1 / 34

Overview

Israel-Korea Conference Numerical Methods for Digital Geometry Processing Bruno Lévy INRIA, ALICE. Overview. 1. Numerical Problems in DGP 2. Linear and Quadratic DGP 3. Non-linear DGP. 1970’s. 2000’s. Motivations:. Need for scalability in DGP.

sorcha
Download Presentation

Overview

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. Israel-Korea Conference Numerical Methodsfor Digital Geometry ProcessingBruno LévyINRIA, ALICE

  2. Overview 1. Numerical Problems in DGP 2. Linear and Quadratic DGP 3. Non-linear DGP

  3. 1970’s 2000’s Motivations: Need for scalability in DGP

  4. 1. Numerical Problems in DGPMesh Parameterization Ui = S ai,jUj j  Ni i  i,j ai,j > 0 j1 The border is mapped to a convex polygon j… j2 [Tutte], [Floater]

  5. 1. Numerical Problems in DGPDiscrete Fairing 2 F(p)=Spi - S ai,jpj i j  Ni i j1 j… j2 [Mallet], [Kobbelt], …

  6. 1. Numerical Problems in DGPStencils F = sum of terms, attached to neighborhoods Parameterization [Haker00] [Levy02] Curv. Estimation [Cohen-Steiner 03] Texture mapping [Levy01] Discrete fairing [Levy03] Discrete fairing [Kobbelt98, Mallet95] Parameterization [Desbrun02] Deformations [CohenOr], [Sorkine] [Eck]

  7. 2. Linear and Quadratic DGPRemoving degrees of freedom 2 F(x) = A x - b 2 xf xl [ Af Al] - b F(xf) =

  8. 2 2 F(xf) = A.x - d = Al.xl + Af.xf - d F(xf) minimum Aft.Af.xf = Aft.d - AftAl.xl } } M.x = b 2. Linear and Quadratic DGPRemoving degrees of freedom The problem: (1) construct a linear system (2) solve a linear system

  9. NlSolve() 2. Linear and Quadratic DGPThe OpenNL approach (http://www.loria.fr/~levy/software) The problem: (1) construct a linear system (2) solve a linear system NlLockVariable(i1, val1) NlLockVariable(i2, val2) … For each stencil instance (one-rings): NlBeginRow(); NlAddCoefficient(i, a); … NlEndRow(); • Need for • Dynamic Matrix DS • Updating formula

  10. U M = n=106 10 centuries n=100 0.01 s (1) solve U x = b Y = b L L L U x = Y (2) solve 2. Linear and Quadratic DGPDirect Solvers (LU) A Textbook solver: LU factorization (and Cholesky) a ‘small’ problem: O(n3) !!

  11. mi,j xfj … … … mn,1 mi,1 m1,1 … … … mn,j m1,j mi,j mi,n m1,n mn,n j = i ) ( w ci- xfi (1-w).xfi + mi,i 2. Linear and Quadratic DGPSuccessive Over-Relaxation (Gauss-Seidel) xf1 …. …. …. …. …. = xfi ci …. …. …. …. …. xfnf [Taubin95] [Levy98] …

  12. 2. Linear and Quadratic DGPSuccessive Over-Relaxation (Gauss-Seidel) 1000 iterations S.O.R.

  13. Complicated ops: Matrix x vector (see paper) 2. Linear and Quadratic DGPWhite Magic: The Conjugate Gradient inline int solve_conjugate_gradient( const SparseMatrix &A, const Vector& b, Vector& x, double eps, int max_iter ){ int N = A.n() ; double t, tau, sig, rho, gam; double bnorm2 = BLAS::ddot(N,b,1,b,1) ; double err=eps*eps*bnorm2 ; mult(A,x,g); BLAS::daxpy(N,-1.,b,1,g,1); BLAS::dscal(N,-1.,g,1); BLAS::dcopy(N,g,1,r,1); while ( BLAS::ddot(N,g,1,g,1)>err && its < max_iter) { mult(A,r,p); rho=BLAS::ddot(N,p,1,p,1); sig=BLAS::ddot(N,r,1,p,1); tau=BLAS::ddot(N,g,1,r,1); t=tau/sig; BLAS::daxpy(N,t,r,1,x,1); BLAS::daxpy(N,-t,p,1,g,1); gam=(t*t*rho-tau)/tau; BLAS::dscal(N,gam,r,1); BLAS::daxpy(N,1.,g,1,r,1); ++its; } return its ; } Only simple vector ops (BLAS) Demo (Constrained Tex Map, Siggraph01)

