JZUS - Journal of Zhejiang University SCIENCE

Frontiers of Information Technology & Electronic Engineering 2016 Vol.17 No.6 P.501-515

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

Author(s): Zhao Wang, Zhong-xuan Luo, Jie-lin Zhang, Emil Saucan
Affiliation(s): School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China; more
Corresponding email(s): wangzhao.ok@163.com, zxluo@dlut.edu.cn, jielinzh@dlut.edu.cn, semil@ee.technion.ac.il
Key Words: Mesh parameterization, Convex combination weights, Stretch operator, Jacobian matrix

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Abstract: mesh parameterization is one of the fundamental operations in computer graphics (CG) and computer-aided design (CAD). In this paper, we propose a novel local/global parameterization approach, ARAP++, for single- and 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.

This paper proposes an extended local/global parameterization (ARAP++) method for single-boundary and multi-boundary triangular meshes. It suggests two main extensions in comparison with ARAP. 1) The current ARAP scheme combines a local mapping from a 3D triangle to the plane with a global ‘stitch’ operation for individual triangles, while the proposed ARAP++ adopts a global operation that stitches the 1-ring patches together; 2) Similar to energy minimization of mesh parameterization, the scheme of this paper is based on optimization of the spring energy, which can cover the ARAP approach (based on the optimization of Dirichlet energy) as a special case. The main interesting point is that each triangle is looked upon 3 different view points: those of the 3 vertices. An interesting property is that each angle can therefore be corrected is order to sum to Pi around each vertex.

ARAP++:一类推广的局部/全局参数化算法

目的：本文针对单边界和多边界的三角网格提出了一种含有凸组合权值的局部/全局参数化方法（ARAP++）。该方法将凸组合权值与三角面片间仿射变换的Jacobian矩阵相结合，得到一种新的线性迭代格式，收敛迅速。它是一种自由边界的参数化方法。可以通过调整Jacobian矩阵的奇异值，使参数化网格保持原始网格的几何性质。同时在迭代格式中引入拉伸算子来处理高曲率的三角网格，减小了其参数化后的面积扭曲与拉伸扭曲。最后通过数值实验将ARAP++方法与其它经典参数化方法进行比较。实验结果表明本文方法所得到的参数化结果较之其它方法在角度、面积和拉伸等方面的扭曲变形有明显改进，从而使得该方法在纹理映射和重网格化等应用中得到较好的视觉效果。
创新点：首先通过优化弹性能量函数推导得到了一类局部/全局线性迭代格式，并阐明该方法与ARAP方法之间的联系。针对保面积的情况，提出了一种快速得到最佳拟合矩阵的方法。最后为提高算法的鲁棒性，对ARAP++方法进行改进，使其可以很好的处理高曲率网格的展平，尽可能防止参数化结果的网格重叠。
方法：ARAP++是一种线性迭代的参数化方法。本文首先将原始网格初始化展平到平面上，然后对初始化网格进行迭代计算。其中每次迭代主要分为两个阶段：（1）局部优化；（2）整体求解。实验证明本文算法收敛迅速，并且可以得到很好的纹理映射和重网格化结果。
结论：本文提出了一种自由边界的网格参数化方法，该方法可以使用不同的凸组合权值得到相应性质的参数化结果（图10），也可以通过改变Jacobian矩阵的奇异值来达到保角、保面积、保刚性的目的（图3）。本文方法针对不同的网格模型也可以通过拉伸算子来协调参数化后角度、面积以及拉伸扭曲之间的关系（图9），从而使得该方法在纹理映射（图13、14）和重网格化（图15）等应用中得到较好的视觉效果。

关键词：网格参数化；凸组合权值；拉伸算子；Jacobian矩阵

Reference

[1]Aigerman, N., Lipman, Y., 2013. Injective and bounded distortion mappings in 3D. ACM Trans. Graph., 32(4), Article 106.

[2]Bouaziz, S., Deuss, M., Schwartzburg, Y., et al., 2012. Shape-up: shaping discrete geometry with projections. Comput. Graph. Forum, 31(5):1657-1667.

[3]Chen, B.M., Gotsman, C., Bunin, G., 2008. Conformal flattening by curvature prescription and metric scaling. Comput. Graph. Forum, 27(2):449-458.

[4]Chen, Z., Liu, L., Zhang, Z., et al., 2007. Surface paramete-rization via aligning optimal local flattening. Proc. Symp. on Solid and Physical Modeling, p.291-296.

[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.

[7]Eck, M., DeRose, T., Duchamp, T., et al., 1995. Multi-resolution analysis of arbitrary meshes. Proc. 22nd Annual Conf. on Computer Graphics and Interactive Techniques, p.173-182.

[8]Floater, M.S., 1997. Parameterization and smooth approximation of surface triangulations. Comput. Aid. Geom. Des., 14(3):231-250.

[9]Floater, M.S., 2003. Mean value coordinates. Comput. Aid. Geom. Des., 20(1):19-27.

[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.

[11]Gortler, S., Gotsman, C., Thurston, D., 2006. Discrete one-forms on meshes and applications to 3D mesh parameterization. Comput. Aid. Geom. Des., 23(2):83-112.

[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.

[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., et al., 2000. Conformal surface parameterization for texture mapping. IEEE Trans. Visual. Comput. Graph., 6(2):181-189.

[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.

[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.

[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., et al., 2012. Fast automatic skinning transformations. ACM Trans. Graph., 31(4), Article 77.

[24]Jin, M., Kim, J., Luo, F., et al., 2008. Discrete surface Ricci flow. IEEE Trans. Visual. Comput. Graph., 14(5):1030-1043.

[25]Kharevych, L., Springborn, B., Schröder, P., 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph., 25(2):412-438.

[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.

[28]Levi, Z., Zorin, D., 2014. Strict minimizers for geometric optimization. ACM Trans. Graph., 33(6), Article 185.

[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.

[30]Lipman, Y., 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph., 31(4), Article 108.

[31]Liu, L., Zhang, L., Xu, Y., et al., 2008. A local/global approach to mesh parameterization. Comput. Graph. Forum, 27(5):1495-1504.

[32]Mullen, P., Tong, Y., Alliez, P., et al., 2008. Spectral conformal parameterization. Comput. Graph. Forum, 27(5):1487-1494.

[33]Pinkall, U., Polthier, K., 1993. Computing discrete minimal surface and their conjugates. Exp. Math., 2(1):15-36.

[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.

[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.

[36]Sheffer, A., de Sturler, E., 2001. Parameterization of faceted surfaces for meshing using angle-based flattening. Eng. Comput., 17(3):326-337.

[37]Sheffer, A., Lévy, B., Mogilnitsky, M., et al., 2005. ABF++: fast and robust angle based flattening. ACM Trans. Graph., 24(2):311-330.

[38]Sheffer, A., Praun, E., Rose, K., 2007. Mesh parameterization methods and their applications. Comput. Graph. Vis., 2(2):105-171.

[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.

[42]Weber, O., Myles, A., Zorin, D., 2012. Computing extremal quasiconformal maps. Comput. Graph. Forum, 31(5):1679-1689.

[43]Yoshizawa, S., Belyaev, A., Seidel, H., 2004. A fast and simple stretch-minimizing mesh parameterization. Proc. Shape Modeling Applications, p.200-208.

[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., et al., 2010. Mesh reconstruction by meshless denoising and parameterization. Comput. Graph., 34(3):198-208.

[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.

[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.

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