ECCV TutorialECCV TutorialMesh ProcessingMesh Processing

NumericsNumerics

Bruno LévyBruno LévyINRIA - ALICEINRIA - ALICE

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.fr/WIKIon http://alice.loria.fr/WIKI

– GraphiteGraphite– CGALCGAL– OpenMeshOpenMesh– OpenNLOpenNL

ConclusionConclusionMesh Processing: a wide topicMesh Processing: a wide topicData structuresData structuresMesh repairMesh repairMesh analysisMesh analysisSmoothingSmoothingParameterizationParameterizationMesh simplificationMesh simplificationRemeshingRemeshingFreeform modeling ...Freeform modeling ...

SIGGRAPH and EUROGRAPHICS tutorials SIGGRAPH and EUROGRAPHICS tutorials (with M. Botsch, M. Pauly, L. Kobbelt and P. Alliez)(with M. Botsch, M. Pauly, L. Kobbelt and P. Alliez)

http://alice.loria.fr/WIKI/http://alice.loria.fr/WIKI/