Mesh Mesh Parameterization:Parameterization:

Theory and PracticeTheory and PracticeMaking it work in practice - NumericsMaking it work in practice - Numerics

Bruno Lévy - INRIABruno Lévy - INRIA

OverviewOverview

1. Numerical Problems 1. Numerical Problems

2. Linear and Quadratic 2. Linear and Quadratic

3. Non-linear 3. Non-linear

Motivations:Motivations:

1970’s1970’s

2000’s2000’s

Need for Need for scalabilityscalability in GP in GP

1. Numerical Problems in GP1. Numerical Problems in GPMesh ParameterizationMesh Parameterization

ii

jj11

jj22jj……

UUii = = a ai,ji,jUUjj

j j N Nii

i,j ai,j ai,ji,j > 0 > 0aai,ii,i = - = - a ai,ji,j

The border is mapped toThe border is mapped toa convex polygona convex polygon

[Tutte], [Floater][Tutte], [Floater]

1. Numerical Problems in GP1. Numerical Problems in GPDiscrete FairingDiscrete Fairing

ii

jj11

jj22jj……

F(p)=F(p)= ppii - - a ai,ji,jppjj

22

j j N Niiii

[Mallet], [Kobbelt], [Sorkine]…[Mallet], [Kobbelt], [Sorkine]…

1. Numerical Problems in GP1. Numerical Problems in GPNeighborhoodsNeighborhoods

F = sum of terms, attached to neighborhoodsF = sum of terms, attached to neighborhoods

Discrete fairingDiscrete fairing[Kobbelt98, Mallet95][Kobbelt98, Mallet95]ParameterizationParameterization[Desbrun02][Desbrun02]DeformationsDeformations[CohenOr], [Sorkine][CohenOr], [Sorkine]

Curv. EstimationCurv. Estimation[Cohen-Steiner 03][Cohen-Steiner 03]Texture mappingTexture mapping[Levy01][Levy01]Discrete fairingDiscrete fairing[Desbrun], ...[Desbrun], ...

ParameterizationParameterization[Haker00][Haker00][Levy02][Levy02]

[Eck][Eck]

2. Linear and Quadratic GP2. Linear and Quadratic GPRemoving degrees of freedomRemoving degrees of freedom

F(xf) = F(xf) = xfxf

xlxl

[ Af [ Af AlAl] - b] - b

22

F(x) = A x - bF(x) = A x - b22

2. Linear and Quadratic GP2. Linear and Quadratic GPRemoving degrees of freedomRemoving degrees of freedom

F(xf) = A.x - d = Al.xl + Af.xf - d F(xf) = A.x - d = Al.xl + Af.xf - d2 2

F(xf) minimumF(xf) minimum Aft.Af.xf = Af

t.d - AftAl.xl Af

t.Af.xf = Aft.d - Af

tAl.xl

M.x = b M.x = b

}} }}

The problem: (1) construct a linear system (2) solve a linear system

2. Linear and Quadratic GP2. Linear and Quadratic GPThe OpenNL approach The OpenNL approach

(http://alice.loria.fr/software)(http://alice.loria.fr/software)

The problem: (1) construct a linear system (2) solve a linear system

NlLockVariable(i1, val1)NlLockVariable(i1, val1)NlLockVariable(i2, val2)NlLockVariable(i2, val2)……For each stencil instance (one-rings):For each stencil instance (one-rings): NlBeginRow();NlBeginRow(); NlAddCoefficient(i, a);NlAddCoefficient(i, a); … … NlEndRow();NlEndRow();

NlSolve()NlSolve()

Need forNeed for• Dynamic Matrix DSDynamic Matrix DS• Updating formulaUpdating formula

2. Linear and Quadratic GP2. Linear and Quadratic GPDirect Solvers (LU)Direct Solvers (LU)

A Textbook solver: LU factorization (and Cholesky)

M =M =UL

UL x = b

L Y = b (1) solve

U x = Y(2) solve

a ‘small’ problem: O(n3) !!n=100 0.01 sn=106 10 centuries

2. Linear and Quadratic GP2. Linear and Quadratic GPSuccessive Over-Relaxation (Gauss-Seidel)Successive Over-Relaxation (Gauss-Seidel)

=

….

ci

….

xfnf

xf1

mi,jmi,1 mi,n… …

m1,jm1,1 m1,n… …

mn,jmn,1 mn,n… …

….

