Convergence of an iterative algorithm for Teichm\"uller maps via generalized harmonic maps

TL;DR

This paper proves the convergence of an iterative algorithm designed to compute extremal Teichmüller maps, which are surface mappings with minimal conformality distortion, with applications in surface registration.

Contribution

It provides a mathematical proof of convergence for a previously proposed iterative algorithm for Teichmüller map computation.

Findings

The iterative algorithm converges to the extremal Teichmüller map.

The method is effective for landmark-matching registration.

Theoretical guarantees support previous numerical results.

Abstract

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichm\"uller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichm\"uller map between Riemann surfaces of finite type are theoretically guaranteed [1]. Recently, a simple iterative algorithm for computing the Teichm\"uller maps between connected Riemann surfaces with given boundary value was proposed in [11]. Numerical results was 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 [11] indeed converges.