ARAP++: an extension of the local/global approach tomesh parameterization<FootNote> 1 Project supported by the National Natural Science Foundation of China (Nos. 61432003, 61572105, 11171052, and 61328206) </FootNote>

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

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/10.1145/2461912.2461931

[2] Bouaziz, S., Deuss, M., Schwartzburg, Y., , 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.x

[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.x

[4] Chen, Z., Liu, L., Zhang, Z., , 2007. Surface parameterization via aligning optimal local flattening. Proc. Symp. on Solid and Physical Modeling, p.291–296. http://dx.doi.org/10.1145/1236246.1236287

[5] Degener, P., Meseth, J., Klein, R., 2003. An adaptable surface parameterization method. Proc. 12th Int. Meshing Roundtable, p.227–237.

[6] Desbrun, M., Meyer, M., Allize, P., 2002. Intrinsic parameterization of surface meshes. Comput. Graph. Forum, 21(2):209–218. http://dx.doi.org/10.1111/1467-8659.00580

[7] Eck, M., DeRose, T., Duchamp, T., , 1995. Multiresolution analysis of arbitrary meshes. Proc. 22nd Annual Conf. on Computer Graphics and Interactive Techniques, p.173–182. http://dx.doi.org/10.1145/218380.218440

[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)00031-3

[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)00002-5

[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, p.157–186. http://dx.doi.org/10.1007/3-540-26808-1_9

[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.2005.05.002

[12] Gower, J.C., Dijksterhuis, G.B., 2004. Procrustes Problems. Oxford University Press, Oxford.

[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.a2

[14] Gu, X., Yau, S., 2003. Global conformal surface parameterization. Proc. Eurographics/ACM SIGGRAPH Symp. on Geometry Processing, p.127–137.

[15] Haker, S., Angenent, S., Tannenbaum, A., , 2000. Conformal surface parameterization for texture mapping. IEEE Trans. Visual. Comput. Graph., 6(2):181–189. http://dx.doi.org/10.1109/2945.856998

[16] Hoppe, H., DeRose, T., Duchamp, T., , 1993. Mesh optimization. Proc. 20th Annual Conf. on Computer Graphics and Interactive Techniques, p.19–26. http://dx.doi.org/10.1145/166117.166119

[17] Hormann, K., Greiner, G., 2000a. MIPS: an efficient global parameterization method. Proc. Curve and Surface, p.153–162.

[18] Hormann, K., Greiner, G., 2000b. Quadrilateral remeshing. Proc. Vision Modeling and Visualization, p.153–162.

[19] Hormann, K., Greiner, G., Campagna, S., 1999. Hierarchical parameterization of triangulated surfaces. Proc. of Vision, Modeling and Visualization, p.219–226.

[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-X

[21] Hormann, K., Lévy, B., Sheffer, A., 2007. Mesh parameterization: theory and practice. Proc. SIGGRAPH, p.1–122.

[22] Horn, R., Johnson, C., 1990. Norms for vectors and matrices. In: Matrix Analysis. Cambridge University Press, England.

[23] Jacobson, A., Baran, I., Kavan, L., , 2012. Fast automatic skinning transformations. ACM Trans. Graph., 31(4), Article 77. http://dx.doi.org/10.1145/2185520.2185573

[24] Jin, M., Kim, J., Luo, F., , 2008. Discrete surface Ricci flow. IEEE Trans. Visual. Comput. Graph., 14(5):1030–1043. http://dx.doi.org/10.1109/TVCG.2008.57

[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/10.1145/1138450.1138461

[26] Lawson, L., 1977. Software for c1 surface interpolation. In: Mathematical Software III. Academic Press, New York.

[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)00123-1

[28] Levi, Z., Zorin, D., 2014. Strict minimizers for geometric optimization. ACM Trans. Graph., 33(6), Article 185. http://dx.doi.org/10.1145/2661229.2661258

[29] Lévy, B., Petitjean, S., Ray, N., , 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph., 21(3):362–371. http://dx.doi.org/10.1145/566570.566590

[30] Lipman, Y., 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph., 31(4), Article 108. http://dx.doi.org/10.1145/2185520.2185604

[31] Liu, L., Zhang, L., Xu, Y., , 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.x

[32] Mullen, P., Tong, Y., Alliez, P., , 2008. Spectral conformal parameterization. Comput. Graph. Forum, 27(5):1487–1494. http://dx.doi.org/10.1111/j.1467-8659.2008.01289.x

[33] Pinkall, U., Polthier, K., 1993. Computing discrete minimal surface and their conjugates. Exp. Math., 2(1):15–36. http://dx.doi.org/10.1080/10586458.1993.10504266

[34] Sander, P., Snyder, J., Gortler, S., , 2001. Texture mapping progressive meshes. Proc. 28th Annual Conf. on Computer Graphics and Interactive Techniques, p.409–416. http://dx.doi.org/10.1145/383259.383307

[35] Saucan, E., Appleboim, E., Barak-Shimron, E., , 2008. Local versus global in quasiconformal mapping for medical imaging. J. Math. Imag. Vis., 32(3):293–311. http://dx.doi.org/10.1007/s10851-008-0101-6

[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/PL00013391

[37] Sheffer, A., Lévy, B., Mogilnitsky, M., , 2005. ABF++: fast and robust angle based flattening. ACM Trans. Graph., 24(2):311–330. http://dx.doi.org/10.1145/1061347.1061354

[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/10.1561/0600000011

[39] Sorkine, O., Alexa, M., 2007. As-rigid-as-possible surface modeling. Proc. Eurographics Symp. on Geometry Processing, p.109–116.

[40] Tutte, W.T., 1963. How to draw a graph. Proc. London Math. Soc., 13(3):743–768..

[41] Weber, O., Zorin, D., 2014. Locally injective parametrization with arbitrary fixed boundaries. ACM Trans. Graph., 33(4), Article 75. http://dx.doi.org/10.1145/2601097.2601227

[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.x

[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.2004.1314507

[44] Zayer, R., Lévy, B., Seidel, H., 2007. Linear angle based parameterization. Proc. 5th Eurographics Symp. on Geometry Processing, p.135–141.

[45] Zhang, L., Liu, L., Gotsman, C., , 2010. Mesh reconstruction by meshless denoising and parameterization. Comput. Graph., 34(3):198–208. http://dx.doi.org/10.1016/j.cag.2010.03.006

[46] Zhao, X., Su, Z., Gu, X., , 2013. Area-preservation mapping using optimal mass transport. IEEE Trans. Visual. Comput. Graph., 19(12):2838–2847. http://dx.doi.org/10.1109/TVCG.2013.135

[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/10.1109/2945.998671

RIGHTS & PERMISSIONS

2016 Zhejiang University and Springer-Verlag Berlin Heidelberg