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

Zhao WANG; Zhong-xuan LUO; Jie-lin ZHANG; Emil SAUCAN

doi:10.1631/FITEE.1500184

Your Location：

Home >

Browse articles >

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

Regular Papers | Updated：2022-05-19

- ARAP++: an extension of the local/global approach to mesh parameterization
- ARAP++:一类推广的局部/全局参数化算法
- Frontiers of Information Technology & Electronic Engineering Vol. 17, Issue 6, Pages: 501-515(2016)
- Affiliations：
  
  1School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
  2School of Software, Dalian University of Technology, Dalian 116620, China
  3Max Planck Institute for Mathematics in the Sciences, Leipzig 70569, Germany
- Author bio：
  
  E-mail:wangzhao.ok@163.com
  E-mail:zxluo@dlut.edu.cn
  E-mail:jielinzh@dlut.edu.cn
  E-mail:semil@ee.technion.ac.il
- Funds：
- DOI：10.1631/FITEE.1500184
  CLC： TP391
- Received：11 June 2015，
  
  Accepted：16 February 2015，
  
  Published：2016-06
- Accepted：
Scan QR Code
Zhao WANG, Zhong-xuan LUO, Jie-lin ZHANG, et al. ARAP++: an extension of the local/global approach to mesh parameterization[J]. Frontiers of Information Technology & Electronic Engineering, 2016, 17(6): 501-515.
DOI：

Zhao WANG, Zhong-xuan LUO, Jie-lin ZHANG, et al. ARAP++: an extension of the local/global approach to mesh parameterization[J]. Frontiers of Information Technology & Electronic Engineering, 2016, 17(6): 501-515. DOI： 10.1631/FITEE.1500184.

摘要

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。

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.

关键词

Keywords

references

N Aigerman , , , Y Lipman . . Injective and bounded distortion mappings in 3D . . ACM Trans. Graph. , , 2013 . . 32 ( ( 4 ): ): Article 106 DOI: 10.1145/2461912.2461931 http://doi.org/10.1145/2461912.2461931 . .

S Bouaziz , , , M Deuss , , , Y Schwartzburg , , , 等 . . Shapeup: shaping discrete geometry with projections . . Comput. Graph. Forum , , 2012 . . 31 ( ( 5 ): ): 1657 - - 1667 . . DOI: 10.1111/j.1467-8659.2012.03171.x http://doi.org/10.1111/j.1467-8659.2012.03171.x . .

BM Chen , , , C Gotsman , , , G Bunin . . Conformal flattening by curvature prescription and metric scaling . . Comput. Graph. Forum , , 2008 . . 27 ( ( 2 ): ): 449 - - 458 . . DOI: 10.1111/j.1467-8659.2008.01142.x http://doi.org/10.1111/j.1467-8659.2008.01142.x . .

Z Chen , , , L Liu , , , Z Zhang , , , 等 . . Surface parameterization via aligning optimal local flattening . . 2007 . . Proc.Symp. on Solid and Physical Modeling . . 291 - - 296 . . DOI: 10.1145/1236246.1236287 http://doi.org/10.1145/1236246.1236287 . .

P Degener , , , J Meseth , , , R Klein . . An adaptable surface parameterization method . . 2003 . . Proc. 12th Int. Meshing Roundtable . . 227 - - 237 . . . .

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

M Eck , , , T DeRose , , , T Duchamp , , , 等 . . Multiresolution analysis of arbitrary meshes . . 1995 . . Proc. 22nd Annual Conf. on Computer Graphics and Interactive Techniques . . 173 - - 182 . . DOI: 10.1145/218380.218440 http://doi.org/10.1145/218380.218440 . .

MS Floater . . Parameterization and smooth approximation of surface triangulations . . Comput. Aid. Geom.Des. , , 1997 . . 14 ( ( 3 ): ): 231 - - 250 . . DOI: 10.1016/S0167-8396(96)00031-3 http://doi.org/10.1016/S0167-8396(96)00031-3 . .

MS Floater . . Mean value coordinates . . Comput. Aid.Geom. Des. , , 2003 . . 20 ( ( 1 ): ): 19 - - 27 . . DOI: 10.1016/S0167-8396(03)00002-5 http://doi.org/10.1016/S0167-8396(03)00002-5 . .

MS Floater , , , K Hormann . . Surface parameterization:a tutorial and survey. In: Dodgson NA, Floater,M.S., Sabin, M.A. (Eds.) . . 2005 . . Advances in Multiresolution for Geometric Modelling . . 157 - - 186 . . DOI: 10.1007/3-540-26808-1_9 http://doi.org/10.1007/3-540-26808-1_9 . .

S Gortler , , , C Gotsman , , , D Thurston . . Discrete oneforms on meshes and applications to 3D mesh parameterization . . Comput. Aid. Geom. Des. , , 2006 . . 23 ( ( 2 ): ): 83 - - 112 . . DOI: 10.1016/j.cagd.2005.05.002 http://doi.org/10.1016/j.cagd.2005.05.002 . .

JC Gower , , , GB Dijksterhuis . . Procrustes Problems , , 2004 . . : Oxford : Oxford University Press , , . .

X Gu , , , S Yau . . Computing conformal structures of surfaces . . Commun. Inform. Syst. , , 2002 . . 2 ( ( 2 ): ): 121 - - 146 . . DOI: 10.4310/CIS.2002.v2.n2.a2 http://doi.org/10.4310/CIS.2002.v2.n2.a2 . .

X Gu , , , S Yau . . Global conformal surface parameterization . . 2003 . . Proc. Eurographics/ACM SIGGRAPH Symp.on Geometry Processing . . 127 - - 137 . . . .

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

H Hoppe , , , T DeRose , , , T Duchamp , , , 等 . . Mesh optimization . . 1993 . . Proc. 20th Annual Conf. on Computer Graphics and Interactive Techniques . . 19 - - 26 . . DOI: 10.1145/166117.166119 http://doi.org/10.1145/166117.166119 . .

K Hormann , , , G Greiner . . MIPS: an efficient global parameterization method . . 2000a . . Proc. Curve and Surface . . 153 - - 162 . . . .

K Hormann , , , G Greiner . . Quadrilateral remeshing . . 2000b . . Proc. Vision Modeling and Visualization . . 153 - - 162 . . . .

K Hormann , , , G Greiner , , , S Campagna . . Hierarchical parameterization of triangulated surfaces . . 1999 . . Proc. of Vision, Modeling and Visualization . . 219 - - 226 . . . .

K Hormann , , , U Labsik , , , G Greiner . . Remeshing triangulated surfaces with optimal parameterizations . . Comput.-Aid. Des. , , 2001 . . 33 ( ( 11 ): ): 779 - - 788 . . DOI: 10.1016/S0010-4485(01)00094-X http://doi.org/10.1016/S0010-4485(01)00094-X . .

K Hormann , , , B Lvy , , , A Sheffer . . Mesh parameterization: theory and practice . . 2007 . . Proc. SIGGRAPH . . 1 - - 122 . . . .

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

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

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

L Kharevych , , , B Springborn , , , P Schrder . . Discrete conformal mappings via circle patterns . . ACM Trans.Graph. , , 2006 . . 25 ( ( 2 ): ): 412 - - 438 . . DOI: 10.1145/1138450.1138461 http://doi.org/10.1145/1138450.1138461 . .

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

Y Lee , , , H Kim , , , S Lee . . Mesh parameterization with a virtual boundary . . Comput. Graph. , , 2002 . . 26 ( ( 5 ): ): 677 - - 686 . . DOI: 10.1016/S0097-8493(02)00123-1 http://doi.org/10.1016/S0097-8493(02)00123-1 . .

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

B Lvy , , , S Petitjean , , , N Ray , , , 等 . . Least squares conformal maps for automatic texture atlas generation . . ACM Trans. Graph. , , 2002 . . 21 ( ( 3 ): ): 362 - - 371 . . DOI: 10.1145/566570.566590 http://doi.org/10.1145/566570.566590 . .

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

L Liu , , , L Zhang , , , Y Xu , , , 等 . . A local/global approach to mesh parameterization . . Comput. Graph. Forum , , 2008 . . 27 ( ( 5 ): ): 1495 - - 1504 . . DOI: 10.1111/j.1467-8659.2008.01290.x http://doi.org/10.1111/j.1467-8659.2008.01290.x . .

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

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

P Sander , , , J Snyder , , , S Gortler , , , 等 . . Texture mapping progressive meshes . . 2001 . . Proc. 28th Annual Conf. on Computer Graphics and Interactive Techniques . . 409 - - 416 . . DOI: 10.1145/383259.383307 http://doi.org/10.1145/383259.383307 . .

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

A Sheffer , , , E de Sturler . . Parameterization of faceted surfaces for meshing using angle-based flattening . . Eng.Comput. , , 2001 . . 17 ( ( 3 ): ): 326 - - 337 . . DOI: 10.1007/PL00013391 http://doi.org/10.1007/PL00013391 . .

A Sheffer , , , B Lvy , , , M Mogilnitsky , , , 等 . . ABF++:fast and robust angle based flattening . . ACM Trans.Graph. , , 2005 . . 24 ( ( 2 ): ): 311 - - 330 . . DOI: 10.1145/1061347.1061354 http://doi.org/10.1145/1061347.1061354 . .

A Sheffer , , , E Praun , , , K Rose . . Mesh parameterization methods and their applications . . Comput. Graph.Vis. , , 2007 . . 2 ( ( 2 ): ): 105 - - 171 . . DOI: 10.1561/0600000011 http://doi.org/10.1561/0600000011 . .

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

WT Tutte . . How to draw a graph . . Proc. London Math. Soc. , , 1963 . . 13 ( ( 3 ): ): 743 - - 768 . . . .

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

O Weber , , , A Myles , , , D Zorin . . Computing extremal quasiconformal maps . . Comput. Graph. Forum , , 2012 . . 31 ( ( 5 ): ): 1679 - - 1689 . . DOI: 10.1111/j.1467-8659.2012.03173.x http://doi.org/10.1111/j.1467-8659.2012.03173.x . .

S Yoshizawa , , , A Belyaev , , , H Seidel . . A fast and simple stretch-minimizing mesh parameterization . . 2004 . . Proc.Shape Modeling Applications . . 200 - - 208 . . DOI: 10.1109/SMI.2004.1314507 http://doi.org/10.1109/SMI.2004.1314507 . .

R Zayer , , , B Lvy , , , H Seidel . . Linear angle based parameterization . . 2007 . . Proc. 5th Eurographics Symp. on Geometry Processing . . 135 - - 141 . . . .

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

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

G Zigelman , , , R Kimmel , , , N Kiryati . . Texture mapping using surface flattening via multidimensional scaling . . IEEE Trans. Visual. Comput. Graph. , , 2002 . . 8 ( ( 2 ): ): 198 - - 207 . . DOI: 10.1109/2945.998671 http://doi.org/10.1109/2945.998671 . .

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

No data

Related Author

No data

Related Institution

No data

Chat

Address：Zhejiang University Press, 148 Tianmushan Road, Hangzhou, China Postal code：310028
Tel：+86-571-88273162 Email：fitee@zju.edu.cn
It is recommended to read the content of this site in Chrome&IE9+. Please switch to extreme mode in browser 360.
Cookies We use cookies to help provide and enhance our service and tailor content. By continuing, you agree to the use of cookies.

⁰