Curve Reconstructj tjon with Many Fe Fewer Samples Stefan - PowerPoint PPT Presentation

“Can we reduce [...]? Yes we can!” “I enjoyed reading this mathematjcally very sound paper.” “... an advance to an important problem ofuen encountered ...” Curve Reconstructj tjon with Many Fe Fewer Samples Stefan Ohrhallinger 1 , Scotu tu A. Mitchell 2 and Michael Wimmer 1 1 TU Wien, Austria, 2 Sandia Natj tjonal Laboratories, U.S.A.

Why Sample Curves with Fe Fewer Points? Each sample costs: €26 57% of €61 S. Ohrhallinger, S.A. Mitchell, M. Wimmer 2

Sampling Conditj tjon ↔ Reconstructj tjon S. Ohrhallinger, S.A. Mitchell, M. Wimmer 3

Algorithm HNN-CRUST S. Ohrhallinger, S.A. Mitchell, M. Wimmer 4

HNN-CRUST Reconstructj tjon Results Sharp angles Open curves Samples CRUST [Amenta et al. ‘98] HNN-CRUST S. Ohrhallinger, S.A. Mitchell, M. Wimmer 5

Earlier Sampling Conditj tjons ε<0.2: CRUST [Amenta et al. ‘98] ε<0.3: NN-CRUST [Dey, Kumar ‘99] ε<0.47: Our HNN-CRUST ρ<0.9: Our HNN-CRUST S. Ohrhallinger, S.A. Mitchell, M. Wimmer 6

What is ε-Sampling? M = medial axis [Blum ’67] lfs = local feature size [Ruppert ‘93] D = disk empty of C ||s,p|| < ε*lfs(p) S. Ohrhallinger, S.A. Mitchell, M. Wimmer 7

The Problem of Large ε Required lfs vanishes at samples → s 1 connects wrongly to s i S. Ohrhallinger, S.A. Mitchell, M. Wimmer 8

So We Designed ρ-Sampling Interval I(p 0 ,p 1 ): reach (I)=min lfs (I) [Federer ‘59] ||s,p|| < ρ*reach(I) reach does not vanish at samples! S. Ohrhallinger, S.A. Mitchell, M. Wimmer 9

Works for Large ρ S. Ohrhallinger, S.A. Mitchell, M. Wimmer 10

Results for ρ<0.9 Sampling ε<0.3 Samples: 61 131 356 ρ<0.9 Samples: 26 58 180 S. Ohrhallinger, S.A. Mitchell, M. Wimmer 11

Bounding Reconstructj tjon Distance ε<0.3: ρ<0.9: ρ<0.9, d=1%: 131 samples 58 samples 60 samples (+2) d=bounded Hausdorf distance (in % of larger axis extent) S. Ohrhallinger, S.A. Mitchell, M. Wimmer 12

Reconstructj tjon Distances Compared ε<0.3 131 131 133 148 204 ρ<0.9 58 60 73 105 173 ∞ 1% 0.3% 0.1% 0.03% d S. Ohrhallinger, S.A. Mitchell, M. Wimmer 13

Improved Bound for ε-Sampling ε<0.3, 131 samples ε<0.47, 94 samples ρ<0.9, 58 samples ε < r-sampling → ρ < r/(1 − r)-sampling Proof: reach(I) ≥ (1-r)lfs(p) ρ<0.9 → ε<0.47 (or ε<0.9 at constant curvature) S. Ohrhallinger, S.A. Mitchell, M. Wimmer 14

Limits of HNN-CRUST Sharp angles Very sharp angles Samples GathanG [Dey, Wenger ‘02] HNN-CRUST S. Ohrhallinger, S.A. Mitchell, M. Wimmer 15

Conclusion and Outlook 1) Simple variant 2) Sampling cond. 3) ρ<0.9 close 4) Corollary: ≡ reconstructjon to tjght bound ε<0.3 → ε<0.47 HNN-CRUST All fjgures/tables reproducible from open source (link in paper) Now extending it to: Contact: Stefan Ohrhallinger TU Wien, Austria noisy samples 3D S. Ohrhallinger, S.A. Mitchell, M. Wimmer 16

Computer-Assisted Proof of ρ<0.9 Blue disks = exclusion zone of C, must contain point z (=farthest connected to s 1 instead of s 2 by HNN-CRUST) z C is defjned by points x, s 1 , y, s 2 s 1 β y α r 1 C is bounded by parameters: x r=|s 0 s 1 |/|s 1 ,s 2 |, in ]0..1] α, β with s 1 -tangent, [0°..27°] s 2 Sample parameter space in tjny steps, worst case combinatjons Case r=1, α=β=27° S. Ohrhallinger, S.A. Mitchell, M. Wimmer 17

Computer-Assisted Proof – More Cases r=1, α=β=27° r=⅓, α=β=27° r= , α=β=27° r= , α=27°, β=0° r= , α=13°, β=27° r= , α=β=0° r=0, α=β=0° r=1, α=0°, β=27° S. Ohrhallinger, S.A. Mitchell, M. Wimmer 18