….

….

….

….

….

xfi

….

….

xfixfimi,imi,i

j = ij = i mi,j xfjmi,j xfjci-ci-(( )) used in [Taubin95]used in [Taubin95]

……

2. Linear and Quadratic DGP2. Linear and Quadratic DGPSuccessive Over-Relaxation (Gauss-Seidel)Successive Over-Relaxation (Gauss-Seidel)

1000 iterations S.O.R.

2. Linear and Quadratic GP2. Linear and Quadratic GPWhite Magic: White Magic: The Conjugate GradientThe Conjugate Gradient

inline int solve_conjugate_gradient(inline int solve_conjugate_gradient( const SparseMatrix &A, const Vector& b, Vector& const SparseMatrix &A, const Vector& b, Vector& x, x, double eps, int max_iterdouble eps, int max_iter ){ ){ int N = A.n() ;int N = A.n() ; double t, tau, sig, rho, gam;double t, tau, sig, rho, gam; double bnorm2 = BLAS::ddot(N,b,1,b,1) ; double bnorm2 = BLAS::ddot(N,b,1,b,1) ; double err=eps*eps*bnorm2 ; double err=eps*eps*bnorm2 ; mult(A,x,g);mult(A,x,g); BLAS::daxpy(N,-1.,b,1,g,1);BLAS::daxpy(N,-1.,b,1,g,1); BLAS::dscal(N,-1.,g,1);BLAS::dscal(N,-1.,g,1); BLAS::dcopy(N,g,1,r,1);BLAS::dcopy(N,g,1,r,1); while ( BLAS::ddot(N,g,1,g,1)>err && its < while ( BLAS::ddot(N,g,1,g,1)>err && its < max_iter) { max_iter) { mult(A,r,p);mult(A,r,p); rho=BLAS::ddot(N,p,1,p,1);rho=BLAS::ddot(N,p,1,p,1); sig=BLAS::ddot(N,r,1,p,1);sig=BLAS::ddot(N,r,1,p,1); tau=BLAS::ddot(N,g,1,r,1);tau=BLAS::ddot(N,g,1,r,1); t=tau/sig;t=tau/sig; BLAS::daxpy(N,t,r,1,x,1);BLAS::daxpy(N,t,r,1,x,1); BLAS::daxpy(N,-t,p,1,g,1);BLAS::daxpy(N,-t,p,1,g,1); gam=(t*t*rho-tau)/tau;gam=(t*t*rho-tau)/tau; BLAS::dscal(N,gam,r,1);BLAS::dscal(N,gam,r,1); BLAS::daxpy(N,1.,g,1,r,1);BLAS::daxpy(N,1.,g,1,r,1); ++its;++its; }} return its ;return its ;}}

Only simpleOnly simplevector ops (BLAS)vector ops (BLAS)

Complicated ops:Complicated ops:Matrix x vectorMatrix x vector(see course notes)(see course notes)

[Shewchuck: CG without the agonizing pain][Shewchuck: CG without the agonizing pain]

Iterative SolversIterative Solvers•Successive Over-Relaxation •Sparse C.G. >100x speedup

5

2.5

0 900 45

- log|G.x + c |

Time(s)

Precond.Conj. Grad.

S.O.R.

Conj. Grad.

Iterative SolversIterative Solvers

j, gi,j

Sparse storage of G = AftAf Sparse storage of G = AftAf

The Sparse Conjugate Gradient Solver

Demo

Interactive solver !!!

White magic: MultigridWhite magic: Multigrid

Sparse Conjugate Gradient is O(n2) !!

Remember: direct solver is O(n3)

That’s much better, but …

… we want even more efficiency !!

White magic: MultigridWhite magic: Multigrid

[Lee],[Schroeder],[Kobbelt],[Hackbuch]

1

1

2

2

3

3

4

4

Cascadic multigrid

Coarse to fine[Bornemann96]

White magic: MultigridWhite magic: Multigrid

Step 2: Cascadic multigrid

[MIPS, HLSCM, ABF++…]

White Magic: MultigridWhite Magic: Multigrid

direct solver : O(n3)Sparse CG : O(n2)

Multigrid : O(n) !!

