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Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory

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Abstract

Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering. However, most of the existing conformal parameterization algorithms only focus on simply-connected surfaces and cannot be directly applied to surfaces with holes. In this work, we propose two novel algorithms for computing the conformal parameterization of multiply-connected surfaces. We first develop an efficient method for conformally parameterizing an open surface with one hole to an annulus on the plane. Based on this method, we then develop an efficient method for conformally parameterizing an open surface with k holes onto a unit disk with k circular holes. The conformality and bijectivity of the mappings are ensured by quasi-conformal theory. Numerical experiments and applications are presented to demonstrate the effectiveness of the proposed methods.

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References

  1. Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, pp. 157–186, Springer (2005)

  2. Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Found. Trends Comput. Graph. Vis. 2(2), 105–171 (2006)

    Article  MATH  Google Scholar 

  3. Hormann, K., Lévy, B., Sheffer, A.: Mesh parameterization: theory and practice. In: ACM SIGGRAPH 2007 Courses. Association for Computing Machinery, New York, NY, USA (2007)

  4. Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.-T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23(8), 949–958 (2004)

    Article  Google Scholar 

  6. Choi, P.T., Lam, K.C., Lui, L.M.: FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces. SIAM J. Imaging Sci. 8(1), 67–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Choi, G.P.-T., Ho, K.T., Lui, L.M.: Spherical conformal parameterization of genus-0 point clouds for meshing. SIAM J. Imaging Sci. 9(4), 1582–1618 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, G.P.-T., Lui, L.M.: A linear formulation for disk conformal parameterization of simply-connected open surfaces. Adv. Comput. Math. 44(1), 87–114 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21(3), 362–371 (2002)

    Article  Google Scholar 

  10. Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. Comput. Graph. Forum 21(3), 209–218 (2002)

    Article  Google Scholar 

  11. Gu, X., Yau, S.-T.: Global conformal surface parameterization. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 127–137 (2003)

  12. Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(05), 765–780 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Sheffer, A., Lévy, B., Mogilnitsky, M., Bogomyakov, A.: ABF++: fast and robust angle based flattening. ACM Trans. Graph. 24(2), 311–330 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), 1–11 (2008)

    Article  Google Scholar 

  17. Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface Ricci flow. IEEE Trans. Vis. Comput. Graph. 14(5), 1030–1043 (2008)

    Article  Google Scholar 

  18. Yang, Y.-L., Guo, R., Luo, F., Hu, S.-M., Gu, X.: Generalized discrete Ricci flow. Comput. Graph. Forum 28(7), 2005–2014 (2009)

    Article  Google Scholar 

  19. Yang, Y.-L., Kim, J., Luo, F., Hu, S.-M., Gu, X.: Optimal surface parameterization using inverse curvature map. IEEE Trans. Vis. Comput. Graph. 14(5), 1054–1066 (2008)

    Article  Google Scholar 

  20. Zhang, M., Zeng, W., Guo, R., Luo, F., Gu, X.D.: Survey on discrete surface Ricci flow. J. Comput. Sci. Technol. 30(3), 598–613 (2015)

    Article  MathSciNet  Google Scholar 

  21. Mullen, P., Tong, Y., Alliez, P., Desbrun, M.: Spectral conformal parameterization. Comput. Graph Forum 27(5), 1487–1494 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  23. Marshall, D.E., Rohde, S.: Convergence of a variant of the zipper algorithm for conformal mapping. SIAM J. Numer. Anal. 45(6), 2577–2609 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Choi, G.P.T., Leung-Liu, Y., Gu, X., Lui, L.M.: Parallelizable global conformal parameterization of simply-connected surfaces via partial welding. SIAM J. Imaging Sci. 13(3), 1049–1083 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sawhney, R., Crane, K.: Boundary first flattening. ACM Trans. Graph. 37(1), 1–14 (2017)

