Numerische Mathematik

, Volume 121, Issue 4, pp 671–703 | Cite as

Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

  • Wei Zeng
  • Lok Ming Lui
  • Feng Luo
  • Tony Fan-Cheong Chan
  • Shing-Tung Yau
  • David Xianfeng Gu


Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmüller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

Mathematics Subject Classification (2000)

65 52 30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlfors L.: Conformality with respect to Riemannian matrices. Ann. Acad. Sci. Fenn. Ser. 206, 1–22 (1955)MathSciNetGoogle Scholar
  2. 2.
    Ahlfors L.: Lectures in Quasiconformal Mappings. Van Nostrand Reinhold, New York (1966)Google Scholar
  3. 3.
    Ben-Chen M., Gotsman C., Bunin G.: Conformal flattening by curvature prescription and metric scaling. Comput. Graph. Forum 27(2), 449–458 (2008)CrossRefGoogle Scholar
  4. 4.
    Bers L., Nirenberg L.: On Linear and Nonlinear Elliptic Boundary Value Problems in the Plane, pp. 141–167. Convegno Internazionale Suelle Equaziono Cremeonese, Roma (1955)Google Scholar
  5. 5.
    Bers L.: Mathematical Aspects of Subcritical and Transonic Gas Dynamics. Wiley, New York (1958)Google Scholar
  6. 6.
    Bers L.: Quasiconformal mappings, with applications to differential equations, function theory and topology. Am. Math. Soc. Bull. 83(6), 1083–1100 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bers L., Nirenberg L.: On a Representation Theorem for Linear Elliptic Systems with Discontinuous Coefficients and its Applications, pp. 111–140. Convegno Internazionale Suelle Equaziono Cremeonese, Roma (1955)Google Scholar
  8. 8.
    Bobenko, A., Springborn, B., Pinkall, U.: Discrete conformal equivalence and ideal hyperbolic polyhedra (2012, in press)Google Scholar
  9. 9.
    Bobenko A.I., Springborn B.A.: Variational principles for circle patterns and koebe’s theorem. Trans. Am. Math. Soc. 356, 659–689 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bowers, P.L., Hurdal, M.: Planar conformal mapping of piecewise flat surfaces. In: Visualization and Mathematics III, pp. 3–34. Springer, Berlin (2003)Google Scholar
  11. 11.
    Bucking, U.: On existence and convergence of conformally equivalent triangle meshes for conformal mappings and regular lattices. In: Barrett Memorial Lectures (May 17–21, 2010)Google Scholar
  12. 12.
    Belinskii P.P., Godunov S.K., Yanenko I.: The use of a class of quasiconformal mappings to construct difference nets in domains with curvilinear boundaries. USSR Comp. Math. Phys. 15, 133–144 (1975)CrossRefGoogle Scholar
  13. 13.
    Carleson L., Gamelin T.: Complex Dynamics. Springer, New York (1993)zbMATHGoogle Scholar
  14. 14.
    Chow B.: The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2), 325–334 (1991)zbMATHGoogle Scholar
  15. 15.
    Chow B., Luo F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63(1), 97–129 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dai J., Luo W., Jin M., Zeng W., He Y., Yau S.T., Gu X.: Geometric accuracy analysis for discrete surface approximation. Comput. Aided Geom. Des. 24(6), 323–338 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Daripa P.: On a numerical method for quasiconformal grid generation. J. Comput. Phys. 96, 229–236 (1991)zbMATHCrossRefGoogle Scholar
  18. 18.
    Daripa P.: A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane. SIAM J. Sci. Stat. Comput. 13(6), 1418–1432 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Daripa P., Masha D.: An efficient and novel numerical method for quasiconformal mappings of doubly connected domains. Numer. Algorithm 18, 159–175 (1998)zbMATHCrossRefGoogle Scholar
  20. 20.
