A Local/Global Approach to Mesh Parameterization

1. A Local/Global Approach to Mesh Parameterization Ligang Liu Lei Zhang Yin Xu Zhejiang University, China Craig Gotsman Technion, Israel Steven J. Gortler Harvard University, USA

2. Mesh Parameterization Output Flattened 2D mesh Input 3D mesh

3. Mesh Parameterization • Isometric mapping • Preserves all the basic geometry properties: length, angles, area, … • For non-developable surfaces, there will always be some distortion • Try to keep the distortion as small as possible

4. Previous Work • Discrete harmonic mappings • Finite element method [Pinkall and Polthier 1993; Eck et al. 1995] • Convex combination maps [Floater 1997] • Mean value coordinates [Floater 2003] • Discrete conformal mappings • MIPS [Hormann and Greiner 1999] • Angle-based flattening [Sheffer and de Sturler 2001; Sheffer et al. 2005] • Linear methods [Lévy et al. 2002; Desbrun et al. 2002] • Curvature based [Yang et al. 2008, Ben-Chen et al. 2008, Springborn et al, 2008] • Discrete equiareal mappings • [Maillot et al.1993; Sander et al. 2001; Degener et al. 2003]

5. Inspiration • Laplacian & Poisson-based editing [Sorkine et al. 2004, Yu et al. 2004] • Deformation transfer [Sumner et al. 2004] • Linear Tangent-Space Alignment [Chen et al. 2007] • As-rigid-as-possible surface modeling [Sorkine and Alexa 2007] “Think globally, act locally”

6. The Key Idea perform local transformations on triangles and stitch them all together consistently to a global solution

7. Local flattening Stitch globally Local/Global Approach Input 3D mesh Output 2D parameterization

8. Triangle Flattening • Each individual triangle is independently flattened into plane without any distortion Isometric Reference triangles

9. (Linear) Intrinsic Deformation Energy (Affine) Reference triangles x Parameterization u : some family of allowed linear transformations Auxiliary linear e.g. similarity or rotation Area of 3D triangle Jacobian matrix of Lt

10. Unknown Target 2D coords Source 2D coords Angles of triangle Unknown linear transformation Extrinsic Deformation Energy [Pinkall and Polthier 1993]

11. As-Similar-As-Possible (Conformality) M family of similarity transformations

12. Conformal Mapping Similarity = Rotation + Scale Preserves angles

13. Auxiliary variables As-Similar-As-Possible (ASAP) At Similarity transformations Linear system in u, a, b

14. As-Similar-As-Possible (ASAP) • Equivalent to LSCM technique [Levy et al. 2002] which minimizes singular values of the Jacobian

15. As-Rigid-As-Possible (Rigidity) M family of rotation transformations

16. As-Rigid-As-Possible (ARAP) At Rotations Non-linear system in u,a,b We will treat u and A as separate sets of variables, to enable a simple optimization process.

17. As-Rigid-As-Possible (ARAP) SVD Poisson equation At Rotations Solve by “local/global” algorithm [Sorkine and Alexa, 2007] : Non-linear system in u,a,b Find an initial guess of u while not converged Fix u and solve locally for each At Fix At and solve globally for u end

18. Optimal Local Rotation A R

19. Advantages • Each iteration decreases the energy • Matrix L of Poisson equation is fixed • Precompute Cholesky factorization • Just back-substitute in each iteration

20. As-Rigid-As-Possible (ARAP) • Equivalent to minimizing:

21. ASAP ARAP

22. 1 2 angle-preserving (conformal) area-preserving (authalic) length-preserving (isometric) ASAP ARAP Most conformal Most isometric

23. ASAP ARAP

24. Tradeoff Between Conformality and Rigidity ? Tradeoff ASAP ARAP Preserves angles, but not preserve area Preserves areas, but damage conformality

25. Hybrid Model Similarity transformation parameter Local Phase: Solve cubic equation for atand bt Global Phase: Poisson equation • = 0  ASAP •  =   ARAP

26. Results ASAP (λ=0) (2.05, 15.6) λ=0.0001 (2.05, 5.74) λ=0.001 (2.07, 2.88) λ=0.1 (2.18, 2.14) ARAP (λ=) (2.19, 2.11) Angular distortion: Area distortion:

27. Effect of  ASAP (λ=0) ARAP (λ=) • = 0  

28. Effect of  ASAP (λ=0) ARAP (λ=) • = 0  

29. Multiple Boundaries ABF++ (2.00, 2.09) ARAP (2.01, 2.01)

30. ASAP (2.01, 30.1) ARAP (2.03, 2.03) ABF++ (2.01, 2.19) Linear ABF [Zayer et al. 2007] (2.01, 2.22) Inverse Curvature Map [Yang et al. 2008] (2.46, 2.51) Curvature Prescription [Ben-Chen et al. 2008] (2.01, 2.18)

31. Comparison ASAP ARAP ABF++ [Sheffer et al. 2005] Inverse Curvature Map [Yang et al. 2008] (2.00, 88.1) (2.06, 2.05) (2.00, 2.64) (2.05, 2.67)

32. Conclusion Simple iterative “local/global” algorithm Converges in a few iterations Low conformal and stretch distortions Generalization of stress majorization (MDS) Can be used for deformable mesh registration

33. Thank you !