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ARAP++: an extension of the local/global approach to mesh parameterization

  • Zhao Wang
  • Zhong-xuan Luo
  • Jie-lin Zhang
  • Emil Saucan
Article

Abstract

Mesh parameterization is one of the fundamental operations in computer graphics (CG) and computeraided design (CAD). In this paper, we propose a novel local/global parameterization approach, ARAP++, for singleand multi-boundary triangular meshes. It is an extension of the as-rigid-as-possible (ARAP) approach, which stitches together 1-ring patches instead of individual triangles. To optimize the spring energy, we introduce a linear iterative scheme which employs convex combination weights and a fitting Jacobian matrix corresponding to a prescribed family of transformations. Our algorithm is simple, efficient, and robust. The geometric properties (angle and area) of the original model can also be preserved by appropriately prescribing the singular values of the fitting matrix. To reduce the area and stretch distortions for high-curvature models, a stretch operator is introduced. Numerical results demonstrate that ARAP++ outperforms several state-of-the-art methods in terms of controlling the distortions of angle, area, and stretch. Furthermore, it achieves a better visualization performance for several applications, such as texture mapping and surface remeshing.

Keywords

Mesh parameterization Convex combination weights Stretch operator Jacobian matrix 

CLC number

TP391 

Notes

Acknowledgements

The authors would like to thank Dr. Zhao-liang MENG, Dr. Xin FAN, and Dr. Wan-feng QI for their constructive recommendations for this work.

