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Mesh Parameterization: Theory and Practice Setting the Boundary Free Mesh Parameterization: Theory and Practice Setting the Boundary Free Bruno Lvy - INRIA Slide 2 Overview 1. Introduction - Motivations 2. Using differential geometry 3. Analytic methods 4. Conclusion Slide 3 1. Introduction Setting the bndy free, why ? n Floater-Tutte: provably correct result for fixed convex boundary for fixed convex boundary Slide 4 1. Introduction Seamster [Sheffer et.al] Cuts the model, ready for "pelting" Slide 5 1. Introduction Fixed boundary - distortion Slide 6 1. Introduction Free boundary - better result Slide 7 1. Introduction Why is this important ? Demo: Normal mapping Slide 8 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 Slide 9 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 ? Slide 10 2. Using Differential Geometry... to minimize deformations n [Hormann and Greiner] MIPS n [Pinkall and Poltier] cotan formula [Do Carmo] for meshes Slide 11 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) Slide 12 2. Geometry of T p (S) Partial derivatives of x (.,. ) v u uu vv x/ux/u x/vx/v P T P (S) Slide 13 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 ) ) ) Slide 14 2. Geometry of T p (S) Jacobian Matrix J P JP =JP = x/ux/u y/uy/u z/uz/u x/vx/v y/vy/v z/vz/v [ ] x/ux/u x/vx/v P dxP(w)dxP(w) uu vv w dx P (w) = w u x/ u + w v x/ v = J P.w u 0,v 0 Slide 15 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 Slide 16 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 Slide 17 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 Slide 18 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 Slide 19 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 || 2. 1 + sin 2 .||w 2 || 2. 2 + sin . cos ( 1.w t 2.w 1 + 2.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 Slide 20 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 Slide 21 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 Slide 22 RI 3 RI 2 u v PiPi PiPi u i,v i 2. Using Differential Geometry Triangulated surfaces Object space (3D) Texture space (2D) Slide 23 2. Using Differential Geometry Triangulated Surfaces X X Y Y u u v v Slide 24 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) Slide 25 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 Slide 26 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 Slide 27 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 Slide 28 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( ) ) Slide 29 3. Analytic methods Stretch optimization [Sander et.al] Slide 30 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 Slide 31 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 Slide 32 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 Slide 33 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) Slide 34 Isotropic Parameterizations Conformal = Harmonic E C ( u ) + A u (T) = E D ( u ) E D ( u ) = . | u | 2 Dirichlet Energy A u (T) = det(J u ) Area of T E C (u) = . || D 90 ( u) - v || 2 where: Conformal Energy [Douglas31] [Rado30] [Courant50] [Brakke90] Slide 35 Application of free boundaries Show 2D domain Segmentation: VSA [Alliez et.al] Slide 36 Epilogue Limits of analytic methods distortions ; validity Geometric methods LSCM ; DNCP Slide 37 Resources n Source code & papers on http://alice.loria.fr on http://alice.loria.fr Graphite OpenNL Slide 38 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 ! Slide 39 Course Evaluations 4 Random Individuals will win an ATI Radeon tm HD2900XT http://www.siggraph.org/course_evaluation