ARAP++: an extension of the local/global approach to mesh parameterization
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 multiboundary triangular meshes. It is an extension of the asrigidaspossible (ARAP) approach, which stitches together 1ring 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 highcurvature models, a stretch operator is introduced. Numerical results demonstrate that ARAP++ outperforms several stateoftheart 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 matrixCLC number
TP3911 Introduction
Mesh parameterization is an important research topic in computer graphics (CG), and it has been widely used in digital geometry processing tasks, such as texture mapping (Haker et al., 2000), surface fitting (Hormann and Greiner, 2000b), and surface remeshing (Hormann et al., 2001). When a discrete surface is directly flattened onto the plane, distortions are inevitable due to the parameterization process. Preserving the geometric properties of the original mesh is essential for a good parameterization.
In this paper, we propose a novel local/global mesh parameterization approach, ARAP++. Our work is inspired mainly by the asrigidaspossible (ARAP) approach (Sorkine and Alexa, 2007; Liu et al., 2008; Bouaziz et al., 2012) and the convex combination approach (Eck et al., 1995; Floater, 1997). ARAP++ adopts the idea of ARAP regarding the approximation of the Jacobian matrix with a fitting matrix, and then achieves the global flattened result by stitching together the local 1ring patches. To optimize the local spring energy, we introduce the convex combination weights and stretch operator (Sander et al., 2001; Yoshizawa et al., 2004) to ARAP++. In consequence, ARAP++ renders lower area and stretch distortions than ARAP. The flattened results of our method are obtained using a free boundary. Therefore, ARAP++ outperforms those of the convex combination approach in processing boundaries of models. These facts show that ARAP++ enhances robustness and adaptiveness relative to both methods above, and achieves a better result in texture mapping (Fig. 1).
 1.
We devise a novel local/global approach to mesh parameterization (ARAP++) based on the optimization of spring energy, and analyze the relation to the ARAP approach which is based on the optimization of the Dirichlet energy. In addition, a broader class of convex combination weights are considered in our method (Sections 4.2 and 4.3).
 2.
Compared with ARAP, ARAP++ improves the local phase and obtains the flattened results by stitching together the 1ring patches, instead of individual triangles (Section 4.2). Moreover, we give a simple and fast calculation method to obtain an authalic fitting matrix (in the Appendix).
 3.
To deal with highcurvature models, we introduce a stretch operator to ARAP++. It enhances the robustness of the method, and eliminates the influence of overlapping and flipping (Section 4.6).
2 Related work
In the past decade, extensive research was conducted regarding the mesh parameterization problem. We refer readers to several survey papers for fundamental theories and methods (Floater and Hormann, 2005; Hormann et al., 2007; Sheffer et al., 2007). Below we briefly review the major techniques which have close relationship to our work.
The convex combination approach (Tutte, 1963; Eck et al., 1995; Floater, 1997; 2003; Desbrun et al., 2002; Lee et al., 2002; Yoshizawa et al., 2004) is a type of linear and fixed boundary parameterization. It is fast and stable without producing overlapping. However, if the boundary is not fixed optimally in advance, then it will lead to higher distortions along the boundary as well as the regions which are away from it.
Some methods rely on angle optimization, both for their intuitiveness and for their effectiveness. The anglebased flattening methods (Sheffer and de Sturler, 2001; Sheffer et al., 2005; Kharevych et al., 2006; Zayer et al., 2007) are defined in the angle space, producing minimal angular distortion. Jin et al. (2008) built a unified framework for discrete surface Ricci flow algorithms. Chen et al. (2008) modified the Gaussian curvature by means of the transition probability matrix, and computed metric scaling as a solution of the Poisson equation. Weber and Zorin (2014) proposed an algorithm with arbitrarily fixed boundary, which guarantees that the result is locally injective.
Moreover, there are a variety of parameterization methods. Hormann et al. (1999) provided a hierarchical representation of discrete surface, which includes two major steps: edge collapse and vertex split. Zigelman et al. (2002) presented multidimensional scaling (MDS) parameterization, which can preserve the geodesic distance of a triangular mesh. Gu and Yau (2002; 2003) approached the global surface parametrization problem by computing conformal structures of general twomanifolds. Gortler et al. (2006) described some simple properties of discrete oneforms, and their applications to threedimensional (3D) mesh embedding. Chen et al. (2007) flattened surfaces with the theory of local tangent space alignment (LTSA). Mullen et al. (2008) defined discrete spectral conformal maps, and obtained a conformal mapping result by minimization of Dirichlet energy. Zhao et al. (2013) presented an authalic flattening method based on the optimal mass transport technique.
3 Preliminaries
In this section, we give a brief overview for the convex combination approach. The basic idea is to map the boundary nodes of a 3D mesh to a convex polygon in the plane. The internal node can be expressed as a weighted average of their 1ring nodes, thus guaranteeing a bijective mapping.
3.1 Spring energy and Dirichlet energy
In addition, there are five classic types of weights ω_{ i,j } that can be applied to Eq. (2), namely uniform (Tutte, 1963), shapepreserving(Floater, 1997), meanvalue (Floater, 2003), cotan (Eck et al., 1995), and intrinsic (Desbrun et al., 2002) weights.
3.2 Error metrics on discrete surface
4 Mesh parameterization
In this section, we describe our mesh parameterization technique (ARAP++). The input is a mesh of disk topology, and the output is a freeboundary flattened result. The algorithm consists of two main steps, local phase and global phase. Our method further extends and improves the ARAP approach (Liu et al., 2008).
4.1 Overview of the ARAP++ approach
We outline the method in Algorithm
, and the details are given in the following subsections.4.2 Local phase
The local flattening of N(p_{ i }) well preserves the shape (angle and area) of N(x_{ i }). In the following process, we wish N(q_{ i }) preserves the shape of N(p_{ i }) as much as possible, so that it can preserve the shape of N(x_{ i }) indirectly.
From the properties of the Jacobian matrix (Floater and Hormann, 2005), we infer the following three conclusions (Fig. 3):
4.3 Global phase
Comparison of distortion measures of five classic types of weights on two standard test models
Method  Balls^{a}  Nefertiti^{b}  

