Post on 30-Dec-2015
description
SGP 2008
A Local/Global Approach to Mesh Parameterization
Ligang Liu Lei Zhang Yin XuZhejiang University, China
Craig GotsmanTechnion, Israel
Steven J. GortlerHarvard University, USA
3
SGP 2008
Mesh Parameterization
• Isometric mapping– Preserves all the basic geometry properties:
length, angles, area, …
• For non-developable surfaces, there will always be some distortion– Try to keep the distortion as small as
possible
4
SGP 2008
Previous Work• Discrete harmonic mappings
– Finite element method [Pinkall and Polthier 1993; Eck et al. 1995]– Convex combination maps [Floater 1997]– Mean value coordinates [Floater 2003]
• Discrete conformal mappings– MIPS [Hormann and Greiner 1999]– Angle-based flattening [Sheffer and de Sturler 2001; Sheffer et al. 2005]– Linear methods [Lévy et al. 2002; Desbrun et al. 2002]– Curvature based [Yang et al. 2008, Ben-Chen et al. 2008, Springborn et al, 2008]
• Discrete equiareal mappings – [Maillot et al.1993; Sander et al. 2001; Degener et al. 2003]
5
SGP 2008
Inspiration
• Laplacian & Poisson-based editing [Sorkine et al. 2004, Yu et al. 2004]
• Deformation transfer [Sumner et al. 2004]
• Linear Tangent-Space Alignment [Chen et al. 2007]
• As-rigid-as-possible surface modeling [Sorkine and Alexa 2007]
“Think globally, act locally”
6
SGP 2008
The Key Idea
perform local transformations on triangles
and stitch them all together consistently
to a global solution
8
SGP 2008
Triangle Flattening
• Each individual triangle is independently flattened into plane without any distortion
Reference triangles
Isometric
9
SGP 2008
Intrinsic Deformation Energy
2
1
( , ) ( ( ))T
t t t Ft
E u A J L u A
( )tL u
tA M : some family of allowed linear transformations
Area of 3D triangle Jacobian matrix of Lt
(Affine)
Reference triangles x Parameterization u
e.g. similarity or rotationAuxiliary linear
tA (Linear)
10
SGP 2008
Unknown lineartransformation
Angles of triangle
Source2D coords
UnknownTarget
2D coords
2 21 1
,1 0
arg min cot ( ) ( )T
i i i i it t t t t tu A
t i
u u u A x x
[Pinkall and Polthier 1993]
Extrinsic Deformation Energy
0u 1u
2u
0x1x
2x
0tA M
tA
13
SGP 2008
Linear system in u, a, b
As-Similar-As-Possible (ASAP)
At Similarity transformations
t t
tt t
a bA
b a
Auxiliaryvariables
2 21 1
,1 0
arg min cot ( ) ( )T
i i i i it t t t t tu A
t i
u u u A x x
14
SGP 2008
As-Similar-As-Possible (ASAP)
• Equivalent to LSCM technique [Levy et al.
2002] which minimizes
singular values of the Jacobian
21 2
1
T
t t tt
16
SGP 2008
As-Rigid-As-Possible (ARAP)
At Rotations
2 2,( 1)t t
t t tt t
a bA a b
b a Non-linear system in u,a,b
2 21 1
,1 0
arg min cot ( ) ( )T
i i i i it t t t t t
u Ai i
u u u A x x
We will treat u and A as separate sets of variables, to enable a simple optimization process.
17
SGP 2008
As-Rigid-As-Possible (ARAP)
At Rotations
2 2,( 1)t t
t t tt t
a bA a b
b a Non-linear system in u,a,b
Solve by “local/global” algorithm [Sorkine and Alexa, 2007] :
Find an initial guess of uwhile not converged
Fix u and solve locally for each At
Fix At and solve globally for uend
Poisson equation
SVD
2 21 1
,1 0
arg min cot ( ) ( )T
i i i i it t t t t t
u Ai i
u u u A x x
19
SGP 2008
Advantages
• Each iteration decreases the energy
• Matrix L of Poisson equation is fixed– Precompute Cholesky factorization– Just back-substitute in each iteration
Lx b
22
SGP 2008
1
2 1t
t
1 2 1t t
1 2 1t t
22
tL
angle-preserving (conformal)
area-preserving (authalic)
length-preserving (isometric)
ASAP ARAP 1 2 2Minimize ( )t t t
t
1 2 2 2Minimize ( 1) ( 1)t t tt
Most conformal Most isometric
SGP 2008
Tradeoff Between Conformality and Rigidity
ASAP ARAPPreserves angles, but not preserve area
?Tradeoff
Preserves areas, but damage conformality
25
SGP 2008
Hybrid Model
Local Phase: Solve cubic equation for at and bt
Global Phase: Poisson equation
= 0 ASAP
= ARAP
parameter
2
21 1 2 2 2
1 0
arg min cot ( ) ( ) ( 1)T
t ti i i i it t t t t t t
i i t t
a bu u u x x a b
b a
Similarity transformation
26
SGP 2008
Results
λ=0.0001(2.05, 5.74)
λ=0.001(2.07, 2.88)
λ=0.1(2.18, 2.14)
ARAP (λ=)(2.19, 2.11)
ASAP (λ=0)(2.05, 15.6)
1 2
2 1Angle t t
tt t t
D
Angular distortion:
1 2
1 2
1Areat t t
t t t
D
Area distortion:
30
SGP 2008
ASAP(2.01, 30.1)
ARAP(2.03, 2.03)
ABF++(2.01, 2.19)
Inverse Curvature Map[Yang et al. 2008]
(2.46, 2.51)
Linear ABF[Zayer et al. 2007]
(2.01, 2.22)
Curvature Prescription[Ben-Chen et al. 2008]
(2.01, 2.18)
31
SGP 2008
Comparison
ASAP ARAP ABF++[Sheffer et al. 2005]
Inverse Curvature Map[Yang et al. 2008]
(2.05, 2.67)(2.00, 2.64)(2.06, 2.05)(2.00, 88.1)
32
SGP 2008
Conclusion
• Simple iterative “local/global” algorithm• Converges in a few iterations• Low conformal and stretch distortions• Generalization of stress majorization (MDS)
• Can be used for deformable mesh registration