An Adaptive MLS Surface for Reconstruction with Guarantees

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An Adaptive MLS Surface for Reconstruction
withGuarantees

• Tamal K. Dey and Jian Sun

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Definitions of MLS surfaces

• Projection MLS (PMLS) ABC01Lev98
  PKKG03AK04
• Stationary points of certain projection operator
• Normal
• Energy function
• Implicit form

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Definitions of MLS surfaces

• Implicit MLS (IMLS) SOS04 Kol05
• The moving lease squares solution to set of
  weighted constrains
• Constrains
• Implicit function in
• the simplest case

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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 104 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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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

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Flow of reasoning
Lemma 4
Surjectivity
Lemma 1
Hom
Isotopic
Injectivity
Theorem 1
Lemma 6
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

• Timing

Function of variant PMLS ZPKG02
Projection procedure