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

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Presentation transcript:

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

 Surface reconstruction of singular surface inputoutput

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

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.  …

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  …

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

Identify feature points Reconstruct feature curves Reconstruct singular surface

Identify feature points Reconstruct feature curves Reconstruct singular surface

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

 Gaussian-weighted graph Laplacian ([BQWZ12])

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

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

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

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

 Gaussian-weighted graph Laplacian highlow

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

Identify feature points Reconstruct feature curves Reconstruct singular surface

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

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

 Reeb graph

a noisy graph feature points Reeb graph

 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

 Graph simplification(denoise) a zigzag graph

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

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

 Graph smoothening  Use snake to smooth out the graph

Identify feature points Reconstruct feature curves Reconstruct singular surface

 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

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

 Weighted cocone un-weighted point weighted point

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

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

SphereCube with mesh

 Robust to noise input with 1% noise result

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

 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

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

 Real scanned data

 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

 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.