1. PolyCube-Maps seamless texture mapping
Marco Tarini · Kai Hormann · Paolo Cignoni · Claudio Montani
Visual Computing Lab · ISTI · CNR · Pisa, Italy

2. Our Goal Texture mapping seamless hardware supported low distortion
general object

3. Texture Atlas (multi-chart approach to parameterization)
disk-like patches
texture charts v + =
Let me start this talk with two concepts that we are all familiar with. u
3D mesh
2D texture image
textured bunny
images courtesy of Lévy, Sylvain, Ray and Maillot , SIGGRAPH 02

4. Cube-Maps Typically used for environment mapping mesh environment
per-fragment projection over the cube
per-fragment texture fetch from the appropriate face
mesh
environment
texture
images from Bubble demonstration program, nVidia

5. with per-vertex 3D texture coord. (u,v,w)
Abusing Cube-Maps
What if we store surface color in a cube-map?
object space
texture space
object space u v w + =
“But we can also think of an ABuse of cubemaps.
Take a spherical objact, say, an apple.
We could then use a cubemap to color it,
rather than the env around it.”
We would assign a 3D texture coord to each vertex.
Color info defined on a 3D texture domain <click> BUT stored in a 2D domain.
The hw would then project, for each produced frgment, the interpolated t.c. on that cube, and
mesh
with per-vertex 3D texture coord. (u,v,w)
cube-map
textured apple

6. texture atlas cube-map seams a triangle cannot span multiple charts
from
chart B
texture
from
chart A
texture atlas
seams
a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined

7. texture atlas cube-map seams a triangle cannot span multiple charts
from
chart B
texture
from
chart A
texture atlas
seams
a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined

8. texture atlas cube-map seams a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined

9. texture atlas cube-map seams a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined

10. texture atlas cube-map seams a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined

11. texture atlas cube-map works (is general) does not (~spheres only!)
seams
a triangle cannot span multiple charts
mesh dependency
mipmapping difficult
chart packing: wasted texels
artifacts at boundaries
no defined neighbors for boundary texels
works (is general)
cube-map
seamless
a triangle can span multiple faces
mesh independent
mipmapping ok
no packing: no wasted texels
no boundaries, no artifacts
texel neighbors always defined
does not (~spheres only!)
so, why don’t we use cubemaps instead of standard texture? Obvoiusly, it is because what we are showing in the right part of this slide is a TOY exsample, that can work only because the texture object is similar to a sphere (actually, to a cube).
Our objective is to have the same set of advantages for a GENERAL MESH. For that, we will extend the Cubemap mechanism. Before we do that, let see more closely what ingredient, which secret souce, make cubemap so ideal to texture an apple.

12. What we Keep from CubeMaps…
from
rasterizer
Texture defined in 3D BUT stored in 2D
interpolated coordinates
in 3D texture space
projection
coordinates
in 3D texture domain
(a 2-manifold similar to the mesh)
Per-fragment
Hardware implemented
coordinates in 3D texture domain
FRAGMENT SHADER
mapping
coordinates
in 2D texture domain
2D texture
fetch…

13.  Going Beyond Apples Cube-Maps work well only for sphere-like objects
world space
texture space 
Let us try to go beyond our toy example.
sphere-like mesh
cubic texture domain

14.  Going Beyond Apples But for more general objects? ? world space
texture space ? 
arbitrary meshes
cubic texture domain

15. incompatible topology
Going Beyond Apples
But for more general objects?
world space ?
texture space 
...huge distortions,
incompatible topology
arbitrary meshes
cubic texture domain
[Praun Hoppe SIGGGRAPH 2003]

16.  Going Beyond Apples But for more general objects? world space
texture space
far too complex (GPU) 
arbitrary mesh
coarse mesh

