Mesh Parameterization: Theory and Practice Making it work in practice Numerics

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Mesh ParameterizationTheory and
PracticeMaking it work in practice - Numerics
Bruno Lévy - INRIA
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Overview

• 1. Numerical Problems
• 2. Linear and Quadratic
• 3. Non-linear

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Motivations
Need for scalability in GP
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1. Numerical Problems in GPMesh Parameterization
Ui S ai,jUj
j ? Ni
i
? i,j ai,j gt 0 ai,i - S ai,j
j1
j
The border is mapped to a convex polygon
j2
Tutte, Floater
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1. Numerical Problems in GPDiscrete Fairing
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F(p) S pi - S ai,jpj
i
j ? Ni
i
j1
j
j2
Mallet, Kobbelt, Sorkine
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1. Numerical Problems in GPNeighborhoods
F sum of terms, attached to neighborhoods
Parameterization Haker00 Levy02
Curv. Estimation Cohen-Steiner 03 Texture
mapping Levy01 Discrete fairing Desbrun, ...
Discrete fairing Kobbelt98, Mallet95 Parameteriz
ation Desbrun02 Deformations CohenOr,
Sorkine
Eck
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2. Linear and Quadratic GPRemoving degrees of
freedom
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F(x) A x - b
2

xf xl
Af Al - b
F(xf)
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2. Linear and Quadratic GPRemoving degrees of
freedom
The problem (1) construct a linear system
(2) solve a linear system
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2. Linear and Quadratic GPThe OpenNL approach
(http//alice.loria.fr/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

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2. Linear and Quadratic GPDirect Solvers (LU)
A Textbook solver LU factorization (and Cholesky)
a small problem O(n3) !!
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2. Linear and Quadratic GPSuccessive
Over-Relaxation (Gauss-Seidel)
xf1
.
.
.
.
.

ci
xfi
.
.
.
.
.
xfnf
used in Taubin95
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2. Linear and Quadratic DGPSuccessive
Over-Relaxation (Gauss-Seidel)
1000 iterations S.O.R.
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2. Linear and Quadratic GPWhite 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
BLASddot(N,b,1,b,1) double
errepsepsbnorm2
mult(A,x,g) BLASdaxpy(N,-1.,b,1,g,1)
BLASdscal(N,-1.,g,1)
BLASdcopy(N,g,1,r,1) while (
BLASddot(N,g,1,g,1)gterr its lt max_iter)
mult(A,r,p)
rhoBLASddot(N,p,1,p,1)
sigBLASddot(N,r,1,p,1)
tauBLASddot(N,g,1,r,1)
ttau/sig BLASdaxpy(N,t,r,1,x,1)
BLASdaxpy(N,-t,p,1,g,1)
gam(ttrho-tau)/tau
BLASdscal(N,gam,r,1)
BLASdaxpy(N,1.,g,1,r,1) its
return its
Only simple vector ops (BLAS)
Shewchuck CG without the agonizing pain
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Iterative Solvers

• Successive Over-Relaxation
• Sparse C.G. gt100x
  speedup

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Iterative Solvers
The Sparse Conjugate Gradient Solver
Sparse storage of G AftAf
Interactive solver !!!
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White magic Multigrid
Remember direct solver is O(n3)
Sparse Conjugate Gradient is O(n2) !!
Thats much better, but
we want even more efficiency !!
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White magic Multigrid
Lee,Schroeder, Kobbelt,Hackbuch
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White magic Multigrid
MIPS, HLSCM, ABF
Step 2 Cascadic multigrid
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White Magic Multigrid
direct solver O(n3) Sparse CG O(n2)
Multigrid O(n) !!
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Black Magic Sparse Direct
We started from O(n3) We achieved O(n)
Can we do better ?
-- Interactivity -- -- Ease of implementation --
Sheffer et.al SuperLU for ABF Botsch et.al
Interactive mesh deformation
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2. Linear and Quadratic DGPBlack Magic Sparse
Direct Solvers
Super-nodal Demmel et.al 96 Multi-frontalL
excellent et.al 98 Toledo
et.al
Direct methods revenge
Super-Nodal data structure
TAUCS, SuperLU, CHOLDMOD
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Black Magic Sparse direct
Interactivity
TAUCS, SuperLU, CHOLDMOD
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2. Linear and Quadratic GPOpenNL architecture
NlLockVariables(i,a) NlBeginRow() NlAddCoeffici
ent(i,a) NlEndRow() NlSolve()
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2. Linear and Quadratic GPApplications
Maya
Gocad Meshing for oil-exploration
Blender (OpenSource)
VSP-Technology ATARI-Infogrammes
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2. Linear and Quadratic GPApplications
OpenNL in SILO
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3. Non-Linear GP

• MIPS Hormann, Stretch Sander
• ABF Sheffer, ABF Sheffer Lévy
• PGP Ray,Levy,Li,Sheffer,Alliez
• Circle Packings/Patterns Bobenko, Karevych

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Conquer the non-linear world
We want to optimize a function F(x) What can we
do when F is non-linear ?
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Conquer the non-linear world
Non-linear shapes functionals
Connectivity shapes Angle Based Flattening

Demos
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Conclusionsa map to the solvers jungle
Numerical Solvers
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Conclusions
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
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ConclusionTake home message

• In most cases, TAUCS OOC will do the job.
• For small datasets, PreCG has a good
  simplicity/mem. requirement/efficientcy ratio.
• Use OpenNL for Matrix Assembly - Solver
  Abstraction

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Resources

• Source code papers
• on http//alice.loria.fr
• Graphite
• OpenNL