References

  1. Aigerman, N., Lipman, Y., 2013. Injective and bounded distortion mappings in 3D. ACM Trans. Graph., 32(4), Article 106. http://dx.doi.org/10.1145/2461912.2461931CrossRefGoogle Scholar
  2. Bouaziz, S., Deuss, M., Schwartzburg, Y., et al., 2012. Shapeup: shaping discrete geometry with projections. Comput. Graph. Forum, 31(5): 1657–1667. http://dx.doi.org/10.1111/j.1467-8659.2012.03171.xCrossRefGoogle Scholar
  3. Chen, B.M., Gotsman, C., Bunin, G., 2008. Conformal flattening by curvature prescription and metric scaling. Comput. Graph. Forum, 27(2): 449–458. http://dx.doi.org/10.1111/j.1467-8659.2008.01142.xCrossRefGoogle Scholar
  4. Chen, Z., Liu, L., Zhang, Z., et al., 2007. Surface parameterization via aligning optimal local flattening. Proc. Symp. on Solid and Physical Modeling, p.291–296. http://dx.doi.org/10.1145/1236246.1236287Google Scholar
  5. Degener, P., Meseth, J., Klein, R., 2003. An adaptable surface parameterization method. Proc. 12th Int. Meshing Roundtable, p.227–237.Google Scholar
  6. Desbrun, M., Meyer, M., Allize, P., 2002. Intrinsic parameterization of surface meshes. Comput. Graph. Forum, 21(2): 209–218. http://dx.doi.org/10.1111/1467-8659.00580CrossRefGoogle Scholar
  7. Eck, M., DeRose, T., Duchamp, T., et al., 1995. Multiresolution analysis of arbitrary meshes. Proc. 22nd Annual Conf. on Computer Graphics and Interactive Techniques, p.173–182. http://dx.doi.org/10.1145/218380.218440Google Scholar
  8. Floater, M.S., 1997. Parameterization and smooth approximation of surface triangulations. Comput. Aid. Geom. Des., 14(3): 231–250. http://dx.doi.org/10.1016/S0167-8396(96)00031-3CrossRefGoogle Scholar
  9. Floater, M.S., 2003. Mean value coordinates. Comput. Aid. Geom. Des., 20(1): 19–27. http://dx.doi.org/10.1016/S0167-8396(03)00002-5MathSciNetCrossRefGoogle Scholar
  10. Floater, M.S., Hormann, K., 2005. Surface parameterization: a tutorial and survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (Eds.), Advances in Multiresolution for Geometric Modelling, 157–186. http://dx.doi.org/10.1007/3-540-26808-1_9CrossRefGoogle Scholar
  11. Gortler, S., Gotsman, C., Thurston, D., 2006. Discrete oneforms on meshes and applications to 3D mesh parameterization. Comput. Aid. Geom. Des., 23(2): 83–112. http://dx.doi.org/10.1016/j.cagd.2005.05.002CrossRefGoogle Scholar
  12. Gower, J.C., Dijksterhuis, G.B., 2004. Procrustes Problems. Oxford University Press, Oxford.CrossRefMATHGoogle Scholar
  13. Gu, X., Yau, S., 2002. Computing conformal structures of surfaces. Commun. Inform. Syst., 2(2): 121–146. http://dx.doi.org/10.4310/CIS.2002.v2.n2.a2MathSciNetCrossRefGoogle Scholar
  14. Gu, X., Yau, S., 2003. Global conformal surface parameterization. Proc. Eurographics/ACM SIGGRAPH Symp. on Geometry Processing, p.127–137.Google Scholar
  15. Haker, S., Angenent, S., Tannenbaum, A., et al., 2000. Conformal surface parameterization for texture mapping. IEEE Trans. Visual. Comput. Graph., 6(2): 181–189. http://dx.doi.org/10.1109/2945.856998CrossRefGoogle Scholar
  16. Hoppe, H., DeRose, T., Duchamp, T., et al., 1993. Mesh optimization. Proc. 20th Annual Conf. on Computer Graphics and Interactive Techniques, p.19–26. http://dx.doi.org/10.1145/166117.166119Google Scholar
  17. Hormann, K., Greiner, G., 2000a. MIPS: an efficient global parameterization method. Proc. Curve and Surface, p.153–162.Google Scholar
  18. Hormann, K., Greiner, G., 2000b. Quadrilateral remeshing. Proc. Vision Modeling and Visualization, p.153–162.Google Scholar
  19. Hormann, K., Greiner, G., Campagna, S., 1999. Hierarchical parameterization of triangulated surfaces. Proc. of Vision, Modeling and Visualization, p.219–226.Google Scholar
  20. Hormann, K., Labsik, U., Greiner, G., 2001. Remeshing triangulated surfaces with optimal parameterizations. Comput.-Aid. Des., 33(11): 779–788. http://dx.doi.org/10.1016/S0010-4485(01)00094-XCrossRefGoogle Scholar
  21. Hormann, K., Lévy, B., Sheffer, A., 2007. Mesh parameterization: theory and practice. Proc. SIGGRAPH, p.1–122.Google Scholar
  22. Horn, R., Johnson, C., 1990. Norms for vectors and matrices. In: Matrix Analysis. Cambridge University Press, England.Google Scholar
  23. Jacobson, A., Baran, I., Kavan, L., et al., 2012. Fast automatic skinning transformations. ACM Trans. Graph., 31(4), Article 77. http://dx.doi.org/10.1145/2185520.2185573CrossRefGoogle Scholar
  24. Jin, M., Kim, J., Luo, F., et al., 2008. Discrete surface Ricci flow. IEEE Trans. Visual. Comput. Graph., 14(5): 1030–1043. http://dx.doi.org/10.1109/TVCG.2008.57CrossRefGoogle Scholar
  25. Kharevych, L., Springborn, B., Schröder, P., 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph., 25(2): 412–438. http://dx.doi.org/10.1145/1138450.1138461CrossRefGoogle Scholar
