Abstract
By the Riemann mapping theorem, one can bijectively map the interior of an ngon P to that of another ngon Q conformally (i.e., in an angle preserving manner). However, when this map is extended to the boundary it need not necessarily map the vertices of P to those of Q. For many applications it is important to find the "best" vertexpreserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the socalled dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps.
There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a (1+epsilon)approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps.
We present two results in this paper. One studies the problem in the continuous setting and another in the discrete setting.
In the continuous setting, we solve the problem of finding a finite time procedure for approximating Teichmüller maps. Our construction is via an iterative procedure that is proven to converge in O(poly(1/epsilon)) iterations to a (1+epsilon)approximation of the Teichmuller map. Our method uses a reduction of the polygon mapping problem to the marked sphere problem, thus solving a more general problem.
In the discrete setting, we reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete 1) compositions and 2) inverses of discretely represented quasiconformal maps. Assuming finitetime solvers for these subroutines we provide a (1+epsilon)approximation algorithm.
BibTeX  Entry
@InProceedings{goswami_et_al:LIPIcs:2015:5099,
author = {Mayank Goswami and Xianfeng Gu and Vamsi P. Pingali and Gaurish Telang},
title = {{Computing Teichm{\"u}ller Maps between Polygons}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {615629},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5099},
URN = {urn:nbn:de:0030drops50997},
doi = {10.4230/LIPIcs.SOCG.2015.615},
annote = {Keywords: Teichm{\"u}ller maps, Surface registration, Extremal Quasiconformal maps, Computer vision}
}
Keywords: 

Teichmüller maps, Surface registration, Extremal Quasiconformal maps, Computer vision 
Collection: 

31st International Symposium on Computational Geometry (SoCG 2015) 
Issue Date: 

2015 
Date of publication: 

12.06.2015 