    Desbrun M., Meyer M., Alliez P.: Intrinsic parameterizations of surface meshes. Comput. Graph. Forum (Proc. Eurographics 2002) 21(3), 209–218 (2002)CrossRefGoogle Scholar
  21. 21.
    Farkas H.M., Kra I.: Riemann Surfaces. Springer, Berlin (2004)Google Scholar
  22. 22.
    Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, pp. 157–186. Springer, Berlin (2005)Google Scholar
  23. 23.
    Gortler S.J., Gotsman C., Thurston D.: Discrete one-forms on meshes and applications to 3D mesh parameterization. Comput. Aided Geom. Des. 23(2), 83–112 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gotsman C., Gu X., Sheffer A.: Fundamentals of spherical parameterization for 3D meshes. ACM Trans. Graph. 22(3), 358–363 (2003)CrossRefGoogle Scholar
  25. 25.
    Grimm, C., Hughes, J.F.: Parameterizing N-holed tori. In: IMA Conference on the Mathematics of Surfaces, pp. 14–29 (2003)Google Scholar
  26. 26.
    Grotzsch H.: Uber die verzerrung bei schlichten nichtkonformen abbildungen und eine damit zusammenh angende erweiterung des picardschen. Rec. Math. 80, 503–507 (1928)Google Scholar
  27. 27.
    Gu X., He Y., Qin H.: Manifold splines. Graph. Models 68(3), 237–254 (2006)zbMATHCrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    Gu, X., Yau, S.T.: Global conformal parameterization. In: Symposium on Geometry Processing, pp. 127–137 (2003)Google Scholar
  30. 30.
    Guggenheimer H.W.: Differential Geometry. Dover Publications, New York (1977)zbMATHGoogle Scholar
  31. 31.
    Hamilton R.S.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hamilton R.S.: The Ricci flow on surfaces. Math. Gen. Relativ. 71, 237–262 (1988)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Hong, W., Gu, X., Qiu, F., Jin, M., Kaufman, A.E.: Conformal virtual colon flattening. In: Symposium on Solid and Physical Modeling, pp. 85–93 (2006)Google Scholar
  34. 34.
    Hormann, K., Levy, B., Sheffer, A.: Mesh parameterization. SIGGRAPH 2007 Course Notes 2 (2007)Google Scholar
  35. 35.
    Jin M., Kim J., Luo F., Gu X.: Discrete surface Ricci flow. IEEE Trans. Vis. Comput. Graph. 14(5), 1030–1043 (2008)CrossRefGoogle Scholar
  36. 36.
    Kharevych L., Springerborn B., Schröder P.: Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25(2), 412–438 (2006)CrossRefGoogle Scholar
  37. 37.
    Kalberer F., Nieser M., Polthicr K.: Quadcover—surface parameterization using branched coverings. Comput. Graph. 26(3), 375–384 (2007)CrossRefGoogle Scholar
  38. 38.
    Lavrentjev M.: Sur une classe de representations continues. Rec. Math. 48, 407–423 (1935)Google Scholar
  39. 39.
    Lehto O., Virtanen K.: Quasiconformal Mapping in the Plane. Springer, Berlin (1973)Google Scholar
  40. 40.
    Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. SIGGRAPH 2002 pp. 362–371 (2002)Google Scholar
  41. 41.
    Lipman Y., Chen X., Daubechies I., Funkhouser T.: Symmetry factored embedding and distance. ACM Trans. Graph. 29(4), 1–12 (2010)CrossRefGoogle Scholar
  42. 42.
    Lui, L., Wong, T., Gu, X., Thompson, P., Chan, T., Yau, S.: Compression of surface diffeomorphism using Beltrami coefficient. IEEE Comput. Vis. Patt. Recogn. (CVPR), pp. 2839–2846 (2010)Google Scholar
  43. 43.
    Lui, L., Wong, T., Gu, X., Thompson, P., Chan, T., Yau, S.: Hippocampal shape registration using Beltrami holomorphic flow. Medical Image Computing and Computer Assisted Intervention(MICCAI), Part II. LNCS 6362, pp. 323–330 (2010)Google Scholar
  44. 44.
    Lui L., Wong T., Zeng W., Gu X., Thompson P., Chan T., Yau S.: Detecting shape deformations using yamabe flow and Beltrami coefficents. J. Inverse Probl. Imaging (IPI) 4(2), 311–333 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Lui, L., Wong, T., Zeng, W., Gu, X., Thompson, P., Chan, T., Yau, S.: Optimization of surface registrations using beltrami holomorphic flow. J. Scientific Comput. (2011)Google Scholar