17. Introducing Polycubes
Po·ly·cube: n. (Geom.) A solid composed by multiple unit cubes attached face to face

18. Choosing a Polycube Polycube should roughly resemble the mesh
world space
texture space
Why polycubes?
1st, polycubes give us enough flexibility to choose one that
roughly resembles the given mesh and captures its large-scale features
-> important to keep parametric distortion under bearable levels
THEN, it is easy to define, and implement, the projection and the mapping to 2D that, as we have seen, we need.
How do we do that? We will not go into detail here, but <click>
mesh
polycube

19. Partition of Texture Space
2D analogue
case A
case B
dual grid
a key ingredient here is
polycube

20. partition of the parameter space
back to 3D
case 3
case 4-a
case 4-b
case 5
case 6-a
case 6-b

21. partition of the parameter space
back to 3D
dual
cells
case 4-a
case 5
case 4-b
case 3

22. partition of the parameter space
dual
cells
case 4-a
case 5
case 4-b
case 3

23. polycube-maps in a nutshell
texture space
(3D!)
the polycube
stored in texture RAM as:
a packed texture image
not necessarily
on the polycube
surface: project
object space
a fragment
with interpolated
texture coord u v w
final texel value
for the fragment
map to 2D
plus a tiny structure
to store polycube layout
the mesh
(with per-vertex text. coord)

24. Fragment shader Input: current texture position (u,v,z)
Find cell (in dual graph)
Access information for that cell
case
orientation
offset in global texture
Apply axis-to-axis rotation inside cell
according to orientation
Apply projection
according to case
Add global offset
Fetch final texel at resulting position
stored in a texel
in the tiny 3D lookup table

25. Examples of poly-cubic parameterizations
in texture RAM
object
polycube-map w v u
world space
texture space

26. Examples of poly-cubic parameterizations
object
polycube-map w v u
world space
texture space

27. Examples of poly-cubic parameterizations
object
polycube-map w u v
world space
texture space

28. Examples of poly-cubic parameterizations
object
polycube-map w v u
world space
texture space

29. An example application: same texture for different LODw + =
Geometry 3 u v + =
Geometry 2
the same polycube-map
texture
Geometry 1 + =

30. How to build a Polycube-Map (for a given mesh)
Not automatic, to this point
get a suitable polycube
warp it around the mesh
project mesh over it
unwarp
global optimization

31. Global optimization mean value MIPS ext. MIPS projection
(not optimized)

32. Discussion Pros Cons Limits truly seamless texture mapping
no patch boundaries
no color bleeding
very low distortion
nearly optimal texture packing
bilinear filtering possible
mipmapping possible
mesh independency
Cons
long fragment program
~60 GPU instruction long
could be improved, with little branching support
3 t-coords per vertex
instead of 2
Limits
cannot handle arbitrary shape/topology complexity
e.g. a tree

33. Thank you! Questions?
or, just Google for “polycube-map”
Marco Tarini · Kai Hormann · Paolo Cignoni · Claudio Montani
Visual Computing Lab · ISTI · CNR · Pisa, Italy

34. Q: why only 6 cases? MC has 15...
A: only polycubes with two-manifold surfaces are considered... no no

35. Q: Aren’t cubemaps just another atlas?v w
A: true, but such chart borders are:
transparent for triangles
invisible for mipmapping...
hidden from the user
texture charts?
chart
boundaries?

36. Q: Similarities with volumetric textures
Yes, a cube map can be seen as a compressed 3D texture…
…which works only for a 2D surface in 3D texture space
(and around it)
texture stored in 1D
values
defined
in a 2D
space
2D analogue

37. Q: isn’t it a lot of work to build a polycube map?
Polycube-maps
Get a fitting polycube
Warp it around mesh
Project mesh over it
Unwarp back
Global optimization
Atlases
Find a good division into disk-like patches
Pack charts togheter
Global optimization

38. Q: isn’t it a lot of work to build a polycube map?
Polycube-maps
New. Not studied at all.
Future work: design tools to make task a lot easier (but still user guided)
In any case, final result will be clean (seamless...)
Atlases
Very much studied problem
Automatic results inferior w.r.t. manual results
In any case, final parameterization bound to have drawbacks (seams...)