Department of Computer Science and Engineering

An Adaptive MLS Surface for Reconstruction with

Guarantees

Tamal K. Dey and Jian Sun

2/10Department of Computer Science and Engineering

• Projection MLS (PMLS) [ABC*01][Lev98] [PKKG03][AK04]

• Stationary points of certain projection operator

• Normal:

• Energy function:

• Implicit form:

Definitions of MLS surfaces

( )( )

( )

p pp P

pp P

v w xn x

w x

2

( , ( ))

1[( ) ( )] ( )

2T

pp P

y n x

y p n x y

( , ( ))( ) ( ) ( | )T

x

y n xJ x n x

y

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• Implicit MLS (IMLS) [SOS04] [Kol05] • The moving lease squares solution to set of

weighted constrains• Constrains:• • Implicit function in

the simplest case:

Definitions of MLS surfaces

( ) ( )Tp px x p v 2

p P [( ( ) ( )) ( )]Min p pI x x x

[( ) ] ( )( )

( )

Tp pp P

pp P

x p v xI x

x

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Theoretical guarantee of MLS

• Uniform sampling condition (USC)• Kolluri [Kol05] for IMLS, Dey et al. [DGS05] for

PMLS• Restriction of USC

• require more than 10^4 points to sample the arc (red)

• No smooth effect at the surface with big features

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Adaptive MLS (AMLS)

• Adaptive sampling condition (ASC)• need only 6 points to sample the arc

• Choice of weighting function

• AMLS function

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Effect of nearby samples

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Effect of distant samples

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Effect of distant samples (cont’)

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Theoretical guarantee of AMLS

• Define map• Map is a homeomorphism

• in blue and in red

is surjective•

is injective•

are isotopic

1: (0) 0.1N

( ) 0N x

( ) [ ] 0 in 0.3z xn N

( ) 0N x

1(0) 0.1 and N

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Flow of reasoning

Lemma 1

Theorem 1

Lemma 4Surjectivity

Lemma 6Injectivity

Hom Isotopic

[CCS04]

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Algorithm

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Justification of Normal estimation

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Results

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Computational Aspects

• Big Convergent Domain (Newton Projection)

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Computational Aspects

• AMLS vs. PMLS

Function of PMLS [AK04]:

Function of AMLS:

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A brief Explanation

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Computational Aspects

• TimingFunction of variant PMLS [ZPKG02]:

Projection procedure: