1. An Adaptive MLS Surface for Reconstruction with Guarantees
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. Effect of distant samples (cont’)…… 27 …… …… 27 …… 27

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

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

12. Algorithm

13. Justification of Normal estimation

14. Results

15. Computational Aspects
Big Convergent Domain (Newton Projection)

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

17. A brief Explanation

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