  14. U M = L j, gi,j 2. Linear and Quadratic DGPBlack Magic: Sparse Direct Solvers Super-nodal: [Demmel et.al 96] Multi-frontal:[Lexcellent et.al 98] [Toledo et.al] Direct method’s revenge: Super-Nodal data structure Demo: Free-form modeling with meshes

  15. LS with reduced degrees of freedom • Built-in (CG, GMRES, BICGSTAB) • SuperLU • MUMPS • TAUCS • … j, gi,j 2. Linear and Quadratic DGPOpenNL architecture NlLockVariables(i,a) … NlBeginRow() NlAddCoefficient(i,a) … NlEndRow() NlSolve()

  16. 2. Linear and Quadratic DGPApplications Maya Gocad: Meshing for oil-exploration Blender (OpenSource) VSP-Technology ATARI-Infogrammes

  17. 3. Non-Linear DGP • MIPS [Hormann], Stretch [Sander] • ABF [Sheffer], ABF++ [Sheffer & Lévy] • PGP [Ray,Levy,Li,Sheffer,Alliez] • Circle Packings [Bobenko], [Schroeder] • Finite elements with dynamic function bases

  18. 3. Non-Linear DGPGalerkin Finite Elements (linear) • Operator equation: Lf = g • e.g. L = ∆ = ∂2./∂x2 + ∂2./∂y2 • without rhs (g = 0): Laplace • with rhs (g = arbitrary function): Poisson • Function basis (fi): f = Saifi • Scalar Product: <f,g> = ∫ f(x) g(x) dx • i, <Lf, fi> = <g, fi>

  19. 3. Non-Linear DGPGalerkin Finite Elements (linear) …… <Lf1, f1> <Lf1, fn> a1 <g, f1> <Lfi, fj> = <Lfn, f1> …… an <g, fn> <Lfn, fn>

  20. 3. Non-Linear DGPFrustation withGalerkin FEM (demo) Fixed function basis DFB

  21. 3. Non-Linear DGPDynamic Function Bases i p, fi( x,y) p = (x1,y1, …, xm, ym)

  22. 3. Non-Linear DGPDynamic Function Bases • f = Saifi(p1, p2, …, pm, x,y) = Saifi(p,x) • Galerkin: i, <Lf, fi> = <g, fi> • DFB: minimize F(p,a) = |Lf – g|2 • Solve for f [a] and for its sampling [p]

  23. 3. Non-Linear DGPOptimization with DFB: general algorithm While | F | > e 2a,a F 2a,p F da a F 2p,a F 2p,p F dp p F solve = - a  a + da p  p + dp End while

  24. 3. Non-Linear DGPInstancing the general algorithm • Lf = g operator L, rhs g, basis (fi) • Numerical estimation of F and 2F ∂./∂. ∫ Lfi x Lfj ∂./∂. ∫ Lfi x g (smoothness of Lloyd, Wenping Wang) • First example: L = Identity, f = g (ARDECO approximation with DFP)

  25. 3. Non-Linear DGPARDECO (Bitmap SVG)

  26. 3. Non-Linear DGPARDECO (Bitmap SVG)

  27. 3. Non-Linear DGPARDECO (Bitmap SVG)

  28. 3. Non-Linear DGPARDECO (Bitmap SVG) (original: 1M pixels) 300 postcript gradients

  29. 3. Non-Linear DGPARDECO (Bitmap SVG)

  30. 3. Non-Linear DGPDFB research program • 2D: Laplace Eqn, Poisson Eqn • 1D+t: Heat eqn (sugg. by F. Durand) • 2D: ARDECO • 2D+t: Navier-Stokes • 3D: Light Simulation, Mesh-to-splines • 3D+t: Misc. physics Available tools: mixed symbolic/numeric solver

  31. 3. Non-Linear DGPFinding a good control mesh

  32. 3. Non-Linear DGPFinding a good control mesh

  33. Conclusions • The DGP community can directly benefit from the advances done by the NA community • Solving for the approximation and the sampling seems to be an interresting research avenue

  34. Aknowledgemes • IK organizing comittee • AIM@Shape • INRIA/GEOREP • Fench ministry of research/ACI SHOW • Wenping Wang (HKU) • Fredo Durand, Matthias Zwicker (MIT) • Students: B. Vallet, N. Ray, W.C. Li, G. Lecot

More Related