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Department of Computer Science and Engineering An Adaptive MLS Surface for Reconstruction with Guarantees Tamal K. Deyand Jian Sun.

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Presentation on theme: "Department of Computer Science and Engineering An Adaptive MLS Surface for Reconstruction with Guarantees Tamal K. Deyand Jian Sun."— Presentation transcript:

1 Department of Computer Science and Engineering An Adaptive MLS Surface for Reconstruction with Guarantees Tamal K. Deyand Jian Sun

2 2/10 Department 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

3 3/10 Department 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

4 4/10 Department 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 5/10 Department 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 6/10 Department of Computer Science and Engineering Effect of nearby samples

7 7/10 Department of Computer Science and Engineering Effect of distant samples

8 8/10 Department of Computer Science and Engineering Effect of distant samples (cont )

9 9/10 Department of Computer Science and Engineering Effect of distant samples (cont ) …… 27

10 10/10 Department 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

11 11/10 Department of Computer Science and Engineering Flow of reasoning Lemma 1 Theorem 1 Lemma 4 Surjectivity Lemma 6 Injectivity HomIsotopic [CCS04]

12 12/10 Department of Computer Science and Engineering Algorithm

13 13/10 Department of Computer Science and Engineering Justification of Normal estimation

14 14/10 Department of Computer Science and Engineering Results

15 15/10 Department of Computer Science and Engineering Computational Aspects Big Convergent Domain (Newton Projection)

16 16/10 Department of Computer Science and Engineering Computational Aspects AMLS vs. PMLS Function of PMLS [AK04] : Function of AMLS:

17 17/10 Department of Computer Science and Engineering A brief Explanation

18 18/10 Department of Computer Science and Engineering Computational Aspects Timing Function of variant PMLS [ZPKG02]: Projection procedure:


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