Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of an iterative algorithm for Teichmüller maps via harmonic energy optimization


Authors: Lok Ming Lui, Xianfeng Gu and Shing-Tung Yau
Journal: Math. Comp. 84 (2015), 2823-2842
MSC (2010): Primary 52C26, 65D18, 65E05; Secondary 52B20, 52C99
DOI: https://doi.org/10.1090/S0025-5718-2015-02962-7
Published electronically: March 24, 2015
MathSciNet review: 3378849
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are
theoretically guaranteed (see Fletcher and Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Math., vol. 11, Oxford University Press, Oxford, 2007). Recently, a simple iterative algorithm for computing the Teichmüller maps between connected Riemann surfaces with given boundary value was proposed by Lui, Lam, Yau, and Gu in Teichmüller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569. Numerical results were reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmark-matching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.


References [Enhancements On Off] (What's this?)

  • [1] A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmüller theory, Oxford Graduate Texts in Mathematics, vol. 11, Oxford University Press, Oxford, 2007. MR 2269887 (2007g:30001)
  • [2] D. G. Crowdy, Conformal slit maps in applied mathematics, Computational Methods and Function Theory 53 (2012), no. 3, 171-189.
  • [3] Darren G. Crowdy, Athanassios S. Fokas, and Christopher C. Green, Conformal mappings to multiply connected polycircular arc domains, Comput. Methods Funct. Theory 11 (2011), no. 2, 685-706. MR 2858967, https://doi.org/10.1007/BF03321882
  • [4] Darren Crowdy and Jonathan Marshall, Conformal mappings between canonical multiply connected domains, Comput. Methods Funct. Theory 6 (2006), no. 1, 59-76. MR 2241034 (2007e:30010), https://doi.org/10.1007/BF03321118
  • [5] Nicholas Hale and T. Wynn Tee, Conformal maps to multiply slit domains and applications, SIAM J. Sci. Comput. 31 (2009), no. 4, 3195-3215. MR 2529786 (2010i:65055), https://doi.org/10.1137/080738325
  • [6] Thomas K. Delillo and Everett H. Kropf, Numerical computation of the Schwarz-Christoffel transformation for multiply connected domains, SIAM J. Sci. Comput. 33 (2011), no. 3, 1369-1394. MR 2813244, https://doi.org/10.1137/100816912
  • [7] Thomas K. DeLillo, Alan R. Elcrat, Everett H. Kropf, and John A. Pfaltzgraff, Efficient calculation of Schwarz-Christoffel transformations for multiply connected domains using Laurent series, Comput. Methods Funct. Theory 13 (2013), no. 2, 307-336. MR 3089960, https://doi.org/10.1007/s40315-013-0023-1
  • [8] Lloyd N. Trefethen, Numerical computation of the Schwarz-Christoffel transformation, SIAM J. Sci. Statist. Comput. 1 (1980), no. 1, 82-102. MR 572542 (81g:30012a), https://doi.org/10.1137/0901004
  • [9] Tobin A. Driscoll and Lloyd N. Trefethen, Schwarz-Christoffel Mapping, Cambridge Monographs on Applied and Computational Mathematics, vol. 8, Cambridge University Press, Cambridge, 2002. MR 1908657 (2003e:30012)
  • [10] S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle, Conformal surface parameterization for texture mapping, IEEE Transaction of Visualization and Computer Graphics 6 (2000), 181-189.
  • [11] B. Fischl, M. Sereno, R. Tootell, and A. Dale, High-resolution intersubject averaging and a coordinate system for the cortical surface, Human Brain Mapping 8 (1999), 272-284.
  • [12] X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, and S.-T. Yau, Genus zero surface conformal mapping and its application to brain, surface mapping, IEEE Transactions on Medical Imaging 23 (2004), no. 8, 949-958.
  • [13] Y. Wang, L. M. Lui, X. Gu, K. M. Hayashi, T. F. Chan, A. W. Toga, P. M. Thompson, and S.-T. Yau, Brain surface conformal parameterization using riemann surface structure, IEEE Transactions on Medical Imaging 26(2007), no. 6, 853-865.
  • [14] Xianfeng Gu and Shing-Tung Yau, Computing conformal structures of surfaces, Commun. Inf. Syst. 2 (2002), no. 2, 121-145. MR 1958012 (2003m:65023)
  • [15] M. K. Hurdal and K. Stephenson, Discrete conformal methods for cortical brain flattening, Neuroimage 45 (2009), 86-98.
  • [16] R. Michael Porter, An interpolating polynomial method for numerical conformal mapping, SIAM J. Sci. Comput. 23 (2001), no. 3, 1027-1041 (electronic). MR 1860975 (2002h:30007), https://doi.org/10.1137/S1064827599355256
  • [17] Nicholas Hale and T. Wynn Tee, Conformal maps to multiply slit domains and applications, SIAM J. Sci. Comput. 31 (2009), no. 4, 3195-3215. MR 2529786 (2010i:65055), https://doi.org/10.1137/080738325
  • [18] Thomas K. Delillo and Everett H. Kropf, Numerical computation of the Schwarz-Christoffel transformation for multiply connected domains, SIAM J. Sci. Comput. 33 (2011), no. 3, 1369-1394. MR 2813244, https://doi.org/10.1137/100816912
  • [19] Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906 (2001d:32016)
  • [20] L. M. Lui, K. C. Lam, S. T. Yau, X. F. Gu, Teichmüller extremal mapping and its applications to landmark matching registration, (submitted & under revision), arXiv:1211.2569 (http://arxiv.org/abs/1210.8025)
  • [21] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463 (49 #9202)
  • [22] Kurt Strebel, On quasiconformal mappings of open Riemann surfaces, Comment. Math. Helv. 53 (1978), no. 3, 301-321. MR 505549 (81i:30041), https://doi.org/10.1007/BF02566081
  • [23] Edgar Reich, Extremal quasiconformal mappings of the disk, Handbook of Complex Analysis: Geometric Function Theory, Vol.1, North-Holland, Amsterdam, 2002, pp. 75-136. MR 1966190 (2004c:30036), https://doi.org/10.1016/S1874-5709(02)80005-1

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 52C26, 65D18, 65E05, 52B20, 52C99

Retrieve articles in all journals with MSC (2010): 52C26, 65D18, 65E05, 52B20, 52C99


Additional Information

Lok Ming Lui
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: lmlui@math.cuhk.edu.hk

Xianfeng Gu
Affiliation: Department of Computer Sciences, State University of New York at Stony Brook, Stony Brook, New York
Email: gu@cs.sunysb.edu

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts
Email: yau@math.harvard.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02962-7
Received by editor(s): September 18, 2013
Published electronically: March 24, 2015
Additional Notes: The first author was supported by RGC GRF (Project ID: 401811), CUHK Direct Grant (Project ID: 2060413), and CUHK FIS Grant (Project ID: 1902036)
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society