Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

Numerische Mathematik

Volumn 121, Issue 4, 2012, Pages 671-703

  Zeng, Wei ^a   Lui, Lok Ming ^b   Luo, Feng ^c   Chan, Tony Fan Cheong ^d   Yau, Shing Tung ^e   Gu, David Xianfeng ^a  

^a USA   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c RUTGERS UNIVERSITY   (United States)

^d HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

^e HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84864356151     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-012-0446-z     Document Type: Article

References (63)

1. Ahlfors L.: Conformality with respect to Riemannian matrices. Ann. Acad. Sci. Fenn. Ser. 206, 1-22 (1955).
2. Ahlfors L.: Lectures in Quasiconformal Mappings. van Nostrand Reinhold, New York (1966).
3. Ben-Chen M., Gotsman C., Bunin G.: Conformal flattening by curvature prescription and metric scaling. Comput. Graph. Forum 27(2), 449-458 (2008).
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).
5. Bers L.: Mathematical Aspects of Subcritical and Transonic Gas Dynamics. Wiley, New York (1958).
6. Bers L.: Quasiconformal mappings, with applications to differential equations, function theory and topology. Am. Math. Soc. Bull. 83(6), 1083-1100 (1977).
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).
8. Bobenko, A., Springborn, B., Pinkall, U.: Discrete conformal equivalence and ideal hyperbolic polyhedra (2012, in press).
9. Bobenko A. I., Springborn B. A.: Variational principles for circle patterns and koebe's theorem. Trans. Am. Math. Soc. 356, 659-689 (2004).
10. Bowers, P. L., Hurdal, M.: Planar conformal mapping of piecewise flat surfaces. In: Visualization and Mathematics III, pp. 3-34. Springer, Berlin (2003).
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).
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).
13. Carleson L., Gamelin T.: Complex Dynamics. Springer, New York (1993).
14. Chow B.: The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2), 325-334 (1991).
15. Chow B., Luo F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63(1), 97-129 (2003).
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).
17. Daripa P.: On a numerical method for quasiconformal grid generation. J. Comput. Phys. 96, 229-236 (1991).
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).
19. Daripa P., Masha D.: An efficient and novel numerical method for quasiconformal mappings of doubly connected domains. Numer. Algorithm 18, 159-175 (1998).
20. Desbrun M., Meyer M., Alliez P.: Intrinsic parameterizations of surface meshes. Comput. Graph. Forum (Proc. Eurographics 2002) 21(3), 209-218 (2002).
21. Farkas H. M., Kra I.: Riemann Surfaces. Springer, Berlin (2004).
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).
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).
24. Gotsman C., Gu X., Sheffer A.: Fundamentals of spherical parameterization for 3D meshes. ACM Trans. Graph. 22(3), 358-363 (2003).
25. Grimm, C., Hughes, J. F.: Parameterizing N-holed tori. In: IMA Conference on the Mathematics of Surfaces, pp. 14-29 (2003).
26. Grotzsch H.: Uber die verzerrung bei schlichten nichtkonformen abbildungen und eine damit zusammenh angende erweiterung des picardschen. Rec. Math. 80, 503-507 (1928).
27. Gu X., He Y., Qin H.: Manifold splines. Graph. Models 68(3), 237-254 (2006).
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).
29. Gu, X., Yau, S. T.: Global conformal parameterization. In: Symposium on Geometry Processing, pp. 127-137 (2003).
30. Guggenheimer H. W.: Differential Geometry. Dover Publications, New York (1977).
31. Hamilton R. S.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255-306 (1982).
32. Hamilton R. S.: The Ricci flow on surfaces. Math. Gen. Relativ. 71, 237-262 (1988).
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).
34. Hormann, K., Levy, B., Sheffer, A.: Mesh parameterization. SIGGRAPH 2007 Course Notes 2 (2007).
35. Jin M., Kim J., Luo F., Gu X.: Discrete surface Ricci flow. IEEE Trans. Vis. Comput. Graph. 14(5), 1030-1043 (2008).
36. Kharevych L., Springerborn B., Schröder P.: Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25(2), 412-438 (2006).
37. Kalberer F., Nieser M., Polthicr K.: Quadcover-surface parameterization using branched coverings. Comput. Graph. 26(3), 375-384 (2007).
38. Lavrentjev M.: Sur une classe de representations continues. Rec. Math. 48, 407-423 (1935).
39. Lehto O., Virtanen K.: Quasiconformal Mapping in the Plane. Springer, Berlin (1973).
40. Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. SIGGRAPH 2002 pp. 362-371 (2002).
41. Lipman Y., Chen X., Daubechies I., Funkhouser T.: Symmetry factored embedding and distance. ACM Trans. Graph. 29(4), 1-12 (2010).
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).
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).
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).
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).
46. Luo F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765-780 (2004).
47. Mastin C., Thompson J.: Discrete quasiconformal mappings. Z. Angew. Math. Phys. 29, 1-11 (1978).
48. Mastin C., Thompson J.: Quasiconformal mappings and grid generation. SIAM J. Sci. Stat. Comput. 5(2), 305-310 (1984).
49. Morrey C.: On the solutions of quasi-linear elliptic differential equations. Trans. Am. Math. Soc. 43, 126-166 (1938).
50. Praun E., Hoppe H.: Spherical parametrization and remeshing. ACM Trans. Graph. 22(3), 340-349 (2003).
51. Ray N., Li W. C., Levy B., Sheffer A., Alliez P.: Periodic global parameterization. ACM Trans. Graph. 25(4), 1460-1485 (2005).
52. Sheffer A., Lévy B., Mogilnitsky M., Bogomyakov A.: ABF++: fast and robust angle based flattening. ACM Trans. Graph. 24(2), 311-330 (2005).
53. ® in Computer Graphics and Vision (2012, in press).
54. Sheffer A., de Sturler E.: Parameterization of faced surfaces for meshing using angle based flattening. Eng. Comput. 17(3), 326-337 (2001).
55. Springborn B., Schröder P., Pinkall U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), 1-11 (2008).
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).
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).
58. Weisel J.: Numerische ermittlung quasikonformer abbildungen mit finiten elementen. Numer. Math. 35, 201-222 (1980).
59. Zayer, R., Levy, B., Seidel, H. P.: Linear angle based parameterization. In: Symposium on Geometry Processing, pp. 135-141 (2007).
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).
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).
62. Zeng W., Samaras D., Gu X.: Ricci flow for 3D shape analysis. IEEE Trans. Pattern Anal. Mach. Intell. 32(4), 662-677 (2010).
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).