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. [GG09] Feature preserving point set surfaces based on non-linear kernel regression, ZTIRELI, et.al [CG06] Delaunay triangulation based surface reconstruction, CAZALS, et.al [FCOS05] Robust moving least-squares tting 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 (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.