1. SGP 2008 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. SGP 2008 2 Mesh Parameterization Input 3D mesh Output Flattened 2D mesh

3. SGP 2008 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. SGP 2008 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 [L é vy 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. SGP 2008 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. SGP 2008 6 The Key Idea perform local transformations on triangles and stitch them all together consistently to a global solution

7. SGP 2008 7 Local/Global Approach Stitch globally Input 3D mesh Output 2D parameterization

8. SGP 2008 8 Triangle Flattening Each individual triangle is independently flattened into plane without any distortion Reference triangles Isometric

9. SGP 2008 9 Intrinsic Deformation Energy : some family of allowed linear transformations Area of 3D triangleJacobian matrix of L t (Affine) Reference triangles xParameterization u e.g. similarity or rotation Auxiliary linear (Linear)

10. SGP 2008 10 Unknown linear transformation Angles of triangle Source 2D coords Unknown Target 2D coords [Pinkall and Polthier 1993] Extrinsic Deformation Energy

11. SGP 2008 As-Similar-As-Possible (Conformality) M  family of similarity transformations

12. SGP 2008 12 Conformal Mapping Similarity = Rotation + Scale Preserves angles

13. SGP 2008 13 Linear system in u, a, b As-Similar-As-Possible (ASAP) A t  Similarity transformations Auxiliary variables

14. SGP 2008 14 As-Similar-As-Possible (ASAP) Equivalent to LSCM technique [Levy et al. 2002] which minimizes singular values of the Jacobian

15. SGP 2008 As-Rigid-As-Possible (Rigidity) M  family of rotation transformations

16. SGP 2008 16 As-Rigid-As-Possible (ARAP) A t  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. SGP 2008 17 As-Rigid-As-Possible (ARAP) A t  Rotations Non-linear system in u,a,b Solve by “local/global” algorithm [Sorkine and Alexa, 2007] : Find an initial guess of u while not converged Fix u and solve locally for each A t Fix A t and solve globally for u end Poisson equation SVD

18. SGP 2008 18 Optimal Local Rotation AR

19. SGP 2008 19 Advantages Each iteration decreases the energy Matrix L of Poisson equation is fixed –Precompute Cholesky factorization –Just back-substitute in each iteration

20. SGP 2008 20 As-Rigid-As-Possible (ARAP) Equivalent to minimizing:

21. SGP 2008 21 A S AP A R AP

22. SGP 2008 22  2222 angle-preserving (conformal) area-preserving (authalic) length-preserving (isometric) A S APA R AP Most conformalMost isometric

23. SGP 2008 23 A S APA R AP

24. SGP 2008 Tradeoff Between Conformality and Rigidity A S APA R AP Preserves angles, but not preserve area ? Tradeoff Preserves areas, but damage conformality

25. SGP 2008 25 Hybrid Model Local Phase: Solve cubic equation for a t and b t Global Phase: Poisson equation = 0  ASAP =   ARAP parameter Similarity transformation

26. SGP 2008 26 Results 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) ASAP (λ=0) (2.05, 15.6) Angular distortion:Area distortion:

27. SGP 2008 27 Effect of  A S AP (λ=0) A R AP (λ=  ) = 0  

28. SGP 2008 28 Effect of  A S AP (λ=0) A R AP (λ=  ) = 0  

29. SGP 2008 29 Multiple Boundaries ABF++ (2.00, 2.09) ARAP (2.01, 2.01)

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

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

32. SGP 2008 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. SGP 2008 33 Thank you !