    Article  Google Scholar 

  26. Yueh, M.-H., Lin, W.-W., Wu, C.-T., Yau, S.-T.: An efficient energy minimization for conformal parameterizations. J. Sci. Comput. 73(1), 203–227 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Choi, P.T., Lui, L.M.: Fast disk conformal parameterization of simply-connected open surfaces. J. Sci. Comput. 65(3), 1065–1090 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Meng, T.W., Choi, G.P.-T., Lui, L.M.: TEMPO: Feature-endowed Teich\({\ddot{{\rm m}}}\)uller extremal mappings of point clouds. SIAM J. Imaging Sci. 9(4), 1922–1962 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lui, L.M., Lam, K.C., Wong, T.W., Gu, X.: Texture map and video compression using Beltrami representation. SIAM J. Imaging Sci. 6(4), 1880–1902 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yung, C.P., Choi, G.P.T., Chen, K., Lui, L.M.: Efficient feature-based image registration by mapping sparsified surfaces. J. Vis. Commun. Image Represent. 55, 561–571 (2018)

    Article  Google Scholar 

  31. Lipman, Y.: Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31(4), 1–13 (2012)

    Article  Google Scholar 

  32. Weber, O., Myles, A., Zorin, D.: Computing extremal quasiconformal maps. Comput. Graph. Forum 31(5), 1679–1689 (2012)

    Article  Google Scholar 

  33. Wong, T.W., Zhao, H.-K.: Computation of quasi-conformal surface maps using discrete Beltrami flow. SIAM J. Imaging Sci. 7(4), 2675–2699 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lui, L.M., Lam, K.C., Yau, S.-T., Gu, X.: Teichmüller mapping (T-map) and its applications to landmark matching registration. SIAM J. Imaging Sci. 7(1), 391–426 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Choi, G.P.-T., Man, M.H.-Y., Lui, L.M.: Fast spherical quasiconformal parameterization of genus-\(0 \) closed surfaces with application to adaptive remeshing. Geom. Imaging Comput. 3(1), 1–29 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Choi, G.P.T., Dudte, L.H., Mahadevan, L.: Programming shape using kirigami tessellations. Nat. Mater. 18(9), 999–1004 (2019)

    Article  Google Scholar 

  37. Zeng, W., Marino, J., Gurijala, K.C., Gu, X., Kaufman, A.: Supine and prone colon registration using quasi-conformal mapping. IEEE Trans. Vis. Comput. Graph. 16(6), 1348–1357 (2010)

    Article  Google Scholar 

  38. Choi, G.P.T., Chen, Y., Lui, L.M., Chiu, B.: Conformal mapping of carotid vessel wall and plaque thickness measured from 3D ultrasound images. Med. Biol. Eng. Comput. 55(12), 2183–2195 (2017)

    Article  Google Scholar 

  39. Choi, G.P.T., Mahadevan, L.: Planar morphometrics using Teichmüller maps. Proc. R. Soc. A 474(2217), 20170905 (2018)

    Article  MATH  Google Scholar 

  40. Choi, G.P.T., Chan, H.L., Yong, R., Ranjitkar, S., Brook, A., Townsend, G., Chen, K., Lui, L.M.: Tooth morphometry using quasi-conformal theory. Pattern Recognit. 99, 107064 (2020)

    Article  Google Scholar 

  41. Choi, G.P.T., Qiu, D., Lui, L.M.: Shape analysis via inconsistent surface registration. Proc. R. Soc. A 476(2242), 20200147 (2020)

    Article  MathSciNet  Google Scholar 

  42. Yin, X., Dai, J., Yau, S.-T., Gu, X.: Slit map: Conformal parameterization for multiply connected surfaces. In: International Conference on Geometric Modeling and Processing, pp. 410–422. Springer (2008)

  43. Wang, Y., Gu, X., Chan, T. F., Thompson, P. M., Yau, S.-T.: Conformal slit mapping and its applications to brain surface parameterization. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 585–593. Springer (2008)

  44. Koebe, P.: Über die konforme abbildung mehrfach zusammenhängender bereiche. Jahresbericht der Deutschen Mathematiker-Vereinigung 19, 339–348 (1910)

    MATH  Google Scholar 

  45. Zeng, W., Yin, X., Zhang, M., Luo, F., Gu, X.: Generalized Koebe’s method for conformal mapping multiply connected domains. In: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling. pp. 89–100 (2009)

  46. Kropf, E., Yin, X., Yau, S.-T., Gu, X.D.: Conformal parameterization for multiply connected domains: combining finite elements and complex analysis. Eng. Comput. 30(4), 441–455 (2014)

