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

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

SGP 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

SGP 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]

SGP 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”

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

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

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

SGP 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)

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

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

SGP Conformal Mapping Similarity = Rotation + Scale Preserves angles

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

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

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

SGP 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.

SGP 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

SGP Optimal Local Rotation AR

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

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

SGP A S AP A R AP

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

SGP A S APA R AP

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

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

SGP Results λ= (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:

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

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

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

SGP 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)

SGP 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)

SGP 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

SGP Thank you !