Angle distortion  Area distortion  Angle distortion  Area distortion  
Uniform  0.195  0.233  0.157  0.213 
Shapepreserving  0.180  0.201  0.125  0.173 
Meanvalue  0.117  0.187  0.116  0.167 
Cotan  0.124  0.184  0.118  0.158 
Intrinsic  0.177  0.211  0.129  0.195 
Comparison of distortion measures and time of ARAP and ARAP++
Method  Iteration index  Gargoyle^{a}  Triceratops^{b}  

Angle distortion  Area distortion  Time (s)  Angle distortion  Area distortion  Time (s)  
ARAP  1  0.147  0.244  9.05  0.177  0.164  9.44 
2  0.140  0.234  13.39  0.147  0.148  14.01  
3  0.137  0.227  18.47  0.134  0.141  18.87  
ARAP++  1  0.168  0.217  9.59  0.156  0.158  10.74 
2  0.150  0.203  14.06  0.129  0.147  14.93  
3  0.141  0.196  18.73  0.116  0.138  19.43 
4.4 Boundary weights
 1.
Uniform weight
The weight of x_{ j } is ω_{ i,j } = 1/d_{ i }.
 2.
Shapepreserving weight
Using an isometric mapping to locate x_{ i } and its 1ring nodes x_{ j }, \({x_{{n_j}}}\), \({x_{{n_k}}}\), x_{ k } on the plane, we obtain p_{ i } and the 1ring nodes p_{ j }, \({p_{{n_j}}}\), \({p_{{n_k}}}\), p_{ k } (Fig. 7). To generate a local 1ring neighborhood, two virtual boundary nodes \({p_{{n_j}}}\), \({p_{{n_k}}}\) are inserted between p_{ j } and p_{ k }. The inserted nodes should satisfy the following equations:Then the weights ω_{ i,j } of p_{ j }, \({p_{{n_j}}}\), \({p_{{n_k}}}\), and p_{ k } can be computed in the local 1ring neighborhood.$$\left\{ {\matrix{ {\text{ang}({p_j},{p_i},{p_{{m_j}}}) = \left( {2\pi  \text{ang}({p_{j,}}{p_{i,}}{p_k})} \right)/3} \cr {\text{ang}({p_{mj}},{p_i},{p_{{m_k}}}) = \left( {2\pi  \text{ang}({p_j},{p_i},{p_k})} \right)/3} \cr {\text{ang}({p_{{m_k}}},{p_i},{p_k}) = \left( {2\pi  \text{ang}({p_j},{p_i},{p_k})} \right)/3} \cr {\left\ {{p_i}  {p_{mj}}} \right\ = \left\ {{p_i}  {p_{mk}}} \right\ = {{\sum {j \in {N_{(i)}}\left\ {{p_i}  {p_j}} \right\} } \over {{d_i}}}.} \cr } } \right.$$  3.
Meanvalue, cotan, and intrinsic weights Fig. 7 shows that the 1ring nodes xj, \({x_{{n_j}}}\), \({x_{{n_k}}}\), x_{ k } can be divided into two parts: \({x_{{n_j}}}\), \({x_{{n_k}}}\) (intermediate nodes), and x_{ j }, x_{ k } (endpoints). The weights of \({x_{{n_j}}}\), \({x_{{n_k}}}\) are consistent with those of the internal nodes. However, the weights of x_{ j }, x_{ k } are computed in \(\Delta {x_i}{x_j}{x_{{n_j}}}\) and \(\Delta {x_i}{x_k}{x_{{n_k}}}\), respectively.
4.5 Multiboundary flattening

Step 1: Add a virtual node to the center of the holes, connecting the virtual nodes with the 1ring nodes, and thus the multiboundary mesh turns into a singleboundary mesh.