    Article  Google Scholar 

  47. Ho, K.T., Lui, L.M.: QCMC: quasi-conformal parameterizations for multiply-connected domains. Adv. Comput. Math. 42(2), 279–312 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Bobenko, A. I., Sechelmann, S., Springborn, B.: Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization. In: Advances in Discrete Differential Geometry, pp. 1–56, Springer, Berlin (2016)

  49. Lehto, O.: Quasiconformal Mappings in the Plane, vol. 126. Springer, Berlin, Heidelberg (1973)

    Book  MATH  Google Scholar 

  50. Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüller Theory, vol. 76. American Mathematical Society, New York (2000)

    MATH  Google Scholar 

  51. Ahlfors, L.V.: Lectures on Quasiconformal Mappings, vol. 38. American Mathematical Society, New York (2006)

    MATH  Google Scholar 

  52. Birdal, T.: Maximum inscribed circle using Voronoi diagram. https://www.mathworks.com/matlabcentral/fileexchange/32543-maximum-inscribed-circle-using-voronoi-diagram

  53. Gu, X.: RiemannMapper: a mesh parameterization toolkit. https://www3.cs.stonybrook.edu/ gu/software/RiemannMapper/

  54. Persson, P.-O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ng, T.C., Gu, X., Lui, L.M.: Teichmüller extremal map of multiply-connected domains using Beltrami holomorphic flow. J. Sci. Comput. 60(2), 249–275 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Crowdy, D.: The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains. Proc. R. Soc. A 461(2061), 2653–2678 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Crowdy, D., Marshall, J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  58. Crowdy, D.: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Camb. Philos. Soc. 142(2), 319 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Nasser, M.M.S.: Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM J. Sci. Comput. 31(3), 1695–1715 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. Nasser, M.M.S.: PlgCirMap: A MATLAB toolbox for computing conformal mappings from polygonal multiply connected domains onto circular domains. SoftwareX 11, 100464 (2020)

    Article  Google Scholar 

  61. Crowdy, D.: Solving problems in multiply connected domains. NSF-CBMS Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia (2020)

  62. Zeng, W., Lui, L.M., Gu, X., Yau, S.-T.: Shape analysis by conformal modules. Methods Appl. Anal. 15(4), 539–556 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhao, J., Qi, X., Wen, C., Lei, N., Gu, X.: Automatic and robust skull registration based on discrete uniformization. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 431–440, (2019)

  64. Li, S., Zeng, W., Zhou, D., Gu, X., Gao, J.: Compact conformal map for greedy routing in wireless mobile sensor networks. IEEE Trans. Mobile Comput. 15(7), 1632–1646 (2015)

    Article  Google Scholar 

  65. Zhao, X., Su, Z., Gu, X.D., Kaufman, A., Sun, J., Gao, J., Luo, F.: Area-preservation mapping using optimal mass transport. IEEE Trans. Vis. Comput. Graph. 19(12), 2838–2847 (2013)

    Article  Google Scholar 

  66. Su, K., Cui, L., Qian, K., Lei, N., Zhang, J., Zhang, M., Gu, X.D.: Area-preserving mesh parameterization for poly-annulus surfaces based on optimal mass transportation. Comput. Aided Geom. Des. 46, 76–91 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Pumarola, A., Sanchez-Riera, J., Choi, G. P. T., Sanfeliu, A., Moreno-Noguer, F.: 3DPeople: modeling the geometry of dressed humans. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 2242–2251 (2019)

  68. Giri, A., Choi, G.P.T., Kumar, L.: Open and closed anatomical surface description via hemispherical area-preserving map. Signal Process. 180, 107867 (2021)

    Article  Google Scholar 

  69. Choi, G.P.T., Rycroft, C.H.: Density-equalizing maps for simply connected open surfaces. SIAM J. Imaging Sci. 11(2), 1134–1178 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  70. Choi, G.P.T., Chiu, B., Rycroft, C.H.: Area-preserving mapping of 3D carotid ultrasound images using density-equalizing reference map. IEEE Trans. Biomed. Eng. 67(9), 1507–1517 (2020)

    Article  Google Scholar 

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Acknowledgements

This work was supported in part by the National Science Foundation under Grant No. DMS-2002103.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Gary P. T. Choi

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Choi, G.P.T. Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory. J Sci Comput 87, 70 (2021). https://doi.org/10.1007/s10915-021-01479-y

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