Issue No.06 - June (2013 vol.19)
pp: 941-952
Muyang Zhang , State Key Lab. of CAD & CG, Zhejiang Univ., Hangzhou, China
Jin Huang , State Key Lab. of CAD & CG, Zhejiang Univ., Hangzhou, China
Xinguo Liu , State Key Lab. of CAD & CG, Zhejiang Univ., Hangzhou, China
Hujun Bao , State Key Lab. of CAD & CG, Zhejiang Univ., Hangzhou, China
Many natural and man-made objects consist of simple primitives, similar components, and various symmetry structures. This paper presents a divide-and-conquer quadrangulation approach that exploits such global structural information. Given a model represented in triangular mesh, we first segment it into a set of submeshes, and compare them with some predefined quad mesh templates. For the submeshes that are similar to a predefined template, we remesh them as the template up to a number of subdivisions. For the others, we adopt the wave-based quadrangulation technique to remesh them with extensions to preserve symmetric structure and generate compatible quad mesh boundary. To ensure that the individually remeshed submeshes can be seamlessly stitched together, we formulate a mixed-integer optimization problem and design a heuristic solver to optimize the subdivision numbers and the size fields on the submesh boundaries. With this divider-and-conquer quadrangulation framework, we are able to process very large models that are very difficult for the previous techniques. Since the submeshes can be remeshed individually in any order, the remeshing procedure can run in parallel. Experimental results showed that the proposed method can preserve the high-level structures, and process large complex surfaces robustly and efficiently.
Solid modeling, Optimization, Vectors, Shape, Computational modeling, Strips, Smoothing methods, mixed-integer optimization, Quad remeshing, divide-and-conquer, segmentation
Muyang Zhang, Jin Huang, Xinguo Liu, Hujun Bao, "A Divide-and-Conquer Approach to Quad Remeshing", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 6, pp. 941-952, June 2013, doi:10.1109/TVCG.2012.301
[1] D. Bommes, H. Zimmer, and L. Kobbelt, "Mixed-Integer Quadrangulation," ACM Trans. Graphics, vol. 28, no. 3, pp. 1-10, 2009.
[2] M. Zhang, J. Huang, X. Liu, and H. Bao, "A Wave-Based Anisotropic Quadrangulation Method," ACM Trans. Graphics, vol. 29, pp. 118:1-118:8, July 2010.
[3] F. Kälberer, M. Nieser, and K. Polthier, "Quadcover—Surface Parameterization Using Branched Coverings," Computer Graphics Forum, vol. 26, no. 3, pp. 375-384, Sept. 2007.
[4] K. Hormann and G. Greiner, "Quadrilateral Remeshing," Proc. Vision, Modeling, and Visualization, B. Girod, G. Greiner, H. Niemann, and H.-P. Seidel, eds., pp. 153-162, Nov. 2000.
[5] M. Marinov and L. Kobbelt, "A Robust Two-Step Procedure for Quad-Dominant Remeshing," Computer Graphics Forum, vol. 25, no. 3, pp. 537-546, Sept. 2006.
[6] Y.-K. Lai, L. Kobbelt, and S.-M. Hu, "An Incremental Approach to Feature Aligned Quad Dominant Remeshing," Proc. ACM Symp. Solid and Physical Modeling, pp. 137-145, 2008.
[7] J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen, and C. Geuzaine, "Blossom-Quad: A Non-Uniform Quadrilateral Mesh Generator Using a Minimum Cost Perfect Matching Algorithm," Int'l J. Numerical Methods in Eng., vol. 89, no. 9, pp. 1102-1119, 2012.
[8] W. Cook and A. Rohe, "Computing Minimum-Weight Perfect Matchings," INFORMS J. Computing, vol. 11, no. 2, pp. 138-148, Feb. 1999.
[9] P. Alliez, D. Cohen-Steiner, O. Devillers, B. Lévy, and M. Desbrun, "Anisotropic Polygonal Remeshing," ACM Trans. Graphics, vol. 22, no. 3, pp. 485-493, 2003.
[10] M. Marinov and L. Kobbelt, "Direct Anisotropic Quad-Dominant Remeshing," Proc. Pacific Conf. Computer Graphics and Applications, pp. 207-216, 2004.
[11] N. Ray, W.C. Li, B. Lévy, A. Sheffer, and P. Alliez, "Periodic Global Parameterization," ACM Trans. Graphics, vol. 25, no. 4, pp. 1460-1485, 2006.
[12] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart, "Spectral Surface Quadrangulation," ACM Trans. Graphics, vol. 25, no. 3, pp. 1057-1066, 2006.
[13] J. Huang, M. Zhang, J. Ma, X. Liu, L. Kobbelt, and H. Bao, "Spectral Quadrangulation with Orientation and Alignment Control," ACM Trans. Graphics, vol. 27, no. 5, pp. 1-9, 2008.
[14] Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun, "Designing Quadrangulations with Discrete Harmonic Forms," Proc. Eurographics Symp. Geometry Processing, pp. 201-210, 2006.
[15] A. Myles, N. Pietroni, D. Kovacs, and D. Zorin, "Feature-Aligned T-Meshes," ACM Trans. Graphics, vol. 29, pp. 117:1-117:11, July 2010.