Step 2: Compute the flattened result according to Algorithm
. 
Step 3: Remove the virtual nodes and connections from the holes.
4.6 Stretch operator
To reduce the area and stretch distortions for highcurvature models, the stretch operator (Sander et al., 2001; Yoshizawa et al., 2004) is employed to improve our scheme (Eq. (6)). In addition, it can make ARAP++ reduce flipping and overlapping during parameterization.
To achieve the valid results, Eq. (7) requires only one step of iteration in computation. If the number of iterations increases, there will be overlapping in the final results. To control stretch distortion, the exponent θ should be adjusted according to different models. Fig. 9 shows the results of parameterization with different θ,where ω_{ i,j } represents the meanvalue weight, and the singular values of L are δ_{1} = δ_{2} = (σ_{1} + σ_{2})/2.
5 Simulations and comparisons
All the experiments were tested under MATLAB on a Pentium® DualCore, 2.5 GHz CPU computer with 4 GB RAM. To confirm the effectiveness of our method, we carried out simulations on several typical models.
5.1 Comparison of five classic weights
5.2 Comparison of ARAP and ARAP++
5.3 Comparison with several stateoftheart methods
In this subsection, we carry out three simulations to compare ARAP++ with several stateoftheart parameterization methods, namely LSCM, LABF, and BDARAP (conformal distortion C = 2).
Comparison of three parameterization methods and ARAP++ on four standard test meshes
Method  Hand^{a}  Kitten^{b}  Camel^{c}  Elephant^{d}  

Angle distortion  Area distortion  Time (s)  Angle distortio  Area n distortion  Time (s)  Angle distortion  Area distortion  Time (s)  Angle distortion  Area distortion  Time (s)  
LSCM  0.009  0.991  8.74  0.044  0.416  3.79  0.099  0.731  3.59  0.063  0.699  12.67 
LABF  0.015  0.659  35.05  0.026  0.359  16.70  0.053  0.703  14.79  0.047  0.715  45.90 
BDARAP  0.019  0.269  80.78  0.039  0.273  68.78  0.064  0.254  141.12  0.065  0.380  239.23 
ARAP++  0.065  0.107  15.80  0.102  0.161  10.14  0.137  0.227  10.05  0.166  0.276  25.25 
Comparison of three parameterization methods and ARAP++ for highcurvature models
Method  Mushroom^{a}  Mannequin^{b}  Lion^{c}  

Angle distortion  Area distortion  Stretch distortion  Angle distortion  Area distortion  Stretch distortion  Angle distortion  Area distortion  Stretch distortion  
LSCM  0.026  1.034  2.604  0.027  1.436  7.436  0.071  1.241  18.806 
LABF  0.027  1.033  2.601  0.028  1.435  7.438  0.073  1.246  11.131 
BDARAP  0.028  1.031  2.565  0.124  1.417  5.719  0.166  1.059  3.282 
ARAP++  0.154  0.807  1.535  0.322  0.887  1.654  0.285  0.866  2.883 
Minimum and maximum facet areas of the remeshed models for three standard test meshes
Method  Mushroom^{a}  Mannequin^{b}  Lion^{c}  