  26. Lawson, L., 1977. Software for c1 surface interpolation. In: Mathematical Software III. Academic Press, New York.Google Scholar
  27. Lee, Y., Kim, H., Lee, S., 2002. Mesh parameterization with a virtual boundary. Comput. Graph., 26(5): 677–686. http://dx.doi.org/10.1016/S0097-8493(02)00123-1CrossRefGoogle Scholar
  28. Levi, Z., Zorin, D., 2014. Strict minimizers for geometric optimization. ACM Trans. Graph., 33(6), Article 185. http://dx.doi.org/10.1145/2661229.2661258CrossRefGoogle Scholar
  29. Lévy, B., Petitjean, S., Ray, N., et al., 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph., 21(3): 362–371. http://dx.doi.org/10.1145/566570.566590CrossRefGoogle Scholar
  30. Lipman, Y., 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph., 31(4), Article 108. http://dx.doi.org/10.1145/2185520.2185604CrossRefGoogle Scholar
  31. Liu, L., Zhang, L., Xu, Y., et al., 2008. A local/global approach to mesh parameterization. Comput. Graph. Forum, 27(5): 1495–1504. http://dx.doi.org/10.1111/j.1467-8659.2008.01290.xCrossRefGoogle Scholar
  32. Mullen, P., Tong, Y., Alliez, P., et al., 2008. Spectral conformal parameterization. Comput. Graph. Forum, 27(5): 1487–1494. http://dx.doi.org/10.1111/j.1467-8659.2008.01289.xCrossRefGoogle Scholar
  33. Pinkall, U., Polthier, K., 1993. Computing discrete minimal surface and their conjugates. Exp. Math., 2(1): 15–36. http://dx.doi.org/10.1080/10586458.1993.10504266MathSciNetCrossRefGoogle Scholar
  34. Sander, P., Snyder, J., Gortler, S., et al., 2001. Texture mapping progressive meshes. Proc. 28th Annual Conf. on Computer Graphics and Interactive Techniques, p.409–416. http://dx.doi.org/10.1145/383259.383307Google Scholar
  35. Saucan, E., Appleboim, E., Barak-Shimron, E., et al., 2008. Local versus global in quasiconformal mapping for medical imaging. J. Math. Imag. Vis., 32(3): 293–311. http://dx.doi.org/10.1007/s10851-008-0101-6CrossRefGoogle Scholar
  36. Sheffer, A., de Sturler, E., 2001. Parameterization of faceted surfaces for meshing using angle-based flattening. Eng. Comput., 17(3): 326–337. http://dx.doi.org/10.1007/PL00013391CrossRefGoogle Scholar
  37. Sheffer, A., Lévy, B., Mogilnitsky, M., et al., 2005. ABF++: fast and robust angle based flattening. ACM Trans. Graph., 24(2): 311–330. http://dx.doi.org/10.1145/1061347.1061354CrossRefGoogle Scholar
  38. Sheffer, A., Praun, E., Rose, K., 2007. Mesh parameterization methods and their applications. Comput. Graph. Vis., 2(2): 105–171. http://dx.doi.org/10.1561/0600000011MATHGoogle Scholar
  39. Sorkine, O., Alexa, M., 2007. As-rigid-as-possible surface modeling. Proc. Eurographics Symp. on Geometry Processing, p.109–116.Google Scholar
  40. Tutte, W.T., 1963. How to draw a graph. Proc. London Math. Soc., 13(3): 743–768.MathSciNetCrossRefGoogle Scholar
  41. Weber, O., Zorin, D., 2014. Locally injective parametrization with arbitrary fixed boundaries. ACM Trans. Graph., 33(4), Article 75. http://dx.doi.org/10.1145/2601097.2601227CrossRefGoogle Scholar
  42. Weber, O., Myles, A., Zorin, D., 2012. Computing extremal quasiconformal maps. Comput. Graph. Forum, 31(5): 1679–1689. http://dx.doi.org/10.1111/j.1467-8659.2012.03173.xCrossRefGoogle Scholar
  43. Yoshizawa, S., Belyaev, A., Seidel, H., 2004. A fast and simple stretch-minimizing mesh parameterization. Proc. Shape Modeling Applications, p.200–208. http://dx.doi.org/10.1109/SMI.2004.1314507Google Scholar
  44. Zayer, R., Lévy, B., Seidel, H., 2007. Linear angle based parameterization. Proc. 5th Eurographics Symp. on Geometry Processing, p.135–141.Google Scholar
  45. Zhang, L., Liu, L., Gotsman, C., et al., 2010. Mesh reconstruction by meshless denoising and parameterization. Comput. Graph., 34(3): 198–208. http://dx.doi.org/10.1016/j.cag.2010.03.006CrossRefGoogle Scholar
  46. Zhao, X., Su, Z., Gu, X., et al., 2013. Area-preservation mapping using optimal mass transport. IEEE Trans. Visual. Comput. Graph., 19(12): 2838–2847. http://dx.doi.org/10.1109/TVCG.2013.135CrossRefGoogle Scholar
  47. Zigelman, G., Kimmel, R., Kiryati, N., 2002. Texture mapping using surface flattening via multidimensional scaling. IEEE Trans. Visual. Comput. Graph., 8(2): 198–207. http://dx.doi.org/10.1109/2945.998671CrossRefGoogle Scholar

Copyright information

© Journal of Zhejiang University Science Editorial Office and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Zhao Wang
    • 1
  • Zhong-xuan Luo
    • 1
    • 2
  • Jie-lin Zhang
    • 1
  • Emil Saucan
    • 3
  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina
  2. 2.School of SoftwareDalian University of TechnologyDalianChina
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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