Mesh Parameterization: Theory and Practice Setting the Boundary Free Mesh Parameterization: Theory and Practice Setting the Boundary Free Bruno Lévy - INRIA

Overview 1. Introduction - Motivations 2. Using differential geometry 3. Analytic methods 4. Conclusion

1. Introduction Setting the bndy free, why ? n Floater-Tutte: provably correct result for fixed convex boundary for fixed convex boundary

1. Introduction Seamster [Sheffer et.al] Cuts the model, ready for "pelting"

1. Introduction Fixed boundary - distortion

1. Introduction Free boundary - better result

1. Introduction Why is this important ? Demo: Normal mapping

2. Using Differential Geometry... to minimize deformations Q1) How can we compare these two mappings ? Q2) How can we design an algorithm that prefers B ? A B

2. Using Differential Geometry... to minimize deformations n [Greiner et.al]: Variational principles for geometric modeling with Splines PDEs for geometric optimization Can we port this principle to the discrete setting ?

2. Using Differential Geometry... to minimize deformations n [Hormann and Greiner] MIPS n [Pinkall and Poltier] cotan formula [Do Carmo] for meshes

2. Using Differential Geometry Notion of parameterization x (.,.) u v  RI 3 RI 2 S Object space (3D) Texture space (2D) u(x,y,z) x(u,v)

2. Geometry of T p (S) Partial derivatives of x (.,. ) v u uu vv x/ux/u x/vx/v P T P (S)

2. Geometry of T p (S) Differential dx P ; directional derivatives u 0,v 0 P w dxP(w)dxP(w) dx P (w) =  /  t ( x ( (u 0,v 0 )+ t.w ) ) )

2. Geometry of T p (S) Jacobian Matrix J P JP =JP = x/ux/u y/uy/u z/uz/u x/vx/v y/vy/v z/vz/v [ ] x/ux/u x/vx/v P dxP(w)dxP(w) uu vv w dx P (w) = w u  x/  u + w v  x/  v = J P.w u 0,v 0

2. Geometry of T p (S) Measuring things, First Fundamental Form I p T P (S) V 1 = dx p (w 1 ) ; V 2 = dx p (w 2 ) u v V 1 t V 2 = (J w 1 ) t J w 2 = w 1 t J t J w 2 = w 1 t I p w 2 V1V1 V2V2 w1w1 w2w2

2. Geometry of T p (S) Measuring things, First Fundamental Form I p Distances : || V 1 || 2 = w 1 t I p w 1 Angles : V 1 t V 2 = w 1 t I p w 2 I p is called the metric tensor

2. Geometry of T p (S) Anisotropy u v dv du  x x  x x  u u  u u  x x  x x  v v  v v T P (S) r 2 (  ) = || dx P ( cos , sin  ) || 2

2. Geometry of T p (S) Anisotropy ; 1 st fundamental form I P || dx P (w) || 2 = || J P.w || 2 = (J P w).(J P w) t = w t.J P t.J P.w = w t.I P.w IP =IP = IP =IP =  x x  x x  u u  u u 2 2  x x  x x  v v  v v 2 2  x x  x x  u u  u u  x x  x x  v v  v v  x x  x x  u u  u u  x x  x x  v v  v v

2. Geometry of T p (S) Anisotropy ; 1 st fundamental form I P a a b b r 2 (  ) = || dx P ( cos  w 1 + sin  w  ) || 2 = (cos .w 1 + sin .w 2 ) t.I p.(cos .w 1 + sin .w 2 ) = cos 2 .||w 1 || sin 2 .||w 2 || sin . cos  ( 1.w t 2.w w t 1.w 2 ) w 1, w 2 unit eigen vectors of Ip 1, 2 associated eigen values r 2 (  )= cos 2 . 1 + sin 2 . 2

2. Geometry of T p (S) Anisotropy ; eigen structure of I P a a b b a = 1 ; b = 2 (eigen values of I p ) IP =IP = IP =IP =  x x  x x  u u  u u 2 2  x x  x x  v v  v v 2 2  x x  x x  u u  u u  x x  x x  v v  v v  x x  x x  u u  u u  x x  x x  v v  v v