Minimum  Maximum  Minimum  Maximum  Minimum  Maximum  
LSCM  2.746×10^{−4}  2.661×10^{−3}  1.345×10^{−4}  2.343×10^{−3}  3.928×10^{−5}  1.807×10^{−3} 
LABF  2.028×10^{−4}  2.421×10^{−3}  1.345×10^{−4}  2.347×10^{−3}  3.576×10^{−5}  1.866×10^{−3} 
BDARAP  3.131×10^{−4}  2.654×10^{−3}  1.536×10^{−4}  2.048×10^{−3}  4.159×10^{−5}  0.892×10^{−3} 
ARAP++  4.229×10^{−4}  1.911×10^{−3}  2.074×10^{−4}  1.229×10^{−3}  7.068×10^{−5}  0.635×10^{−3} 
6 Conclusions and outlook
Our future work will focus on extending the present method to mesh deformation and spherical parameterization, which will undoubtedly shed some new light on the improvement of computer graphics technology. Another natural direction would be to extend our results to larger patches (not just 1rings), such as those produced by the flattening algorithm introduced in Saucan et al. (2008).
Notes
Acknowledgements
The authors would like to thank Dr. Zhaoliang MENG, Dr. Xin FAN, and Dr. Wanfeng QI for their constructive recommendations for this work.
References
 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
 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.14678659.2012.03171.xCrossRefGoogle Scholar
 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.14678659.2008.01142.xCrossRefGoogle Scholar
 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
 Degener, P., Meseth, J., Klein, R., 2003. An adaptable surface parameterization method. Proc. 12th Int. Meshing Roundtable, p.227–237.Google Scholar
 Desbrun, M., Meyer, M., Allize, P., 2002. Intrinsic parameterization of surface meshes. Comput. Graph. Forum, 21(2): 209–218. http://dx.doi.org/10.1111/14678659.00580CrossRefGoogle Scholar
 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
 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/S01678396(96)000313CrossRefGoogle Scholar
 Floater, M.S., 2003. Mean value coordinates. Comput. Aid. Geom. Des., 20(1): 19–27. http://dx.doi.org/10.1016/S01678396(03)000025MathSciNetCrossRefGoogle Scholar
 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/10.1007/3540268081_9CrossRefGoogle Scholar
 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
 Gower, J.C., Dijksterhuis, G.B., 2004. Procrustes Problems. Oxford University Press, Oxford.CrossRefzbMATHGoogle Scholar
 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
 Gu, X., Yau, S., 2003. Global conformal surface parameterization. Proc. Eurographics/ACM SIGGRAPH Symp. on Geometry Processing, p.127–137.Google Scholar
 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
 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
 Hormann, K., Greiner, G., 2000a. MIPS: an efficient global parameterization method. Proc. Curve and Surface, p.153–162.Google Scholar
 Hormann, K., Greiner, G., 2000b. Quadrilateral remeshing. Proc. Vision Modeling and Visualization, p.153–162.Google Scholar
 Hormann, K., Greiner, G., Campagna, S., 1999. Hierarchical parameterization of triangulated surfaces. Proc. of Vision, Modeling and Visualization, p.219–226.Google Scholar
 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/S00104485(01)00094XCrossRefGoogle Scholar
 Hormann, K., Lévy, B., Sheffer, A., 2007. Mesh parameterization: theory and practice. Proc. SIGGRAPH, p.1–122.Google Scholar
 Horn, R., Johnson, C., 1990. Norms for vectors and matrices. In: Matrix Analysis. Cambridge University Press, England.Google Scholar
 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
 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
 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
 Lawson, L., 1977. Software for c1 surface interpolation. In: Mathematical Software III. Academic Press, New York.Google Scholar
 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/S00978493(02)001231CrossRefGoogle Scholar
 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
 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
 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
 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.14678659.2008.01290.xCrossRefGoogle Scholar
 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.14678659.2008.01289.xCrossRefGoogle Scholar
 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
 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
 Saucan, E., Appleboim, E., BarakShimron, 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/s1085100801016CrossRefGoogle Scholar
 Sheffer, A., de Sturler, E., 2001. Parameterization of faceted surfaces for meshing using anglebased flattening. Eng. Comput., 17(3): 326–337. http://dx.doi.org/10.1007/PL00013391CrossRefGoogle Scholar
 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
 Sheffer, A., Praun, E., Rose, K., 2007. Mesh parameterization methods and their applications. Comput. Graph. Vis., 2(2): 105–171. http://dx.doi.org/10.1561/0600000011zbMATHGoogle Scholar
 Sorkine, O., Alexa, M., 2007. Asrigidaspossible surface modeling. Proc. Eurographics Symp. on Geometry Processing, p.109–116.Google Scholar
 Tutte, W.T., 1963. How to draw a graph. Proc. London Math. Soc., 13(3): 743–768.MathSciNetCrossRefGoogle Scholar
 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
 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.14678659.2012.03173.xCrossRefGoogle Scholar
 Yoshizawa, S., Belyaev, A., Seidel, H., 2004. A fast and simple stretchminimizing mesh parameterization. Proc. Shape Modeling Applications, p.200–208. http://dx.doi.org/10.1109/SMI.2004.1314507Google Scholar
 Zayer, R., Lévy, B., Seidel, H., 2007. Linear angle based parameterization. Proc. 5th Eurographics Symp. on Geometry Processing, p.135–141.Google Scholar
 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
 Zhao, X., Su, Z., Gu, X., et al., 2013. Areapreservation mapping using optimal mass transport. IEEE Trans. Visual. Comput. Graph., 19(12): 2838–2847. http://dx.doi.org/10.1109/TVCG.2013.135CrossRefGoogle Scholar
 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