“Fields on symmetric surfaces” by Panozzo, Lipman, Puppo and Zorin

  • ©Daniele Panozzo, Yaron Lipman, Enrico Puppo, and Denis Zorin




    Fields on symmetric surfaces



    Direction fields, line fields and cross fields are used in a variety of computer graphics applications ranging from non-photorealistic rendering to remeshing. In many cases, it is desirable that fields adhere to symmetry, which is predominant in natural as well as man-made shapes. We present an algorithm for designing smooth N-symmetry fields on surfaces respecting generalized symmetries of the shape, while maintaining alignment with local features. Our formulation for constructing symmetry fields is based on global symmetries, which are given as input to the algorithm, with no isometry assumptions. We explore in detail the properties of generalized symmetries (reflections in particular), and we also develop an algorithm for the robust computation of such symmetry maps, based on a small number of correspondences, for surfaces of genus zero.


    1. Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., and Davis, J. 2005. Scape: shape completion and animation of people. ACM Trans. Graph. 24 (July), 408–416. Google ScholarDigital Library
    2. Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, 77. Google ScholarDigital Library
    3. Bommes, D., Lempfer, T., and Kobbelt, L. 2011. Global structure optimization of quadrilateral meshes. Comput. Graph. Forum 30, 2, 375–384.Google ScholarCross Ref
    4. Bronstein, A. M., Bronstein, M. M., and Kimmel, R. 2006. Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. Natl. Acad. Sci. USA 103, 5, 1168–1172.Google ScholarCross Ref
    5. Cailliere, D., Denis, F., Pele, D., and Baskurt, A. 2008. 3d mirror symmetry detection using hough transform. In Image Processing, 2008. ICIP 2008. 15th IEEE International Conference on, IEEE, 1772–1775.Google Scholar
    6. Crane, K., Desbrun, M., and Schröder, P. 2010. Trivial connections on discrete surfaces. Computer Graphics Forum 29, 5 (July), 1525–1533.Google ScholarCross Ref
    7. Farb, B., and Margalit, D. 2011. A primer on mapping class groups. Princeton Univ Press.Google Scholar
    8. Ghosh, D., Amenta, N., and Kazhdan, M. 2010. Closed-form blending of local symmetries. Computer Graphics Forum 29, 5, 1681–1688.Google ScholarCross Ref
    9. Golovinskiy, A., Podolak, J., and Funkhouser, T. 2009. Symmetry-aware mesh processing. Mathematics of Surfaces XIII, 170–188. Google ScholarDigital Library
    10. Hertzmann, A., and Zorin, D. 2000. Illustrating smooth surfaces. In Proceedings of SIGGRAPH 2000, 517–526. Google ScholarDigital Library
    11. Kälberer, F., Nieser, M., and Polthier, K. 2007. Quad-Cover: Surface Parameterization using Branched Coverings. Computer Graphics Forum 26, 3, 375–384.Google ScholarCross Ref
    12. Kazhdan, M., Amenta, N., Gu, S., Wiley, D., and Hamann, B. 2009. Symmetry restoration by stretching. In Canadian Conference on Computational Geometry, Citeseer.Google Scholar
    13. Kim, V. G., Lipman, Y., Chen, X., and Funkhouser, T. 2010. Mbius Transformations For Global Intrinsic Symmetry Analysis. Computer Graphics Forum 29, 5, 1689–1700.Google ScholarCross Ref
    14. Kim, V. G., Lipman, Y., and Funkhouser, T. 2011. Blended intrinsic maps. ACM Trans. Graph. 30 (Aug.), 79:1–79:12. Google ScholarDigital Library
    15. Koszul, J. 1965. Lectures on groups of transformations, vol. 32 of Lectures on Mathematics. Tata Institute of Fundamental Research, Bombay, India.Google Scholar
    16. Kraevoy, V., and Sheffer, A. 2004. Cross-parameterization and compatible remeshing of 3d models. ACM Transactions on Graphics (Proc. SIGGRAPH 2004). Google ScholarDigital Library
    17. Lai, Y., Jin, M., Xie, X., He, Y., Palacios, J., Zhang, E., Hu, S., and Gu, X. 2010. Metric-driven rosy field design and remeshing. Visualization and Computer Graphics, IEEE Transactions on 16, 1, 95–108. Google ScholarDigital Library
    18. Lipman, Y., Chen, X., Daubechies, I., and Funkhouser, T. 2010. Symmetry factored embedding and distance. In ACM SIGGRAPH 2010 papers, ACM, 1–12. Google ScholarDigital Library
