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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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Surface reconstruction of singular surface inputoutput
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Singular surface A collection of smooth surface patches with boundaries. glue intersect boundary
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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. …
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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 …
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Need a simple yet effective reconstruction algorithm for all three singular surfaces.
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Identify feature points Reconstruct feature curves Reconstruct singular surface
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Identify feature points Reconstruct feature curves Reconstruct singular surface
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Gaussian-weighted graph Laplacian ( [BN02], Belkin-Niyogi, 2002)
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Gaussian-weighted graph Laplacian ([BQWZ12])
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Gaussian-weighted graph Laplacian, scaling ([BQWZ12]) boundary lowhigh
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surf B surf A intersection lowhigh Gaussian-weighted graph Laplacian, scaling ([BQWZ12])
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surf A surf B glue (sharp feature) lowhigh Gaussian-weighted graph Laplacian, scaling ([BQWZ12])
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surf A surf B Gaussian-weighted graph Laplacian (scaling, [BQWZ12]) boundary surf B surf A intersection sharp feature
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Gaussian-weighted graph Laplacian highlow
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Gaussian-weighted graph Laplacian Advantage: Simple Unified approach Robust to noise
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Identify feature points Reconstruct feature curves Reconstruct singular surface
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Graph method proposed by [GSBW11] [ Data skeletonization via reeb graphs, Ge, et.al, 2011]
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Reeb graph ( from Rips-complex [DW11] ) Rips complex Reeb graph (abstract) Reeb graph (abstract) Reeb graph (augmented) Reeb graph (augmented)
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Reeb graph
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a noisy graph feature points Reeb graph
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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
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Graph simplification(denoise) a zigzag graph
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Graph smoothening [KWT88] Use snake to smooth out the graph graph energy graph Laplacian
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Graph smoothening Use snake to smoothen graph graph Laplacian graph energy align along feature min() smoothen graph
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Graph smoothening Use snake to smooth out the graph
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Identify feature points Reconstruct feature curves Reconstruct singular surface
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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
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Weighted cocone cocone weighted Delaunay [ACDL00] A simple algorithm for homeomorphic surface reconstruction, Amenta,-Choi-Dey -Leekha
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Weighted cocone un-weighted point weighted point
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Reconstruction Voronoi cell size ∝ weight Give higher weight to points on the feature curve
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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
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SphereCube with mesh
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Robust to noise input with 1% noise result
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Perform much better than un-weighted cocone Cocone Our method
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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
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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.
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Real scanned data
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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
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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.
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