1. with Tamal Dey, Qichao Que, Issam Safa, Lei Wang, Yusu Wang Computer science and Engineering The Ohio State University Xiaoyin Ge

2.  Surface reconstruction of singular surface inputoutput

3. Singular surface A collection of smooth surface patches with boundaries. glue intersect boundary

4. 2D manifold reconstruction  [AB99] Surface reconstruction by Voronoi filtering. AMENTA N., BERN M.  [ACDL02] A simple algorithm for homeomorphic surface reconstruction. AMENTA N., et. al.  [BC02] Smooth surface reconstruction via natural neighbor interpolation of distance functions. BOISSONNAT et. Al  [ABCO01] Point set surfaces. ALEXA et. al.  …

5. Feature aware method  [LCOL07] Data dependent MLS for faithful surface approximation. LIPMAN, et. al.  [ÖGG09] Feature preserving point set surfaces based on non-linear kernel regression, ÖZTIRELI, et.al  [CG06] Delaunay triangulation based surface reconstruction, CAZALS, et.al  [FCOS05] Robust moving least-squares fitting with sharp features, FLEISHMAN, et.al  …

6. Need a simple yet effective reconstruction algorithm for all three singular surfaces.

7. Identify feature points Reconstruct feature curves Reconstruct singular surface

8. Identify feature points Reconstruct feature curves Reconstruct singular surface

9.  Gaussian-weighted graph Laplacian ( [BN02], Belkin-Niyogi, 2002)

10.  Gaussian-weighted graph Laplacian ([BQWZ12])

11.  Gaussian-weighted graph Laplacian, scaling ([BQWZ12]) boundary lowhigh

12. surf B surf A intersection lowhigh  Gaussian-weighted graph Laplacian, scaling ([BQWZ12])

13. surf A surf B glue (sharp feature) lowhigh  Gaussian-weighted graph Laplacian, scaling ([BQWZ12])

14. surf A surf B  Gaussian-weighted graph Laplacian (scaling, [BQWZ12]) boundary surf B surf A intersection sharp feature

15.  Gaussian-weighted graph Laplacian highlow

16.  Gaussian-weighted graph Laplacian  Advantage:  Simple  Unified approach  Robust to noise

17. Identify feature points Reconstruct feature curves Reconstruct singular surface

18.  Graph method proposed by [GSBW11] [ Data skeletonization via reeb graphs, Ge, et.al, 2011]

19.  Reeb graph ( from Rips-complex [DW11] ) Rips complex Reeb graph (abstract) Reeb graph (abstract) Reeb graph (augmented) Reeb graph (augmented)

20.  Reeb graph

21. a noisy graph feature points Reeb graph

22.  Graph simplification (denoise) noisy branch noisy loop d b c d e a b c a e a b c d e f a b c d e f

23.  Graph simplification(denoise) a zigzag graph

24.  Graph smoothening [KWT88]  Use snake to smooth out the graph graph energy graph Laplacian

25.  Graph smoothening  Use snake to smoothen graph graph Laplacian graph energy align along feature min() smoothen graph

26.  Graph smoothening  Use snake to smooth out the graph

27. Identify feature points Reconstruct feature curves Reconstruct singular surface

28.  Reconstruction [CDR07][CDL07] [CDL07] A Practical Delaunay Meshing Algorithm for a Large Class of Domains, Cheng, et.al [CDR07] Delaunay Refinement for Piecewise Smooth Complexes, Cheng-Dey-Ramos, 2007

29.  Weighted cocone cocone weighted Delaunay [ACDL00] A simple algorithm for homeomorphic surface reconstruction, Amenta,-Choi-Dey -Leekha

30.  Weighted cocone un-weighted point weighted point

31.  Reconstruction  Voronoi cell size ∝ weight  Give higher weight to points on the feature curve

32. a a b b c c d d a. Octaflower 107K a. Octaflower 107K b. Fandisk 114K b. Fandisk 114K c. SphCube 65K c. SphCube 65K d. Beetle 63K d. Beetle 63K

33. SphereCube with mesh

34.  Robust to noise input with 1% noise result

35.  Perform much better than un-weighted cocone Cocone Our method

36.  Conclusion  Unified and simple method to handle all three types of singular surfaces  Robust to noise  Future work  More robust system for real data  Concave corner

37. We thank all people who have helped us to demonstrate this method ! Most of the models used in this paper are courtesy of AIM@SHAPE Shape Repository. The authors acknowledge the support of NSF under grants CCF-1048983, CCF- 1116258 and CCF-0915996.

39.  Real scanned data

40.  Weighted Delaunay ▪ Two points: p w =(p,w p ) and z w =(z,w z ) ▪ their power product Π(p w, z w ) = |p-z| 2 -w p -w z

41.  Timing Stg 1: Building KD tree; Stg 2: computation of graph Laplacian and feature points detection; Stg 3: feature curve construction; Stg 4: feature curve refinement; Stg 5: surface reconstruction.