Transcript Mesh Parameterization: Theory and Practice Setting the Boundary Free Bruno Lévy - INRIA.

Mesh Parameterization:
Theory and Practice
Setting the Boundary Free
Bruno Lévy - INRIA
Overview
1. Introduction - Motivations
2. Using differential geometry
3. Analytic methods
4. Conclusion
1. Introduction
Setting the bndy free, why ?
Floater-Tutte: provably correct result
for fixed convex boundary

1. Introduction
Seamster [Sheffer et.al]
Cuts the model,
ready for "pelting"
1. Introduction
Fixed boundary - distortion
1. Introduction
Free boundary - better result
1. Introduction
Why is this important ?
Demo: Normal mapping
2. Using Differential Geometry
... to minimize deformations
A
B
Q1) How can we compare these two mappings ?
Q2) How can we design an algorithm that prefers B ?
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 ?
2. Using Differential Geometry
... to minimize deformations
[Hormann and Greiner] MIPS
 [Pinkall and Poltier] cotan formula

[Do Carmo] for meshes
2. Using Differential Geometry
Notion of parameterization x(.,.)
u
u(x,y,z)
x(u,v)
IR
3
S
Object space (3D)
IR
2
W
Texture space (2D)
v
2. Geometry of Tp(S)
Partial derivatives of x(.,.)
v
dv du
x/v
P x/u
u
TP(S)
2. Geometry of Tp(S)
Differential dxP ; directional derivatives
dxP(w)
w
u0,v0
P
dxP(w) =  / t ( x( (u0,v0)+ t.w )) )
2. Geometry of Tp(S)
Jacobian Matrix JP
w
dv
u0,v0 du
x/v dxP(w)
P
x/u
x/u x/v
JP = y/u y/v
z/u z/v
dxP(w) = wu x/u + wv x/v = JP.w
[
]
2. Geometry of Tp(S)
Measuring things, First Fundamental Form Ip
V1t V2 = (J w1)t J w2 = w1t Jt J w2 = w1t Ip w2
V2
w2
w1
TP(S)
v
u
V1 = dxp(w1) ; V2 = dxp(w2)
V1
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
2. Geometry of Tp(S)
Anisotropy
dv
TP(S)
x
v
du
v
u
r2(q) = || dxP(cos q, sin q) ||2
x
u
2. Geometry of Tp(S)
Anisotropy ; 1st fundamental form IP
|| dxP(w) ||2 = || JP.w ||2
= (JPw).(JPw)t
x
u
IP =
x
u
2
x
v
x
u
x
v
x
v
2
= wt.JPt.JP.w
= wt.IP.w
2. Geometry of Tp(S)
Anisotropy ; eigen structure of IP
b
a
x
u
IP =
a = l1 ; b = l2
x
u
2
x
v
(eigen values of Ip)
x
u
x
v
x
v
2
2. Geometry of Tp(S)
Anisotropy ; eigen structure of IP
b
a
Jp =
x
u
y
u
z
u
x
v
y
v
z
v
a 0
=U
0 b
t
V
0 0
Singular values decomposition (SVD) of J
Rem: Ip = Jt.J
a = l1 ; b = l2
2. Using Differential Geometry
Triangulated surfaces
Pi
IR
u
3
IR
Object space (3D)
ui ,vi
2
v
Texture space (2D)
2. Using Differential Geometry
Triangulated Surfaces
v
u
2. Using Differential Geometry
Anisotropy - See Kai's diff. geo. primer

first fundamental form

eigenvalues of

singular values of
(anisotropy ellipse axes)
3. Analytic methods
General 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
3. Analytic methods
The first fundamental form I is the metric tensor
Minimize a matrix norm of I - Id
[Maillot, Yahia & Verroust, 1993]
3. Analytic methods
MIPS [Hormann et. al]
Principle: F should be invariant by similarity
and shoud punish collapsing triangles
[Hormann & Greiner]
3. Analytic methods
Stretch optimization [Sander et.al]
w(q)
dxP(w(q))
TP(S)
v
u
r2(q) = ||dxp(w(q))||2 = || dxP(cos q, sin q) ||2
Stretch L2 = 1/2p ∫ r2(q)dq
L∞ = max(r(q))
3. Analytic methods
Stretch optimization [Sander et.al]
3. Analytic Methods
Conformal Parameterization
x
v
x
u
l2 = l1
x N = x
^
v
u
3. Analytic Methods
Conformal Parameterization
v
u
Cauchy-Riemann:
{
v

u
= x
y
v = u
y
x
No Piecewise Linear solution in general
3. Analytic Methods
LSCM [Levy et.al]
Minimize
S
T
v
x
v
y
-
u
- y
u
x
2
Fix two vertices to
determine rot,transl,scaling
+ easy to implement - overlaps, deformations
3. Analytic Methods
DNCP [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)
Isotropic Parameterizations
Conformal
= Harmonic
EC(u) + Au(T) = ED(u)
where:
Au(T) =
D
ED(u) = ½ . |
u |2
det(Ju)
Dirichlet Energy
Area of T
EC(u) = ½ . || D90(u) - v ||2 Conformal Energy
[Douglas31] [Rado30] [Courant50] [Brakke90]
Application of free boundaries
Show 2D domain
Segmentation:
VSA [Alliez et.al]
Epilogue
Limits of analytic methods
LSCM ; DNCP
distortions ; validity
Geometric methods
Resources

Source code & papers
on http://alice.loria.fr
– Graphite
– OpenNL
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 !
Course Evaluations
4 Random Individuals will win an ATI Radeontm HD2900XT
http://www.siggraph.org/course_evaluation