[16] J. Tierny, J. DanielsII, L.G. Nonato, V. Pascucci, and C.T. Silva, "Inspired Quadrangulation," Computer Aided Design, vol. 43, no. 11, pp. 1516-1526, 2011.
[17] D. Cohen-Steiner and J.-M. Morvan, "Restricted Delaunay Triangulations and Normal Cycle," Proc. Symp. Computational Geometry, pp. 312-321, June 2003.
[18] J. Palacios and E. Zhang, "Rotational Symmetry Field Design on Surfaces," ACM Trans. Graphics, vol. 26, no. 3,article 55, 2007.
[19] N. Ray, B. Vallet, W.C. Li, and B. Lévy, "N-Symmetry Direction Field Design," ACM Trans. Graphics, vol. 27, no. 2, pp. 1-13, 2008.
[20] S. Dong, S. Kircher, and M. Garland, "Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds," Computer Aided Geometric Design, vol. 22, no. 5, pp. 392-423, 2005.
[21] D. Kovacs, A. Myles, and D. Zorin, "Anisotropic Quadrangulation," Computer Aided Geometric Design, vol. 28, no. 8, pp. 449-462, 2011.
[22] J. Daniels, C.T. Silva, J. Shepherd, and E. Cohen, "Quadrilateral Mesh Simplification," ACM Trans. Graphics, vol. 27, pp. 148:1-148:9, Dec. 2008.
[23] M. Tarini, N. Pietroni, P. Cignoni, D. Panozzo, and E. Puppo, "Practical Quad Mesh Simplification," Computer Graphics Forum, vol. 29, no. 2, pp. 407-418, 2010.
[24] D. Bommes, T. Lempfer, and L. Kobbelt, "Global Structure Optimization of Quadrilateral Meshes," Computer Graphics Forum, vol. 30, pp. 375-384, 2011.
[25] Y. Li, W. Wang, R. Ling, and C. Tu, "Shape Optimization of Quad Mesh Elements," Computers & Graphics, vol. 35, no. 3, pp. 444-451, 2011.
[26] J. DanielsII, M. Lizier, M. Siqueira, C. Silva, and L. Nonato, "Template-Based Quadrilateral Meshing," Computers & Graphics, vol. 35, no. 3, pp. 471-482, 2011.
[27] S.J. Owen, "A Survey of Unstructured Mesh Generation Technology," Proc. Int'l Conf. Meshing Roundtable, pp. 239-267, 1998.
[28] Gurobi, Gurobi Optimizer 4.5, http:/, 2013.
[29] L. Shapira, A. Shamir, and D. Cohen-Or, "Consistent Mesh Partitioning and Skeletonisation Using the Shape Diameter Function," Visual Computer, vol. 24, no. 4, pp. 249-259, 2008.
[30] J. Sun, M. Ovsjanikov, and L.J. Guibas, "A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion." Computer Graphics Forum, vol. 28, no. 5, pp. 1383-1392, 2009.
[31] V. Jain and H. Zhang, "Robust 3D Shape Correspondence in the Spectral Domain," Proc. Int'l Conf. Shape Modeling and Applications, pp. 118-129, 2006.
[32] C.D. Team, S. Benzley, C.E. Department, R. Kerr, S.R. Jankovich, and D.B. Mcrae, "Cubit Mesh Generation Environment Volume 1: Users Manual," technical report, 1994.
[33] S.A. Mitchell, "High Fidelity Interval Assignment," Proc. Int'l Conf. Meshing Roundtable, pp. 33-44, 1997.
[34] J. Shepherd, S. Benzley, and S. Mitchell, "Interval Assignment for Volumes with Holes," Int'l J. Numerical Methods in Eng., pp. 49-277, 2000.
[35] Y. Bazilevs, V. Calo, J. Cottrell, J. Evans, T. Hughes, S. Lipton, M. Scott, and T. Sederberg, "Isogeometric Analysis Using T-Splines," Computer Methods in Applied Mechanics and Eng., vol. 199, nos. 5-8, pp. 229-263, 2010.
[36] S. Katz and A. Tal, "Hierarchical Mesh Decomposition Using Fuzzy Clustering and Cuts," ACM Trans. Graphics, vol. 22, pp. 954-961, July 2003.
[37] X. Chen, A. Golovinskiy, and T. Funkhouser, "A Benchmark for 3D Mesh Segmentation," ACM Trans. Graphics, vol. 28, pp. 73:1-73:12, July 2009.
[38] E. Kalogerakis, A. Hertzmann, and K. Singh, "Learning 3D Mesh Segmentation and Labeling," ACM Trans. Graphics, vol. 29, pp. 102:1-102:12, July 2010.
[39] N.J. Mitra, L.J. Guibas, and M. Pauly, "Partial and Approximate Symmetry Detection for 3D Geometry," ACM Trans. Graphics, vol. 25, pp. 560-568, July 2006.
[40] J. Podolak, P. Shilane, A. Golovinskiy, S. Rusinkiewicz, and T. Funkhouser, "A Planar-Reflective Symmetry Transform for 3D Shapes," ACM Trans. Graphics, vol. 25, pp. 549-559, July 2006.