Black Magic: Black Magic: Sparse DirectSparse Direct[ ]

We started from O(n3)We achieved O(n)

Can we do better ?

-- Interactivity ---- Ease of implementation --

[Sheffer et.al] SuperLU for ABF[Sheffer et.al] SuperLU for ABF[Botsch et.al] Interactive mesh deformation[Botsch et.al] Interactive mesh deformation

2. Linear and Quadratic DGP2. Linear and Quadratic DGPBlack Magic: Black Magic: Sparse Direct SolversSparse Direct Solvers

Direct method’s revenge: Super-Nodal data structure

M =M =UL

j, gi,j

Super-nodal: [Demmel et.al 96]Multi-frontal:[Lexcellent et.al 98] [Toledo et.al]

TAUCS, SuperLU, CHOLDMODTAUCS, SuperLU, CHOLDMOD

Black Magic: Black Magic: Sparse directSparse direct[ ]

Demos

Interactivity

TAUCS, SuperLU, CHOLDMODTAUCS, SuperLU, CHOLDMOD

2. Linear and Quadratic GP2. Linear and Quadratic GPOpenNL architectureOpenNL architecture

NlLockVariables(i,a)NlLockVariables(i,a)……

NlBeginRow()NlBeginRow()NlAddCoefficient(i,a)NlAddCoefficient(i,a)……NlEndRow()NlEndRow()

NlSolve()NlSolve()

j, gi,j

LS with LS with reducedreduceddegrees ofdegrees offreedomfreedom

•Built-in (CG, GMRES, BICGSTAB)Built-in (CG, GMRES, BICGSTAB)•SuperLUSuperLU•MUMPSMUMPS•TAUCSTAUCS•……

2. Linear and Quadratic GP2. Linear and Quadratic GPApplicationsApplications

Gocad:Gocad:Meshing forMeshing foroil-explorationoil-exploration

MayaMaya

VSP-TechnologyVSP-TechnologyATARI-InfogrammesATARI-Infogrammes

Blender Blender (OpenSource)(OpenSource)

2. Linear and Quadratic GP2. Linear and Quadratic GPApplicationsApplications

OpenNL in SILOOpenNL in SILO

3. Non-Linear GP3. Non-Linear GP

MIPS MIPS [Hormann], Stretch [Sander][Hormann], Stretch [Sander] ABF ABF [Sheffer][Sheffer], ABF++ , ABF++ [Sheffer & Lévy][Sheffer & Lévy] PGP [Ray,Levy,Li,Sheffer,Alliez]PGP [Ray,Levy,Li,Sheffer,Alliez] Circle Packings/Patterns Circle Packings/Patterns [Bobenko], [Bobenko],

[Karevych][Karevych]

Conquer the non-linear worldConquer the non-linear world

We want to optimize a function F(x)What can we do when F is non-linear ?

Newton’s method:

Conquer the non-linear worldConquer the non-linear world

Non-linear shapes functionals Connectivity shapes Angle Based Flattening ++

Demos

ConclusionsConclusionsa map to the solvers junglea map to the solvers jungle

Numerical Solvers

Direct Iterative

Multi-grid

Sparse C.G.

S.O.R. [ ] Naive

sparse direct

ConclusionsConclusions

Sparse C.G.

Multi-grid

Sparse direct

Non-linear

100x speedup w.r.t. S.O.R.simple to implement

Linear O(n) !!! (1000x)

best for huge objects

Ultra-fast (best for interactivity) (10000x) Big memory overhead

Difficult to tune …

ConclusionConclusionTake home messageTake home message

In most cases, TAUCS + OOC will do the job.In most cases, TAUCS + OOC will do the job.

For small datasets, PreCG has a good For small datasets, PreCG has a good simplicity/mem. requirement/efficientcy ratio.simplicity/mem. requirement/efficientcy ratio.

Use OpenNL for Matrix Assembly - Solver Use OpenNL for Matrix Assembly - Solver AbstractionAbstraction

ResourcesResources

Source code & papersSource code & papers

on http://alice.loria.fron http://alice.loria.fr

– GraphiteGraphite

– OpenNLOpenNL