1. Parameterization

2. Introduction
The goal of parameterization is to attach a coordinate system to the object
In particular, assign (2D) texture coordinates to the 3D vertices
One application of mesh parameterization is texture mapping

3. UVMapper in Blender

4. Introduction (cont)
Another class of application concerns remeshing algorithm
converting from a mesh representation into an alternative one

5. Introduction (cont)
In summary, constructing a parameterization of a triangulated surface means finding a set of coordinates (ui,vi) associated to each vertex i.
Moreover, the parameter space does not self-intersect.
Self-intersect

6. Mappings in the PMP Book
5.3 Barycentric mapping
Tutte-Floater
Discrete Laplacian (Laplace-Beltrami)
5.4 Conformal mapping
5.4.2 Least square conformal maps
5.4.4 (Geometric-based) ABF (angle-based flattening)
[5.5 Distortion analysis based methods]

7. Barycentric Mapping One of the most widely used
Based on Tutte’s barycentric mapping theorem from graph theory [Tutte60]

8. Wikipedia
Convex combination
Barycentric coordinate system

9. Barycentric Mapping (cont)
Fixing the vertices of the boundary on a convex polygon. The coordinates at the internal vertices are found by solving the equation.
One simple way:
(without considering mesh geometry)

10. 1 3 2 4 5 7 6
Example

11. Ax = b, Solved by Gauss Seidel
DETAILS

12. Other Alternatives Discrete harmonic coordinates [Eck et al 1995]
Mean value coordinates [Floater 2003]

13. Mean Value Coordinate

14. For some surfaces, fixing the boundary on a convex polygon may be problematic
“Free” boundary

15. Conformal Mapping Iso-u and iso-v lines are orthogonal
Minimize mesh distorsion
A conformal parameterization transforms a small circle into a small circle. It is locally a similarity transform.

16. Types of Distortion L stretch, L shear, AngD
Isometric (length-preserving)
Conformal (angle-preserving)
Equi-areal (area-preserving)
Global isometric paramterization only exists for developable surfaces, with vanishing Gaussian curvature K(p) = 0 at all surface points
a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing"). 

17. [Differential Geometry Primer]
Gaussian curvature

18. Gradient in a Triangle  X, Y: orthonormal basis of the triangle
xi, xj, xk: vertex coordinates
in the XY basis
Study the inverse of parameterization: maps (X,Y) of the triangle to a point (u,v)
DETAILS
u intersects the iso-u lines; v intersects the iso-v lines
Conformality condition
iso-u lines  iso-v lines 

20. Least Square Conformal Map
Only developable surfaces admit a conformal paramterization.
For general (non-developable) surface, LSCM minimizes an energy ELSCM that corresponds to the non-conformality of the application
Mimizing a quadratic form
ELSCM is invariant to translation and rotation in the parametric space. To have a unique minimizer, it is required to fixed at least two vertices.
From [Levy et al.2002], if the pinned vertices are chosen on the boundary, all the triangles are consistently oriented (no flips).

21. Quadratic Optimization
Quadratic form:
a polynomial function that the degree is not larger than two.
G is symmetric
Minimizer occurs at its stationary point:

22. Least Square Minimize the sum of residuals: F is a quadratic form
(m > n)
Minimize the sum of residuals:
F is a quadratic form
Minimizer found at stationary point

23. Least Square with Reduced D.O.F.
Free parameters
Lock parameters

24. 1 3 2 2 1 a 2 3 b 3 1 c
Fixed vertices 1 & 2
(u1,v1) & (u2,v2) locked
Variable change columns swap
-AaRMa
AaMa

25. Af Al Af Al

26. ELSCM is invariant to translation and rotation in the parametric space
ELSCM is invariant to translation and rotation in the parametric space. To have a unique minimizer, it is required to fixed at least two vertices.

28. Angle-Based Flattening (ABF) [Sheffer & de Sturler 2000]
Constrained quadratic optimization with equality constraints
Nonlinear optimization
Constraints:
(wheel consistency)
Finding (ui,vi) coordinates, in terms of angles, a
Stable (ui,vi) to ai conversion

29. Wheel Consistency b3 g2 c a b b2 g3 b1 g1

30. 1963
1995
2003
2002

31. Segmentation and atlas
Other Issues
Segmentation and atlas

33. Model Segmentation
Planar parameterization is only applicable to surfaces with disk topology
Closed surfaces and surfaces with genus greater than zero have to be cut prior to planar parameterization
Cut to reduce complexity (to reduce distortion)
Cut introduce cross-cut discontinuities
Segmentation technique (partition the surface into multiple charts) and seam generaton technique (introduce cuts into the surface but keep it as a single chart)

34. Numerical Optimization
From “Mesh Parameterization,
Theory and Practice”,
Siggraph 2007 Coursenote

35. Sparse Linear System SOR (successive over-relaxation)
Simplest, both from the conceptual and the implementation points of view

36. SOR (cont)
New Update Scheme:
Successive Over-Relaxation
1w<2

37. Other Methods Conjugate gradient method Sparse direct solvers (LU)
Summary
SOR-like methods are easy to understand and implement, but do not perform well for more than 10K variables
Direct methods are most efficient, but consume considerable amounts of memory

38. References
“Polygon Mesh Processing”, Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez and Bruno Levy, AK Peters, 2010
“Mesh Parameterization: Theory and Practice”, Kai Hormann, Bruno Lévy and Alla Sheffer, ACM SIGGRAPH Course Notes, 2007
“Least Squares Conformal Maps for Automatic Texture Atlas Generation”, Bruno Lévy, Sylvain Petitjean, Nicolas Ray and Jérome Maillot, ACM SIGGRAPH conference proceedings, 2002

39. Ai Ab Ai Ab
BACK

40. [From Siggraph Course 2007]
Study inverse of parameterization (X,Y)  (u,v)
(li, lj, lk) barycentric coordinates, computed as:

41. MT solely depends on the geometry of the triangle T
Similarly, we can get
u=
MT solely depends on the geometry of the triangle T
BACK