Consolidation of Unorganized Point Clouds for Surface Reconstruction Hui Huang 1 Dan Li 1 Hao Zhang 2 Uri Ascher 1 Daniel Cohen-Or 3 1 University of British Columbia 2 Simon Fraser University 3 Tel-Aviv University 1

2 Raw Scan Data

3 Data Consolidation

4 Surface Reconstruction Delaunay techniques [Amenta & Bern 1998], Power-crust [Amenda et al. 2001], Cocone [Dey & Giesen 2001], [Cazals & Giesen 2006] …… Approximate reconstructions [Hoppe et al. 1992], RBF [Carr et al. 2001], Poisson [Kazhdan et al. 2006] ……

5 Raw Scan Data

6 RBF Reconstruction

7 Difficulties Direct surface reconstruction may fail on challenging datasets Normals are crucial for surface reconstruction noise outliers close-by surface sheets missing normal information  not always available  not always reliable

8 Unsigned Directions by PCA Thick cloudNon-uniform distribution Close-by surface sheets

9 Normal Consistency [Hoppe et al. 1992] Based on angles between unsigned normals May produce errors on close-by surface sheets

10 Point Cloud Consolidation Unorganized Noisy Thick Outliers Non-uniform Un-oriented Input Consolidated Clean Thin Outlier-free Uniform Oriented Output InputOutput

11 Contributions Weighted locally optimal projection operator (WLOP) To consolidate point clouds: Robust normal estimation

12 Locally Optimal Projection LOP operator [Lipman et al. 2007] defines a point set by a fixed point iteration where, for each point x, given the current iterate, the next iterate is to minimize The repulsion function here is

13 New Repulsion Function More locally regular point distribution

14 New Repulsion Function Better convergence behavior

15 Non-uniformity The first term of LOP, an L 1 median, tends to follow the trend of non-uniformity if input is highly non-uniform. Raw scan LOP (old η)LOP (new η) σ = 0.24 σ = 0.18

16 Improved Weighted LOP Define the weighted local densities for each point in the input set and projection set as Then the projection becomes

17 WLOP vs. LOP More globally regular point distribution Raw ScanLOP (old η)LOP (new η)WLOP σ = 0.24 σ = 0.18 σ = 0.09

18 WLOP vs. LOP Better convergence

19 Normal Propagation Select a source Detect thin surface features Normal flipping Propagate

20 Source Selection 

21 Distance Measure

22 Thin Features and Normal Flipping Outside the convex hull Limitation: cannot distinguish between flat and concave Remedy: normal flipping

23 Orientation-aware PCA Propagate Predictor Corrector Loop OPCA PCA

24 Noisy inputTraditional result Without flip With flip After correction One Example

25 Up-sampling Raw scanWith consolidationWithout consolidation

26 Surface Generation LOP WLOP RBF

27 RBFPoisson

28 NormFet+AMLS+Cocone [Dey et al.] TraditionalOur

29 TraditionalWith OPCAWithout iteration

30 Limitations

31 Back-cullingFront-cullingSparse setPoisson surface

32 Theoretical guarantee for the correctness of normal estimation under sampling Rigorous theoretical analysis of the predictor- corrector iteration Better handling of missing data Recovery and enhancement of sharp features Future Work

33 Federico Ponchio Anonymous Reviewers NSERC (No and No ) The Israel Science Foundation Acknowledgements

34 Point-Consolidation API is available