andrew nealen tu berlin takeo igarashi the university of tokyo / presto jst olga sorkine

Andrew Nealen, TU Berlin, 2006 1

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Andrew NealenTU Berlin

Takeo IgarashiThe University of Tokyo / PRESTO JST

Olga SorkineMarc Alexa

TU Berlin

Laplacian Mesh Optimization

Laplacian Mesh Optimization


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What is it ?


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Overview

Motivation• Problem formulation• Laplacian mesh processing basics

Laplacian mesh optimization framework Applications

• Triangle shape optimization• Mesh smoothing

Discussion


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Motivation

Local detail preserving triangle optimization• A Sketch-Based Interface for Detail Preserving

Mesh Editing [Nealen et al. 2005]


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Motivation

Local detail preserving triangle optimization• A Sketch-Based Interface for Detail Preserving

Mesh Editing [Nealen et al. 2005]

Can we perform global optimization this way ?

=L x


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Laplacian Mesh Processing

Discrete Laplacians

=L x

n

cotangent : wij = cot ij + cot ij

uniform : wij = 1

( , )( , )

1i i ij j

i j Eiji j E

ww

δ x x


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Surface reconstructionn

cotangent : wij = cot ij + cot ij

( , )( , )

1i i ij j

i j Eiji j E

ww

δ x x

uniform : wij = 1

=L x

L

L

y

z

x

z

y

x


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Surface reconstructionn

z

y

x

y

z

x

=L

L

L

c1

fixedit

c2


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Least-squares solutionn

z

y

x

y

z

x

=L

L

L

c1

fixedit

c2

w1 w1

w2 w2

wLi wLi

A x = bATA x = bAT

(ATA)-1x = bAT

Normal Equations


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Tangential smoothingn

z

y

x

y

z

x

=L

L

L

fixc1

L

L

L


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L

L

L



z

y

x

y

z

x

=

fixc1


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L

L

L



z

y

x

y

z

x

=

fixc1


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More motivation…

So: can we use such a system for globaloptimization ?

=L x


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Our Solution

All vertices are (weighted) anchors

Preserves global shape Uses existing LS framework Anchor + Laplacian weights determine

result


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Framework

Detail preserving tri shape optimization for L = Luni and f = cot(similar to local optimization)

Mesh smoothing L = Lcot (outer fairness) or L = Luni (outer and inner fairness) and f = 0

=L x fWL WL

pWP WP


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Tri Shape Optimization

Detail preserving tri shape optimization

=Luni x

pWP WP


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Positional Weights


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Constant Weights


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Linear Weights


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CDF Weights


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CDF Weights


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Sharp Features


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Sharp Features


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Sharp Features


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Mesh Smoothing

Mesh smoothing L = Lcot (outer fairness) or L = Lumb (outer and inner fairness) and f = 0

Controlled by WP and WL (Intensity, Features) Similar to Least-Squares Meshes [Sorkine et al. 04]

=L x 0WL WL

pWP WP


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Using WP


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Using WP and WL


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Results


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Noisy


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Smoothed


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Original


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Smoothing Outer and Inner Fairness (Lumb)


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Original


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SmoothingOuter Fairness only (Lcot)


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Discussion

The good...• Easily controllable tri shape optimization and

smoothing• Leverages existing least squares framework• Can replace tangential smoothing step for

general remeshers ... and the not so good

• Euclidean distance is not Hausdorff distance, so error control is indirect

• Does rely on some (user) parameter tweaking


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Thank you !

Contact info

Andrew [email protected]

Takeo [email protected]

Olga [email protected]

Marc [email protected]

andrew nealen tu berlin takeo igarashi the university of tokyo / presto jst olga sorkine

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