    19. Mémoli, F., and Sapiro, G. 2004. Comparing Point Clouds. In Proceedings Symposium on Geometry Processing 2004, Eurographics, 33–42. Google ScholarDigital Library
    20. Mitra, N., Guibas, L., and Pauly, M. 2006. Partial and approximate symmetry detection for 3d geometry. ACM Transactions on Graphics (TOG) 25, 3, 560–568. Google ScholarDigital Library
    21. Mitra, N., Guibas, L., and Pauly, M. 2007. Symmetrization. ACM Transactions on Graphics 26, 3. Google ScholarDigital Library
    22. Montgomery, D., and Zippin, L. 1955. Topological transformation groups, vol. 1. Interscience Publishers New York.Google Scholar
    23. Nieser, M., Palacios, J., Polthier, K., and Zhang, E. 2012. Hexagonal global parameterization of arbitrary surfaces. Visualization and Computer Graphics, IEEE Transactions on 18, 6 (june), 865–878. Google ScholarDigital Library
    24. Osada, R., Funkhouser, T., Chazelle, B., and Dobkin, D. 2002. Shape distributions. ACM Trans. Graph. 21 (October), 807–832. Google ScholarDigital Library
    25. Ovsjanikov, M., Sun, J., and Guibas, L. 2008. Global intrinsic symmetries of shapes. Computer graphics forum 27, 5, 1341–1348. Google ScholarDigital Library
    26. Ovsjanikov, M., Mérigot, Q., Mémoli, F., and Guibas, L. 2010. One Point Isometric Matching with the Heat Kernel. Computer Graphics Forum 29, 5, 1555–1564.Google ScholarCross Ref
    27. Palacios, J., and Zhang, E. 2007. Rotational symmetry field design on surfaces. ACM Trans. Graph. 26, 3, 55. Google ScholarDigital Library
    28. Palacios, J., and Zhang, E. 2011. Interactive visualization of rotational symmetry fields on surfaces. IEEE Transactions on Visualization and Computer Graphics 17, 7 (July), 947–955. Google ScholarDigital Library
    29. Peng, C.-H., Zhang, E., Kobayashi, Y., and Wonka, P. 2011. Connectivity editing for quadrilateral meshes. ACM Trans. Graph. 30 (Dec.), 141:1–141:12. Google ScholarDigital Library
    30. Podolak, J., Shilane, P., Golovinskiy, A., Rusinkiewicz, S., and Funkhouser, T. 2006. A planar-reflective symmetry transform for 3d shapes. ACM Transactions on Graphics 25, 3, 549–559. Google ScholarDigital Library
    31. Podolak, J., Golovinskiy, A., and Rusinkiewicz, S. 2007. Symmetry-enhanced remeshing of surfaces. In Proceedings of the fifth Eurographics symposium on Geometry processing, Eurographics Association, 235–242. Google ScholarDigital Library
    32. Praun, E., Sweldens, W., and Schröder, P. 2001. Consistent mesh parameterizations. Proc. of SIGGRAPH 2001. Google ScholarDigital Library
    33. Raviv, D., Bronstein, A. M., Bronstein, M. M., and Kimmel, R. 2007. Symmetries of non-rigid shapes. In Proc. Non-rigid Registration and Tracking (NRTL) workshop. See Proc. of International Conference on Computer Vision (ICCV).Google Scholar
    34. Raviv, D., Bronstein, A., Bronstein, M., and Kimmel, R. 2010. Full and partial symmetries of non-rigid shapes. International journal of computer vision 89, 1, 18–39. Google ScholarDigital Library
    35. Ray, N., Li, W., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460–1485. Google ScholarDigital Library
    36. Ray, N., Vallet, B., Li, W., and Lévy, B. 2008. N-Symmetry Direction Field Design. ACM Trans. Graph. 27, 2. Google ScholarDigital Library
    37. Ray, N., Vallet, B., Alonso, L., and Levy, B. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1, 1–11. Google ScholarDigital Library
    38. Schreiner, J., Asirvatham, A., Praun, E., and Hoppe, H. 2004. Inter-surface mapping. ACM Transactions on Graphics (Proc. SIGGRAPH). Google ScholarDigital Library
    39. Schwerdtfeger, H. 1979. Geometry of complex numbers: circle geometry, Moebius transformation, non-euclidean geometry. Dover Books on Mathematics Series. Dover.Google Scholar
    40. Tarini, M., Puppo, E., Panozzo, D., Pietroni, N., and Cignoni, P. 2011. Simple quad domains for field aligned mesh parametrization. ACM Trans. Graph. 30 (Dec.), 142:1–142:12. Google ScholarDigital Library
    41. Xu, K., Zhang, H., Tagliasacchi, A., Liu, L., Li, G., Meng, M., and Xiong, Y. 2009. Partial intrinsic reflectional symmetry of 3d shapes. ACM Transactions on Graphics (TOG) 28, 5, 138. Google ScholarDigital Library

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