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
3/10Department of Computer Science and Engineering
• 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
4/10Department of Computer Science and Engineering
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
5/10Department of Computer Science and Engineering
Adaptive MLS (AMLS)
• Adaptive sampling condition (ASC)• need only 6 points to sample the arc
• Choice of weighting function
• AMLS function
6/10Department of Computer Science and Engineering
Effect of nearby samples
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Effect of distant samples
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Effect of distant samples (cont’)
10/10Department of Computer Science and Engineering
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
11/10Department of Computer Science and Engineering
Flow of reasoning
Lemma 1
Theorem 1
Lemma 4Surjectivity
Lemma 6Injectivity
Hom Isotopic
[CCS04]
12/10Department of Computer Science and Engineering
Algorithm
13/10Department of Computer Science and Engineering
Justification of Normal estimation
14/10Department of Computer Science and Engineering
Results
15/10Department of Computer Science and Engineering
Computational Aspects
• Big Convergent Domain (Newton Projection)
16/10Department of Computer Science and Engineering
Computational Aspects
• AMLS vs. PMLS
Function of PMLS [AK04]:
Function of AMLS:
17/10Department of Computer Science and Engineering
A brief Explanation
18/10Department of Computer Science and Engineering
Computational Aspects
• TimingFunction of variant PMLS [ZPKG02]:
Projection procedure: