October  2017, 11(5): 825-855. doi: 10.3934/ipi.2017039

Subdivision connectivity remeshing via Teichmüller extremal map

1. 

Department of Statistics, University of California, Davis, USA

2. 

Department of Computer Science, State University of New York at Stony Brook, USA

3. 

Department of Mathematics, The Chinese University of Hong Kong, China

Received  March 2016 Revised  June 2017 Published  July 2017

Curvilinear surfaces in 3D Euclidean spaces are commonly represented by triangular meshes. The structure of the triangulation is important, since it affects the accuracy and efficiency of the numerical computation on the mesh. Remeshing refers to the process of transforming an unstructured mesh to one with desirable structures, such as the subdivision connectivity. This is commonly achieved by parameterizing the surface onto a simple parameter domain, on which a structured mesh is built. The 2D structured mesh is then projected onto the surface via the parameterization. Two major tasks are involved. Firstly, an effective algorithm for parameterizing, usually conformally, surface meshes is necessary. However, for a highly irregular mesh with skinny triangles, computing a folding-free conformal parameterization is difficult. The second task is to build a structured mesh on the parameter domain that is adaptive to the area distortion of the parameterization while maintaining good shapes of triangles. This paper presents an algorithm to remesh a highly irregular mesh to a structured one with subdivision connectivity and good triangle quality. We propose an effective algorithm to obtain a conformal parameterization of a highly irregular mesh, using quasi-conformal Teichmüller theories. Conformality distortion of an initial parameterization is adjusted by a quasi-conformal map, resulting in a folding-free conformal parameterization. Next, we propose an algorithm to obtain a regular mesh with subdivision connectivity and good triangle quality on the conformal parameter domain, which is adaptive to the area distortion, through the landmark-matching Teichmüller map. A remeshed surface can then be obtained through the parameterization. Experiments have been carried out to remesh surface meshes representing real 3D geometric objects using the proposed algorithm. Results show the efficacy of the algorithm to optimize the regularity of an irregular triangulation.

Citation: Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039
References:
[1]

P. AlliezE. C. de VerdiereO. Devillers and M. Isenburg, Isotropic surface remeshing, Graphical Models, 67 (2005), 204-231. doi: 10.1109/SMI.2003.1199601. Google Scholar

[2]

P. AlliezD. Cohen-SteinerO. DevillersB. Lévy and M. Desbrun, Anisotropic polygonal remeshing, ACM Transactions on Graphics (TOG), 22 (2003), 485-493. doi: 10.1145/1201775.882296. Google Scholar

[3]

Z. ChenJ. Cao and W. Wang, Isotropic surface remeshing using constrained centroidal delaunay mesh, Computer Graphics Forum, 31 (2012), 2077-2085. doi: 10.1111/j.1467-8659.2012.03200.x. Google Scholar

[4]

M. Desbrun, P. Alliez and M. Meyer, Interactive Geometry Remeshing Proceeding of ACM SIGGRAPH, 2002.Google Scholar

[5]

Q. DuV. Faber and M. Gunzburger, Centroidal Voronoi tessellations: Applications and algorithms, SIAM Review, 41 (1999), 637-676. doi: 10.1137/S0036144599352836. Google Scholar

[6]

M. EckT. DeRoseT. DuchampH. HoppeM. Lounsbery and W. Stuetzle, Multiresolution analysis of arbitrary meshes, ACM Computer Graphics (SIGGRAPH 95 Proceedings), (1995), 173-182. doi: 10.1145/218380.218440. Google Scholar

[7]

P. J. Frey, About surface remeshing, Proceedings of 9th International Meshing Roundtable, (2000), 123-136. Google Scholar

[8]

P. J. Frey and H. Borouchaki, Geometric surface mesh optimization, Computing and Visualization in Science, 1 (1998), 113-121. doi: 10.1007/s007910050011. Google Scholar

[9]

F. Gardiner and N. Lakic, Quasiconformal Teichmuller Theory American Mathematics Society, 2000. Google Scholar

[10]

A. Gersho, Asymptotically optimal block quantization, IEEE Transactions on Information Theory, 25 (1979), 373-380. doi: 10.1109/TIT.1979.1056067. Google Scholar

[11]

X. F. GuS. J. Gortler and H. Hoppe, Geometry images, ACM Transactions on Graphics (TOG) -Proceedings of ACM SIGGRAPH, 21 (2002), 355-361. doi: 10.1145/566570.566589. Google Scholar

[12]

X. Gu and S. T. Yau, Computing conformal structures of surfaces, Communication in Information System, 2 (2002), 121-145. doi: 10.4310/CIS.2002.v2.n2.a2. Google Scholar

[13]

X. GuY. WangT. F. ChanP. Thompson and S. T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, Information Processing in Medical Imaging, (2003), 172-184. doi: 10.1007/978-3-540-45087-0_15. Google Scholar

[14]

S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro and M. Halle, Conformal surface parameterization for texture mapping, IEEE Trans. Visualization and Computer Graphics, 6 (2000), 181–189. doi: 10.1109/2945.856998. Google Scholar

[15]

H. Hoppe, Progressive meshes, Computer Graphics. SIGGRAPH 96 Proceedings, 30 (1996), 99-108. doi: 10.1145/237170.237216. Google Scholar

[16]

H. HoppeT. DeRoseT. DuchampJ. McDonald and W. Stuetzle, Mesh optimization, Computer Graphics. SIGGRAPH 93 Proceedings, 27 (1993), 19-26. doi: 10.1145/166117.166119. Google Scholar

[17]

K. HormannU. Labsik and G. Greiner, Remeshing triangulated surfaces with optimal parameterizations, Computer-Aided Design, 33 (2001), 779-788. doi: 10.1016/S0010-4485(01)00094-X. Google Scholar

[18]

M. K. Hurdal and K. Stephenson, Cortical cartography using the discrete conformal approach of circle packings, NeuroImage, 23 (2004), S119-S128. doi: 10.1016/j.neuroimage.2004.07.018. Google Scholar

[19]

M. JinJ. KimF. Luo and X. Gu, Discrete surface Ricci flow, IEEE Transaction on Visualization and Computer Graphics, 14 (2008), 1030-1043. doi: 10.1109/TVCG.2008.57. Google Scholar

[20]

M. JinJ. KimF. Luo and X. Gu, Combinatorial Yamabe flow on surfaces, Communication in Contemporary Mathematics, 6 (2004), 765-780. doi: 10.1142/S0219199704001501. Google Scholar

[21]

U. LabsikL. KobbeltR. Schneider and H. P. Seidel, Progressive transmission of subdivision surfaces, Computational Geometry, 15 (2000), 25-39. doi: 10.1016/S0925-7721(99)00045-0. Google Scholar

[22]

R. LaiZ. WenW. YinX. Gu and L. M. Lui, Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization, Journal of Scientific Computing, 58 (2014), 705-725. doi: 10.1007/s10915-013-9752-6. Google Scholar

[23]

B. Lévy and J. Maillot, Least Squares Conformal Maps for Automatic Texture Atlas Generation ACM SIGGRAPH Proceedings, 2002.Google Scholar

[24]

M. LounsberyT. DeRose and J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, ACM Transactions on Graphics (TOG) TOG Homepage Archive, 16 (1997), 34-73. doi: 10.1145/237748.237750. Google Scholar

[25]

L. M. LuiT. W. WongW. ZengX. F. GuP. M. ThompsonT. F. Chan and S. T. Yau, Optimization of surface registrations using beltrami holomorphic flow, Journal of Scientific Computing, 50 (2012), 557-585. doi: 10.1007/s10915-011-9506-2. Google Scholar

[26]

L. M. LuiT. W. WongX. F. GuP. M. ThompsonT. F. Chan and S. T. Yau, Hippocampal Shape Registration using Beltrami Holomorphic flow, Medical Image Computing and Computer Assisted Intervention(MICCAI), Part Ⅱ, LNCS, 6362 (2010), 323-330. Google Scholar

[27]

L. M. LuiK. C. LamS. T. Yau and X. F. Gu, Teichmuller mapping (T-Map) and its applications to landmark matching registrations, SIAM Journal on Imaging Sciences, 7 (2014), 391-426. doi: 10.1137/120900186. Google Scholar

[28]

L. M. LuiK. C. LamT. W. Wong and X. F. Gu, Texture map and video compression using Beltrami representation, SIAM Journal on Imaging Sciences, 6 (2013), 1880-1902. doi: 10.1137/120866129. Google Scholar

[29]

L. M. LuiX. F. Gu and S. T. Yau, Convergence analysis of an iterative algorithm for Teichmuller maps via harmonic energy optimization, Mathematics of Computation, 84 (2015), 2823-2842. doi: 10.1090/S0025-5718-2015-02962-7. Google Scholar

[30]

T. C. NgX. F. Gu and L. M. Lui, Teichmuller extremal map of multiply-connected domains using Beltrami holomorphic flow, Journal of Scientific Computing, 60 (2014), 249-275. doi: 10.1007/s10915-013-9791-z. Google Scholar

[31]

P. Pébay and T. Baker, Analysis of triangle quality measures, Mathematics of Computation, 72 (2003), 1817-1839. doi: 10.1090/S0025-5718-03-01485-6. Google Scholar

[32]

G. Peyré and L. Cohen, Geodesic computations for fast and accurate surface remeshing and parameterization, Progress in Nonlinear equation and applications, 63 (2005), 157-171. doi: 10.1007/3-7643-7384-9_18. Google Scholar

[33]

E. Praun and H. Hoppe, Spherical parametrization and remeshing, ACM Transactions on Graphics (TOG) -Proceedings of ACM SIGGRAPH, 22 (2003), 340-349. doi: 10.1145/1201775.882274. Google Scholar

[34]

A. RassineuxP. VillonJ.-M. Savignat and O. Stab, Surface remeshing by local Hermite diffuse interpolation, International Journal for Numerical Methods in Engineering, 49 (2000), 31-49. Google Scholar

[35]

G. RongM. JinL. Shuai and X. Guo, Centroidal Voronoi tessellation in universal covering space of manifold surfaces, Computer Aided Geometric Design (CAGD), 28 (2011), 475-496. doi: 10.1016/j.cagd.2011.06.005. Google Scholar

[36]

P. Schroder and W. Sweldens, Spherical wavelets: Efficiently representing functions on the sphere, ACM Computer Graphics (SIGGRAPH 95 Proceedings), (1995), 161-172. Google Scholar

[37]

V. Surazhsky and C. Gotsman, Explicit surface remeshing, Computer Graphics. SIGGRAPH 92 Proceedings, 26 (1992), 55-64. Google Scholar

[38]

G. Fejes Tóth, A stability criterion to the moment theorem, Studia Scientiarum Mathematicarum Hungarica, 38 (2001), 209-224. doi: 10.1556/SScMath.38.2001.1-4.14. Google Scholar

[39]

G. Turk, Re-tiling polygonal surfaces, Computer Graphics. SIGGRAPH 92 Proceedings, 26 (1992), 55-64. doi: 10.1145/133994.134008. Google Scholar

[40]

O. WeberA. Myles and D. Zorin, Computing extremal quasiconformal maps, Computer Graphics Forum, 31 (2012), 1679-1689. doi: 10.1111/j.1467-8659.2012.03173.x. Google Scholar

[41]

D. YanB. LévyY. LiuF. Sun and W. Wang, Isotropic remeshing with fast and exact computation of restricted Voronoi diagram, Eurographics Symposium on Geometry Processing, 28 (2009), 1445-1454. doi: 10.1111/j.1467-8659.2009.01521.x. Google Scholar

[42]

Y. L. YangR. GuoF. LuoS. M. Hu and X. F. Gu, Generalized discrete ricci flow, Computer Graphics Forum, 28 (2009), 2005-2014. Google Scholar

[43]

W. ZengL. M. LuiL. ShiD. WangW. C. ChuJ. C. ChengJ. HuaS. T. Yau and X. F. Gu, Shape Analysis of Vestibular Systems in Adolescent Idiopathic Scoliosis Using Geodesic Spectra, Medical Image Computing and Computer Assisted Intervation, 13 (2010), 538-546. doi: 10.1007/978-3-642-15711-0_67. Google Scholar

[44]

W. ZengL. M. LuiF. LuoT. F. ChanS. T. Yau and X. F. Gu, Computing quasiconformal maps using an auxiliary metric and discrete curvature flow, Numerische Mathematik, 121 (2012), 671-703. doi: 10.1007/s00211-012-0446-z. Google Scholar

[45]

W. Zeng and X. Gu, Registration for 3D surfaces with large deformations using quasi-conformal curvature flow IEEE Conference on Computer Vision and Pattern Recognition (CVPR11) Colorado Springs, Colorado, USA, Jun 20-25,2011. doi: 10.1109/CVPR.2011.5995410. Google Scholar

[46]

D. ZorinP. Schroder and W. Sweldens, Interactive multiresolution mesh editing, ACM Computer Graphics (SIGGRAPH 97 Proceedings), (1997), 259-268. doi: 10.1145/258734.258863. Google Scholar

show all references

References:
[1]

P. AlliezE. C. de VerdiereO. Devillers and M. Isenburg, Isotropic surface remeshing, Graphical Models, 67 (2005), 204-231. doi: 10.1109/SMI.2003.1199601. Google Scholar

[2]

P. AlliezD. Cohen-SteinerO. DevillersB. Lévy and M. Desbrun, Anisotropic polygonal remeshing, ACM Transactions on Graphics (TOG), 22 (2003), 485-493. doi: 10.1145/1201775.882296. Google Scholar

[3]

Z. ChenJ. Cao and W. Wang, Isotropic surface remeshing using constrained centroidal delaunay mesh, Computer Graphics Forum, 31 (2012), 2077-2085. doi: 10.1111/j.1467-8659.2012.03200.x. Google Scholar

[4]

M. Desbrun, P. Alliez and M. Meyer, Interactive Geometry Remeshing Proceeding of ACM SIGGRAPH, 2002.Google Scholar

[5]

Q. DuV. Faber and M. Gunzburger, Centroidal Voronoi tessellations: Applications and algorithms, SIAM Review, 41 (1999), 637-676. doi: 10.1137/S0036144599352836. Google Scholar

[6]

M. EckT. DeRoseT. DuchampH. HoppeM. Lounsbery and W. Stuetzle, Multiresolution analysis of arbitrary meshes, ACM Computer Graphics (SIGGRAPH 95 Proceedings), (1995), 173-182. doi: 10.1145/218380.218440. Google Scholar

[7]

P. J. Frey, About surface remeshing, Proceedings of 9th International Meshing Roundtable, (2000), 123-136. Google Scholar

[8]

P. J. Frey and H. Borouchaki, Geometric surface mesh optimization, Computing and Visualization in Science, 1 (1998), 113-121. doi: 10.1007/s007910050011. Google Scholar

[9]

F. Gardiner and N. Lakic, Quasiconformal Teichmuller Theory American Mathematics Society, 2000. Google Scholar

[10]

A. Gersho, Asymptotically optimal block quantization, IEEE Transactions on Information Theory, 25 (1979), 373-380. doi: 10.1109/TIT.1979.1056067. Google Scholar

[11]

X. F. GuS. J. Gortler and H. Hoppe, Geometry images, ACM Transactions on Graphics (TOG) -Proceedings of ACM SIGGRAPH, 21 (2002), 355-361. doi: 10.1145/566570.566589. Google Scholar

[12]

X. Gu and S. T. Yau, Computing conformal structures of surfaces, Communication in Information System, 2 (2002), 121-145. doi: 10.4310/CIS.2002.v2.n2.a2. Google Scholar

[13]

X. GuY. WangT. F. ChanP. Thompson and S. T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, Information Processing in Medical Imaging, (2003), 172-184. doi: 10.1007/978-3-540-45087-0_15. Google Scholar

[14]

S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro and M. Halle, Conformal surface parameterization for texture mapping, IEEE Trans. Visualization and Computer Graphics, 6 (2000), 181–189. doi: 10.1109/2945.856998. Google Scholar

[15]

H. Hoppe, Progressive meshes, Computer Graphics. SIGGRAPH 96 Proceedings, 30 (1996), 99-108. doi: 10.1145/237170.237216. Google Scholar

[16]

H. HoppeT. DeRoseT. DuchampJ. McDonald and W. Stuetzle, Mesh optimization, Computer Graphics. SIGGRAPH 93 Proceedings, 27 (1993), 19-26. doi: 10.1145/166117.166119. Google Scholar

[17]

K. HormannU. Labsik and G. Greiner, Remeshing triangulated surfaces with optimal parameterizations, Computer-Aided Design, 33 (2001), 779-788. doi: 10.1016/S0010-4485(01)00094-X. Google Scholar

[18]

M. K. Hurdal and K. Stephenson, Cortical cartography using the discrete conformal approach of circle packings, NeuroImage, 23 (2004), S119-S128. doi: 10.1016/j.neuroimage.2004.07.018. Google Scholar

[19]

M. JinJ. KimF. Luo and X. Gu, Discrete surface Ricci flow, IEEE Transaction on Visualization and Computer Graphics, 14 (2008), 1030-1043. doi: 10.1109/TVCG.2008.57. Google Scholar

[20]

M. JinJ. KimF. Luo and X. Gu, Combinatorial Yamabe flow on surfaces, Communication in Contemporary Mathematics, 6 (2004), 765-780. doi: 10.1142/S0219199704001501. Google Scholar

[21]

U. LabsikL. KobbeltR. Schneider and H. P. Seidel, Progressive transmission of subdivision surfaces, Computational Geometry, 15 (2000), 25-39. doi: 10.1016/S0925-7721(99)00045-0. Google Scholar

[22]

R. LaiZ. WenW. YinX. Gu and L. M. Lui, Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization, Journal of Scientific Computing, 58 (2014), 705-725. doi: 10.1007/s10915-013-9752-6. Google Scholar

[23]

B. Lévy and J. Maillot, Least Squares Conformal Maps for Automatic Texture Atlas Generation ACM SIGGRAPH Proceedings, 2002.Google Scholar

[24]

M. LounsberyT. DeRose and J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, ACM Transactions on Graphics (TOG) TOG Homepage Archive, 16 (1997), 34-73. doi: 10.1145/237748.237750. Google Scholar

[25]

L. M. LuiT. W. WongW. ZengX. F. GuP. M. ThompsonT. F. Chan and S. T. Yau, Optimization of surface registrations using beltrami holomorphic flow, Journal of Scientific Computing, 50 (2012), 557-585. doi: 10.1007/s10915-011-9506-2. Google Scholar

[26]

L. M. LuiT. W. WongX. F. GuP. M. ThompsonT. F. Chan and S. T. Yau, Hippocampal Shape Registration using Beltrami Holomorphic flow, Medical Image Computing and Computer Assisted Intervention(MICCAI), Part Ⅱ, LNCS, 6362 (2010), 323-330. Google Scholar

[27]

L. M. LuiK. C. LamS. T. Yau and X. F. Gu, Teichmuller mapping (T-Map) and its applications to landmark matching registrations, SIAM Journal on Imaging Sciences, 7 (2014), 391-426. doi: 10.1137/120900186. Google Scholar

[28]

L. M. LuiK. C. LamT. W. Wong and X. F. Gu, Texture map and video compression using Beltrami representation, SIAM Journal on Imaging Sciences, 6 (2013), 1880-1902. doi: 10.1137/120866129. Google Scholar

[29]

L. M. LuiX. F. Gu and S. T. Yau, Convergence analysis of an iterative algorithm for Teichmuller maps via harmonic energy optimization, Mathematics of Computation, 84 (2015), 2823-2842. doi: 10.1090/S0025-5718-2015-02962-7. Google Scholar

[30]

T. C. NgX. F. Gu and L. M. Lui, Teichmuller extremal map of multiply-connected domains using Beltrami holomorphic flow, Journal of Scientific Computing, 60 (2014), 249-275. doi: 10.1007/s10915-013-9791-z. Google Scholar

[31]

P. Pébay and T. Baker, Analysis of triangle quality measures, Mathematics of Computation, 72 (2003), 1817-1839. doi: 10.1090/S0025-5718-03-01485-6. Google Scholar

[32]

G. Peyré and L. Cohen, Geodesic computations for fast and accurate surface remeshing and parameterization, Progress in Nonlinear equation and applications, 63 (2005), 157-171. doi: 10.1007/3-7643-7384-9_18. Google Scholar

[33]

E. Praun and H. Hoppe, Spherical parametrization and remeshing, ACM Transactions on Graphics (TOG) -Proceedings of ACM SIGGRAPH, 22 (2003), 340-349. doi: 10.1145/1201775.882274. Google Scholar

[34]

A. RassineuxP. VillonJ.-M. Savignat and O. Stab, Surface remeshing by local Hermite diffuse interpolation, International Journal for Numerical Methods in Engineering, 49 (2000), 31-49. Google Scholar

[35]

G. RongM. JinL. Shuai and X. Guo, Centroidal Voronoi tessellation in universal covering space of manifold surfaces, Computer Aided Geometric Design (CAGD), 28 (2011), 475-496. doi: 10.1016/j.cagd.2011.06.005. Google Scholar

[36]

P. Schroder and W. Sweldens, Spherical wavelets: Efficiently representing functions on the sphere, ACM Computer Graphics (SIGGRAPH 95 Proceedings), (1995), 161-172. Google Scholar

[37]

V. Surazhsky and C. Gotsman, Explicit surface remeshing, Computer Graphics. SIGGRAPH 92 Proceedings, 26 (1992), 55-64. Google Scholar

[38]

G. Fejes Tóth, A stability criterion to the moment theorem, Studia Scientiarum Mathematicarum Hungarica, 38 (2001), 209-224. doi: 10.1556/SScMath.38.2001.1-4.14. Google Scholar

[39]

G. Turk, Re-tiling polygonal surfaces, Computer Graphics. SIGGRAPH 92 Proceedings, 26 (1992), 55-64. doi: 10.1145/133994.134008. Google Scholar

[40]

O. WeberA. Myles and D. Zorin, Computing extremal quasiconformal maps, Computer Graphics Forum, 31 (2012), 1679-1689. doi: 10.1111/j.1467-8659.2012.03173.x. Google Scholar

[41]

D. YanB. LévyY. LiuF. Sun and W. Wang, Isotropic remeshing with fast and exact computation of restricted Voronoi diagram, Eurographics Symposium on Geometry Processing, 28 (2009), 1445-1454. doi: 10.1111/j.1467-8659.2009.01521.x. Google Scholar

[42]

Y. L. YangR. GuoF. LuoS. M. Hu and X. F. Gu, Generalized discrete ricci flow, Computer Graphics Forum, 28 (2009), 2005-2014. Google Scholar

[43]

W. ZengL. M. LuiL. ShiD. WangW. C. ChuJ. C. ChengJ. HuaS. T. Yau and X. F. Gu, Shape Analysis of Vestibular Systems in Adolescent Idiopathic Scoliosis Using Geodesic Spectra, Medical Image Computing and Computer Assisted Intervation, 13 (2010), 538-546. doi: 10.1007/978-3-642-15711-0_67. Google Scholar

[44]

W. ZengL. M. LuiF. LuoT. F. ChanS. T. Yau and X. F. Gu, Computing quasiconformal maps using an auxiliary metric and discrete curvature flow, Numerische Mathematik, 121 (2012), 671-703. doi: 10.1007/s00211-012-0446-z. Google Scholar

[45]

W. Zeng and X. Gu, Registration for 3D surfaces with large deformations using quasi-conformal curvature flow IEEE Conference on Computer Vision and Pattern Recognition (CVPR11) Colorado Springs, Colorado, USA, Jun 20-25,2011. doi: 10.1109/CVPR.2011.5995410. Google Scholar

[46]

D. ZorinP. Schroder and W. Sweldens, Interactive multiresolution mesh editing, ACM Computer Graphics (SIGGRAPH 97 Proceedings), (1997), 259-268. doi: 10.1145/258734.258863. Google Scholar

Figure 1.  Examples of irregular meshes. Conventional conformal parameterization methods fail on these examples
Figure 2.  Original surface meshes: Foot, hand, human face and venus
Figure 3.  Parameterization of Foot: (A) Tutte's embedding and (B) conformal parameterization
Figure 4.  Parameterization of Venus: (A) Tutte's embedding and (B) conformal parameterization
Figure 5.  Histogram for $\| \mu (\phi) \|_\infty$c
Figure 6.  Remeshed surface with a uniform mesh on the parameter domain
Figure 7.  Remeshing on the parameterization domain of the hand surface: (A) Parameterization mesh; (B) base mesh; (C) subdivision mesh; (D) Teichm¨uller adaptive mesh
Figure 8.  Remeshing results for the foot surface with different meshes on the parameter domain
Figure 9.  Surface remeshing results of the foot surface
Figure 10.  More surface remeshing results of the foot surface at different viewpoint angles
Figure 11.  Surface remeshing results of the hand surface at different viewpoint angles
Figure 12.  Surface remeshing results of the human face
Figure 13.  Surface remeshing of the venus surface
Figure 14.  Surface remeshing of the lion vase surface
Figure 15.  Surface remeshing of the mask surface
Figure 16.  Triangle quality of Foot
Figure 17.  Triangle quality of Venus
Figure 18.  Surface remeshing of the foot surface with subdivision connectivity at various levels
Figure 19.  Surface remeshing of the mask surface with subdivision connectivity at various levels
Table 1.  The conformality distortion of different parameterization methods for meshes with various triangle quality
Mesh mean of $\tau_i$ std of $\tau_i$ min of $\tau_i$ Ricci Yamabe IDRF Double Ours
Foot 0.7769 0.1666 0.0126 0.2653 0.0238 0.0238 0.0304 0.0247
Hand 0.7117 0.1983 0.0079 0.3328 fail fail fail 0.0976
Face 0.7847 0.1951 0.0317 0.2420 fail fail 0.0679 0.0109
Venus 0.8036 0.0712 0.0014 0.2837 fail fail fail 0.0064
Mask 0.8328 0.1603 0.0010 0.2379 fail fail 0.1104 0.0057
Lion 0.6697 0.2220 0.0157 0.2029 fail fail 0.0959 0.0323
Mesh mean of $\tau_i$ std of $\tau_i$ min of $\tau_i$ Ricci Yamabe IDRF Double Ours
Foot 0.7769 0.1666 0.0126 0.2653 0.0238 0.0238 0.0304 0.0247
Hand 0.7117 0.1983 0.0079 0.3328 fail fail fail 0.0976
Face 0.7847 0.1951 0.0317 0.2420 fail fail 0.0679 0.0109
Venus 0.8036 0.0712 0.0014 0.2837 fail fail fail 0.0064
Mask 0.8328 0.1603 0.0010 0.2379 fail fail 0.1104 0.0057
Lion 0.6697 0.2220 0.0157 0.2029 fail fail 0.0959 0.0323
Table 2.  Comparison between the direct CVT method and Teichm¨uller remeshing
# = 2557 # = 11065
Direct Teichmüuller Direct Teichmüuller
CVT Remeshing CVT Remeshing
Triangle Quality mean = 0.9189
std = 0.0704
min = 0.1471
mean 0.9031 =
std = 0.0476
min = 0.5252
mean = 0.9096
std = 0.0759
min = 0.0122
mean = 0.8786
std = 0.0368
min = 0.3240
Time 79s 21s 839s 61s
# = 2557 # = 11065
Direct Teichmüuller Direct Teichmüuller
CVT Remeshing CVT Remeshing
Triangle Quality mean = 0.9189
std = 0.0704
min = 0.1471
mean 0.9031 =
std = 0.0476
min = 0.5252
mean = 0.9096
std = 0.0759
min = 0.0122
mean = 0.8786
std = 0.0368
min = 0.3240
Time 79s 21s 839s 61s
[1]

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