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An Adaptive MLS Surface for Reconstruction with Guarantees

An Adaptive MLS Surface for Reconstruction with Guarantees. Tamal K. Dey and Jian Sun. Definitions of MLS surfaces. Projection MLS (PMLS) [ABC*01][Lev98] [PKKG03][AK04] Stationary points of certain projection operator Normal: Energy function: Implicit form:. Definitions of MLS surfaces.

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An Adaptive MLS Surface for Reconstruction with Guarantees

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  1. An Adaptive MLS Surface for Reconstruction withGuarantees Tamal K. Dey and Jian Sun

  2. Definitions of MLS surfaces • Projection MLS (PMLS)[ABC*01][Lev98] [PKKG03][AK04] • Stationary points of certain projection operator • Normal: • Energy function: • Implicit form:

  3. 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:

  4. 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. Adaptive MLS (AMLS) • Adaptive sampling condition (ASC) • need only 6 points to sample the arc • Choice of weighting function • AMLS function

  6. Effect of nearby samples

  7. Effect of distant samples

  8. Effect of distant samples (cont’)

  9. Theoretical guarantee of AMLS • Define map • Map is a homeomorphism • in blue and in red  is surjective  is injective are isotopic

  10. Flow of reasoning Lemma 4 Surjectivity Lemma 1 Hom Isotopic Injectivity Theorem 1 Lemma 6 [CCS04]

  11. Algorithm

  12. Justification of Normal estimation

  13. Results

  14. Computational Aspects • Big Convergent Domain (Newton Projection)

  15. Computational Aspects • AMLS vs. PMLS Function of PMLS[AK04]: Function of AMLS:

  16. A brief Explanation

  17. Computational Aspects • Timing Function of variant PMLS[ZPKG02]: Projection procedure:

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