1. Bounded-distortion Piecewise Mesh Parameterization
Olga Sorkine, Daniel Cohen-Or
Tel-Aviv University
Rony Goldenthal, Dani Lischinski
The Hebrew University of Jerusalem

2. Overview Parameterization – definition Applications
Distortion and partition
Previous work
Our approach
Results

3. Parameterization - definition
S  R given surface
D  R2 - parameter domain
s : D  S and onto

4. Mesh parameterization
Uniquely defined by mapping the mesh vertices to the parameter domain:
U : {v1, v2, …, vn} → D  R2
U(vi) = (ui, vi)
No two edges cross in the plane
U is piecewise linear (linear inside each face)
Mesh parameterization  mesh embedding

5. Applications
Texture mapping, 3D painting
Resampling, remeshing
Digital geometry processing
Multi-resolution analysis
Using parameterization, we can operate on the 3D surface as if it were flat.
Remeshing images taken from “Interactive Geometry Remeshing”, P. Alliez, M. Meyer and M. Desbrun, SIGGRAPH 2002

6. Distortion and partition
Ideal parameterization is an isometry
General surfaces cannot be embedded without distortion
Surfaces with non-zero Gaussian curvature
Non-disk topology
Partitioning the surface reduces distortion
But: the parameterization is not continuous over the boundaries between the patches of the partition

7. Previous work Partition/cut the mesh in pre-process
Interactive user input
Normals bucketing
Region growing from feature curves (Lévy et al. 02)
Flatten each patch by energy minimization
Convex mapping (Floater 97)
Harmonic mapping (Maillot et al. 93, Eck et al 95)
Conformal mapping (Lévy et al. 02, Desbrun et al. 02)
Non-linear Jacobian energy minimization (Sander et al. 01)

8. Previous work – partial list
Maillot et al. 93
Eck et al. 95
Floater 97
Lévy and Mallet 98
Lee et al. 98
Haker et al. 00
Sander et al. 01
Gu et al. 02
Sheffer 02
Lévy et al. 02
Desbrun et al. 02
Zigelman et al. 02
Bennis et al. 91

9. Previous framework - discussion
A-priori partition sets lower bound on the distortion  cannot comply with preset upper bound on the distortion.
If the distortion is too high, need to subdivide the partition and recompute the parameterization.
Most of the methods cannot prevent triangle flips and global self-intersections (overlaps).
High computational cost (for non-linear optimizations).

10. If distortion > threshold stop
Bennis et al. 91
If distortion > threshold stop  
Surface
Plane
Works on C2 surfaces sampled on regular grid, even geodesic spacing in both u and v directions.

11. Our contribution Parameterization with bounded distortion
Simultaneous partition and parameterization
Valid parameterization – no self-intersections
Simple and fast algorithm
Generic scheme

12. Algorithm overview
Greedy algorithm: grow one patch at a time, until no more vertices can be added.
At each step, attempts to flatten the “best” vertex adjacent to the current patch – local criteria.
The distortion of each mesh triangle is guaranteed to be below specified threshold.

13. Algorithm overview Select random seed triangle, flatten it.
Maintain a priority queue of the vertices adjacent to the current patch.
Flatten triangles adjacent to current patch:
At each step, take the best vertex off the queue
Check for self-intersections
Stop when no triangles can be added to the patch, and start a new one.

14. Patch Growth
The 3D surface

15. Patch Growth
The 3D surface
The planar patch

16. Patch Growth
The 3D surface
The planar patch

17. Patch Growth
The 3D surface
The planar patch

18. Patch Growth
The 3D surface
The planar patch

19. Patch Growth
The 3D surface
The planar patch

20. Patch Growth
The 3D surface
The planar patch

21. Patch Growth
The 3D surface
The planar patch

22. Patch Growth
The 3D surface
The planar patch

23. Patch Growth
The 3D surface
The planar patch

24. Patch Growth
The 3D surface
The planar patch

25. The Jacobian distortion metric
In 2D
In 3D
D(T, T’) = max{max , 1/ min }
max and min are the singular values of the Jacobian [S/s S/t]:
The values max and min are the maximal stretching and shrinking caused to a unit-length vector by the mapping S
We want to equally “punish” stretch and shrink
Any other reasonable metric can be used!

26. Flattening a single vertex
We can apply local relaxation to optimize the vertex position, but it’s slower,
the initial guess performs well, and we wanted to keep the algorithm as simple as we can.
t1 t2
t1 t2

27. Vertex grade components
Maximal distortion caused to the triangles flattened with the vertex. If it’s greater than the threshold, the grade is set to zero and the vertex can’t be flattened in the current patch!
The ratio between patch area and squared perimeter (to create round patches with small boundary length).
Crease angles or other segmentation information.
More criteria…

28. Checking self-intersections
Local self-intersection – triangle flipping V
1  2 1 2

29. Checking self-intersections
Global self-intersections: maintain space partition data structure, check the new triangles with the existing boundary edges.

30. Adding seams
After the patch is finished, we may add seams as a post-process, to benefit from cylinder-like structure.

31. Adding seams
After the patch is finished, we may add seams as a post-process, to benefit from cylinder-like structure.

32. Results
11,000 triangles, 1.3 seconds, unoptimized code

33. Results
1.0
1.5
2.0
3.0
1.0
1.5
2.0
3.0
1.0
1.5
2.0
3.0

34. Results Bounding the area/perimeter2 ratio...
40,000 triangles, 4 seconds, unoptimized code

35. Results
100,000 triangles, 9 seconds, unoptimized code

36. Results

37. Conclusions from the comparison:
Comparison with global relaxation technique, normal bucketing partition
Conclusions from the comparison:
Much faster
Lower average distortion
Boundary length – sometimes the same, sometimes longer…

38. Texture mapping with our parameterization

39. 3D painting with our parameterization

40. Summary A simple and fast method for surface parameterization
Simultaneously computes the partition and the parameterization
Employs local (generic) criteria rather than global ones
No self-intersections
No explicit control on the number and the size of the patches
Future work: to incorporate segmentation information to gain some “global” properties

41. Acknowledgements
Israel Science Foundation founded by the Israel Academy of Sciences and Humanities
Israeli Ministry of Science
German Israel Foundation (GIF)
Deutsch Institute

42. Thank you!

43. Normal partition Doesn’t detect developable structures
Prone to self-intersection problems

44. The Jacobian distortion metricp3 q3 S T T’ p1 q1 q2 p2
In 2D
In 3D

45. Flattening a single vertex
The obtained position v can be used as initial guess for a relaxation procedure that finds better position v’ (for example, minimizes the average distortion of triangles incident to the vertex).
However, it is slower, and the initial guess performs well in practice.