
Spherical parameterization and remeshing
[Praun, Hoppe 2003]
IMA Workshop on Computer Graphics

16. Spherical parameterization and remeshing
Surface parametrizations
2001/05/18
Spherical parameterization and remeshing
[Praun, Hoppe 2003]
IMA Workshop on Computer Graphics

17. Spherical geometry images
Surface parametrizations
2001/05/18
Spherical geometry images
IMA Workshop on Computer Graphics

18. Steps
mesh M
sphere S
domain D
image I
demo

19. Spherical parametrization
Surface parametrizations
2001/05/18
Spherical parametrization
mesh M
sphere S
[Kent et al. 1992]
[Haker et al. 2000]
[Alexa 2002]
[Grimm 2002]
[Sheffer et al. 2003]
[Gotsman et al. 2003]
Two challenges:
robustness
good sampling
 coarse-to-fine
 stretch metric
To parameterize a model onto the sphere we want to assign to each vertex a coordinate on the sphere. Edges will correspond to great circle arcs, and mesh triangles to spherical triangles. There are two principal goals for such a map: robustness and good sampling rate. Robustness implies that the resulting map is guaranteed to be one-to-one, meaning that no triangles overlap on the sphere. By good sampling rate we mean that interesting features of the model receive enough real estate on the sphere in order to be accurately sampled. We achieve these goals by using a coarse-to-fine strategy, and minimizing map distortion using Sander et al.’s stretch metric.
[Hormann et al. 1999]
[Sander et al. 2001]
[Sander et al. 2002]
IMA Workshop on Computer Graphics

20. Coarse-to-fine algorithm
Surface parametrizations
2001/05/18
Coarse-to-fine algorithm
Convert to progressive mesh
We start by converting the model to a progressive mesh sequence, successively simplifying it using half-edge collapses, until we reach a tetrahedron. We embed the tetrahedron on the sphere and then we refine the model back to its original resolution. For each vertex split operation we assign a spherical coordinate to the newly introduced vertex, and also optimize the location of its neighbors in order to minimize map distortion.
Parametrize coarse-to-fine
(maintain embedding & minimize stretch)
IMA Workshop on Computer Graphics

21. Traditional conformal metric
Surface parametrizations
2001/05/18
Traditional conformal metric
Preserve angles but “area compression”
Bad for sampling using regular grids
As I mentioned, to evaluate map distortion we use Sander’s stretch metric. The most popular metric for parametrization is conformality. This metric is not well suited for regular resampling since it suffers from area compression. In this example, the horse’s legs and head receive a very small area in the map, and therefore the remesh does a poor job at sampling those features.
IMA Workshop on Computer Graphics

22. Surface parametrizations
2001/05/18
[Sander et al. 2001]
Stretch metric
[Sander et al. 2002]
Penalizes undersampling
Better samples the surface
By contrast, the stretch metric penalizes undersampling, by trying to map large distances on the sphere (and therefore many samples) to short distances on the mesh, so it doesn’t miss features. As you can see, this metric does a better job of capturing the horse geometry.
IMA Workshop on Computer Graphics

23. Applications of spherical remeshing
Surface parametrizations
2001/05/18
Applications of spherical remeshing
Level-of-detail control
Morphing
Geometry amplification
Shape compression
IMA Workshop on Computer Graphics

24. Level-of-detail control

25. Morphing Align meshes on the sphere.
Interpolate the resulting geometry images.

26. Geometry amplification
Surface parametrizations
2001/05/18
Geometry amplification
[Losasso et al. SGP 2003]
“smooth geometry images”
simulation
CPU
GPU
33x33
65x65
129x129
floating-point geometry image
257x257 +
257x257
scalar displacements
demo
IMA Workshop on Computer Graphics

27. Shape compression (Genus-zero shapes) Spherical image topology
Infinite 2D tiling
Wavelets on regular 2D grid

28. Spherical image topology

29. Spherical image topology
Surface parametrizations
2001/05/18
Spherical image topology
Not only do samples along the boundary correspond to each other, but there is a simple way to extend neighborhood queries outside the geometry image. For example, if we are at the blue dot on the left of the image and we want the neighbor located a few samples above it, that coresponds on the uncut octahedron to walking across the orange edge that was cut into the adjacent face. In the domain, we leave the image on the left going up and enter it on the right going down.
IMA Workshop on Computer Graphics

30. Spherical image topology
Surface parametrizations
2001/05/18
Spherical image topology
Another way to look at this is that we keep going up into another copy of the geometry image, obtained by rotating the original about the midpoint of the top edge.
IMA Workshop on Computer Graphics

31. Surface parametrizations
2001/05/18
Infinite 2D tiling
We can apply this rotation process to all the edge midpoints to obtain an infinite lattice of geometry images. This lattice is continuous everywhere and derivativative continuous everywhere except at the repeated instances of the four edge midpoints. Derivative continuity is very important for many processing applications, and we make use of it in our compression results.
IMA Workshop on Computer Graphics

32. Wavelets on regular 2D grid
spherical wavelets
image wavelets
[Schröder & Sweldens 1995]
[Davis 1995] [Antonini et al 1992]

33. Test models

34. Compression results

35. Compression results

36. Compression results

37. Compression results

38. Compression results

39. Summary Geometry image Applications Simplicity of 2D grid Rendering
LOD
Morphing
Geometry amplification
Shape compression

40. Future work
Visual error metrics [Touma & Gotsman 1998] [Sorkine et al 2003]
Attenuation of rippling artifacts
Surface boundaries
Animated meshes “geometry videos” [Briceño et al 2003]