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

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

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

Effect of nearby samples

Effect of distant samples

Effect of distant samples (cont’)

Effect of distant samples (cont’) …… 27 …… …… 27 …… 27

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

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

Algorithm

Justification of Normal estimation

Results

Computational Aspects Big Convergent Domain (Newton Projection)

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

A brief Explanation

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