2. Geometry of T p (S) Anisotropy ; eigen structure of I P a a b b J p =  x x  x x  u u  u u  x x  x x  v v  v v  y y  y y  u u  u u  y y  y y  v v  v v  z z  z z  u u  u u  z z  z z  v v  v v = U V t a 0 0 b 0 0 a 0 0 b 0 0 Singular values decomposition (SVD) of J Rem: I p = J t.J a = 1 ; b = 2

RI 3 RI 2 u v PiPi PiPi u i,v i 2. Using Differential Geometry Triangulated surfaces Object space (3D) Texture space (2D)

2. Using Differential Geometry Triangulated Surfaces X X Y Y u u v v

2. Using Differential Geometry Anisotropy - See Kai's diff. geo. primer n first fundamental form n eigenvalues of n singular values of (anisotropy ellipse axes)

3. Analytic methods General Principle Define some energy functional F in function of J p, I p, 1, 2 Define some energy functional F in function of J p, I p, 1, 2 n Expand their expression in F in function of the unknown u i, v i n Design an algorithm to find the u i,v i 's that minimizes F

3. Analytic methods 3. Analytic methods [Maillot, Yahia & Verroust, 1993] The first fundamental form I is the metric tensor Minimize a matrix norm of I - Id

3. Analytic methods MIPS [Hormann et. al] [Hormann & Greiner] Principle: F should be invariant by similarity and shoud punish collapsing triangles and shoud punish collapsing triangles

3. Analytic methods Stretch optimization [Sander et.al] r 2 (  ) =  dx p (w(  ))  2 = || dx P ( cos , sin  ) || 2 u v w(  ) T P (S) dx P ( w(  ) ) Stretch L 2 = 1/2  ∫ r 2 (  )d  L ∞ = max ( r(  ) )

3. Analytic methods Stretch optimization [Sander et.al]

3. Analytic Methods Conformal Parameterization  x x  x x  u u  u u  x x  x x  v v  v v  x x  x x  v v  v v  x x  x x  u u  u u ^ ^ N N = = 2 = 1

u u v v x x y y  u u  u u  x x  x x = =  v v  v v  y y  y y  u u  u u  y y  y y = -  v v  v v  x x  x x   Cauchy-Riemann: 3. Analytic Methods Conformal Parameterization No Piecewise Linear solution in general

3. Analytic Methods LSCM [Levy et.al] Minimize2  u u  u u  x x  x x  v v  v v  y y  y y  u u  u u  y y  y y - -  v v  v v  x x  x x -  T Fix two vertices to determine rot,transl,scaling + easy to implement - overlaps, deformations

3. Analytic Methods DNCP [Desbrun et.al] Tutte-Floater with harmonic weights (cotan) + additional equation for natural boundaries Bndry point i, grad of Dirichlet energy Natural idea for "setting the bndry free" (Laplace eqn with Neumman bndry)

Isotropic Parameterizations Conformal = Harmonic E C ( u ) + A u (T) = E D ( u )   E D ( u ) = ½. | u | 2 Dirichlet Energy A u (T) = det(J u ) Area of T E C (u) = ½. || D 90 (  u) -  v || 2 where: Conformal Energy [Douglas31] [Rado30] [Courant50] [Brakke90]

Application of free boundaries Show 2D domain Segmentation: VSA [Alliez et.al]

Epilogue Limits of analytic methods distortions ; validity Geometric methods LSCM ; DNCP

Resources n Source code & papers on on – Graphite – OpenNL

Calls for papers n Eurographics 2008 –Abstracts: Sept 21, papers: Sept 26 n SPM / SPMI 2008 –Abstracts: Nov 27, papers: Dec 4 n SGP 2008 –Abstracts: April 20, papers: April 27 n Special issue Computing - eigenfunctions –Abstracts: Nov 1st, Papers: Nov, 15 Paper copies of CfP available, ask us !

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