Parameterization

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. 7 6 Example 3 2 0 1 4 5

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 (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. 0 0 0 0 a b c 3 1 2 3 3 1 2 1 2 Fixed vertices 1 & 2 (u1,v1) & (u2,v2) locked AaMa -AaRMa Variable change columns swap

25. Al Af Af Al

26. 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 b a b2 g3 g1 b1

30. 1963 1995 2003 2002

31. 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. Ab Ai Ab Ai BACK

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

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