  46. 46.
    Luo F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Mastin C., Thompson J.: Discrete quasiconformal mappings. Z. Angew. Math. Phys. 29, 1–11 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Mastin C., Thompson J.: Quasiconformal mappings and grid generation. SIAM J. Sci. Stat. Comput. 5(2), 305–310 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Morrey C.: On the solutions of quasi-linear elliptic differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Praun E., Hoppe H.: Spherical parametrization and remeshing. ACM Trans. Graph. 22(3), 340–349 (2003)CrossRefGoogle Scholar
  51. 51.
    Ray N., Li W.C., Levy B., Sheffer A., Alliez P.: Periodic global parameterization. ACM Trans. Graph. 25(4), 1460–1485 (2005)CrossRefGoogle Scholar
  52. 52.
    Sheffer A., Lévy B., Mogilnitsky M., Bogomyakov A.: ABF++: fast and robust angle based flattening. ACM Trans. Graph. 24(2), 311–330 (2005)CrossRefGoogle Scholar
  53. 53.
    Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Foundations and Trends® in Computer Graphics and Vision (2012, in press)Google Scholar
  54. 54.
    Sheffer A., de Sturler E.: Parameterization of faced surfaces for meshing using angle based flattening. Eng. Comput. 17(3), 326–337 (2001)zbMATHCrossRefGoogle Scholar
  55. 55.
    Springborn B., Schröder P., Pinkall U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), 1–11 (2008)CrossRefGoogle Scholar
  56. 56.
    Vlasyuk, A.: Automatic construction of conformal and quasiconformal mapping of doubly connected and triple connected domains. Akad. Nauk Ukrainy Inst. Mat., preprint (Akademiya Nauk Ukrainy Institut Matematiki, preprint) 57, 1–57 (1991)Google Scholar
  57. 57.
    Wang S., Wang Y., Jin M., Gu X.D., Samaras D.: Conformal geometry and its applications on 3D shape matching, recognition, and stitching. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1209–1220 (2007)CrossRefGoogle Scholar
  58. 58.
    Weisel J.: Numerische ermittlung quasikonformer abbildungen mit finiten elementen. Numer. Math. 35, 201–222 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Zayer, R., Levy, B., Seidel, H.P.: Linear angle based parameterization. In: In Symposium on Geometry Processing, pp. 135–141 (2007)Google Scholar
  60. 60.
    Zeng, W., Jin, M., Luo, F., Gu, X.: Computing canonical homotopy class representative using hyperbolic structure. In: IEEE International Conference on Shape Modeling and Applications (SMI 2009) (2009)Google Scholar
  61. 61.
    Zeng W., Marino J., Gurijala K., Gu X., Kaufman A.: Supine and prone colon registration using quasi-conformal mapping. IEEE Trans. Vis. Comput. Graph. (IEEE TVCG) 16(6), 1348–1357 (2010)CrossRefGoogle Scholar
  62. 62.
    Zeng W., Samaras D., Gu X.: Ricci flow for 3D shape analysis. IEEE Trans. Pattern Anal. Mach. Intell. 32(4), 662–677 (2010)CrossRefGoogle Scholar
  63. 63.
    Zeng, W., Zeng, Y., Wang, Y., Yin, X., Gu, X., Samaras, D.: 3D non-rigid surface matching and registration based on holomorphic differentials. In: The 10th European Conference on Computer Vision (ECCV) 2008, pp. 1–14 (2008)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Wei Zeng
    • 1
  • Lok Ming Lui
    • 2
  • Feng Luo
    • 3
  • Tony Fan-Cheong Chan
    • 4
  • Shing-Tung Yau
    • 5
  • David Xianfeng Gu
    • 1
  1. 1.Department of Computer ScienceStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  3. 3.Department of MathematicsRutgers UniversityPiscatawayUSA
  4. 4.The Hong Kong University of Science and TechnologyKowloonHong Kong
  5. 5.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations