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

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Mesh Parameterization

Input3D mesh

OutputFlattened 2D mesh

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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

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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]

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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”

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SGP 2008

The Key Idea

perform local transformations on triangles

and stitch them all together consistently

to a global solution

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Local/Global Approach

Sti

tch g

lobally

Input 3D mesh

Output 2D parameterization

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Triangle Flattening

• Each individual triangle is independently flattened into plane without any distortion

Reference triangles

Isometric

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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)

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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

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As-Similar-As-Possible (Conformality)

M family of similarity transformations

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Conformal Mapping

Similarity = Rotation + Scale

Preserves angles

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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

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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

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As-Rigid-As-Possible (Rigidity)

M family of rotation transformations

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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.

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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

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Optimal Local Rotation

A

1

2

0

0TA U SV S

R

TR U V

2

argminR F

R A

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Advantages

• Each iteration decreases the energy

• Matrix L of Poisson equation is fixed– Precompute Cholesky factorization– Just back-substitute in each iteration

Lx b

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As-Rigid-As-Possible (ARAP)

• Equivalent to minimizing:

22 21 2

1

1 1T

t t tt

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ASAP ARAP

1 2 2Minimize ( )t t tt

1 2 2 2Minimize ( 1) ( 1)t t tt

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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

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ASAP ARAP

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Tradeoff Between Conformality and Rigidity

ASAP ARAPPreserves angles, but not preserve area

?Tradeoff

Preserves areas, but damage conformality

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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

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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:

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Effect of

ASAP(λ=0)

ARAP(λ=) = 0

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Effect of

ASAP(λ=0)

ARAP(λ=)

= 0

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Multiple Boundaries

ABF++(2.00, 2.09)

ARAP(2.01, 2.01)

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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)

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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)

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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

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Thank you !