1 / 38

Mesh Parameterization: Theory and Practice Setting the Boundary Free

This overview discusses the theory and practice of mesh parameterization, including fixed and free boundary settings, analytic methods, and applications. It explores concepts such as differential geometry, anisotropy, stretch optimization, conformal parameterization, and more. The paper concludes with a discussion on the limitations of analytic methods and available resources.

madsen
Download Presentation

Mesh Parameterization: Theory and Practice Setting the Boundary Free

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Mesh Parameterization:Theory and PracticeSetting the Boundary Free Bruno Lévy - INRIA

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

  3. 1. IntroductionSetting the bndy free, why ? • Floater-Tutte: provably correct result for fixed convex boundary

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

  5. 1. IntroductionFixed boundary - distortion

  6. 1. IntroductionFree boundary - better result

  7. 1. IntroductionWhy is this important ? Demo: Normal mapping

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

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

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

  11. u(x,y,z) x(u,v) 3 I R 2 I R 2. Using Differential Geometry Notion of parameterization x(.,.) u W v S Object space (3D) Texture space (2D)

  12. dv du x/v P x/u TP(S) 2. Geometry of Tp(S)Partial derivatives of x(.,.) v u

  13. dxP(w) w 2. Geometry of Tp(S)Differential dxP ; directional derivatives u0,v0 P dxP(w) = / t (x( (u0,v0)+ t.w )))

  14. w dxP(w) x/v x/u [ ] JP= y/v y/u z/v z/u 2. Geometry of Tp(S)Jacobian Matrix JP dv x/v u0,v0 du P x/u dxP(w) = wux/u + wvx/v = JP.w

  15. TP(S) V1 = dxp(w1) ; V2 = dxp(w2) 2. Geometry of Tp(S)Measuring things,First Fundamental Form Ip V1t V2 = (J w1)t J w2 = w1tJt J w2 = w1tIp w2 V2 V1 w1 w2 v u

  16. 2. Geometry of Tp(S)Measuring things,First Fundamental Form Ip Distances : || V1 ||2 = w1t Ip w1 Angles : V1t V2 = w1t Ip w2 Ip is called the metric tensor

  17.  x  x v TP(S) u dv du 2. Geometry of Tp(S)Anisotropy v u r2(q) = || dxP(cos q, sinq) ||2

  18. 2 x x  x u v u IP= x 2 x x u v v 2. Geometry of Tp(S)Anisotropy ; 1st fundamental form IP || dxP(w) ||2 = || JP.w ||2 = (JPw).(JPw)t = wt.JPt.JP.w = wt.IP.w

  19. b 2 a x x  x u v u IP= x 2 x x u v v a = l1 ; b = l2 (eigen values of Ip) 2. Geometry of Tp(S)Anisotropy ; eigen structure of IP

  20. x  x v u a 0 0 b 0 0 y  y Jp = = U Vt v u z  z v u Singular values decomposition (SVD) of J a = l1 ; b = l2 Rem: Ip = Jt.J 2. Geometry of Tp(S)Anisotropy ; eigen structure of IP b a

  21. ui ,vi Pi 3 I R 2 I R 2. Using Differential GeometryTriangulated surfaces u v Object space (3D) Texture space (2D)

  22. Y v X u 2. Using Differential GeometryTriangulated Surfaces

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

  24. 3. Analytic methodsGeneral Principle • Define some energy functional F in function of Jp, Ip, l1,l2 • Expand their expression in F in function of the unknown ui, vi • Design an algorithm to find the ui,vi's that minimizes F

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

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

  27. dxP(w(q)) TP(S) Stretch L2 = 1/2p ∫ r2(q)dq 3. Analytic methodsStretch optimization [Sander et.al] w(q) v u r2(q) = ||dxp(w(q))||2= || dxP(cos q, sinq) ||2 L∞ = max(r(q))

  28. 3. Analytic methodsStretch optimization [Sander et.al]

  29.  x  x  x v v u  x N ^ = u 3. Analytic MethodsConformal Parameterization l2 = l1

  30.  u  v  v  u y x y y x v x u { = - Cauchy-Riemann: = 3. Analytic MethodsConformal Parameterization No Piecewise Linear solution in general

  31. 2  v  u S - x y -  u  v T x y 3. Analytic MethodsLSCM [Levy et.al] Fix two vertices to determine rot,transl,scaling Minimize + easy to implement - overlaps, deformations

  32. 3. Analytic MethodsDNCP [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)

  33. ED(u) = ½ . | u |2 D Au(T) = det(Ju) Isotropic ParameterizationsConformal = Harmonic EC(u) + Au(T) = ED(u) where: Dirichlet Energy Area of T Conformal Energy EC(u) = ½ . || D90(u) - v ||2 [Douglas31] [Rado30] [Courant50] [Brakke90]

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

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

  36. Resources • Source code & papers on http://alice.loria.fr • Graphite • OpenNL

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

  38. Course Evaluations 4 Random Individuals will win an ATI Radeontm HD2900XT http://www.siggraph.org/course_evaluation

More Related